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Applied Conformal Field Theory

요약 생성 중 오류 발생

Applied Conformal Field Theory

arXiv:hep-th/9108028v1 11 Nov 1988HUTP-88/A054Applied Conformal Field TheoryPaul Ginsparg†Lyman Laboratory of PhysicsHarvard UniversityCambridge, MA 02138Lectures given at Les Houches summer session, June 28 – Aug. 5, 1988.To appear in Les Houches, Session XLIX, 1988, Champs, Cordes et Ph´enom`enes Critiques/ Fields, Strings andCritical Phenomena, ed. by E. Br´ezin and J. Zinn-Justin, c⃝Elsevier Science Publishers B.V. (1989).9/88 (with corrections, 11/88)† (ginsparg@huhepl.hepnet, ginsparg@huhepl.bitnet, or ginsparg@huhepl.harvard.edu)

Applied Conformal Field TheoryLes Houches Lectures, P. Ginsparg, 1988Contents1. Conformal theories in d dimensions.

. .

. .31.1.

Conformal group in d dimensions. .

. .

. .41.2.

Conformal algebra in 2 dimensions. .

. .

. .61.3.

Constraints of conformal invariance in d dimensions. .

. .

. .92.

Conformal theories in 2 dimensions . .

. .

122.1. Correlation functions of primary ﬁelds.

. .

. 122.2.

Radial quantization and conserved charges. .

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152.3. Free boson, the example.

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. 212.4.

Conformal Ward identities. .

. .

. 243.

The central charge and the Virasoro algebra . .

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263.1. The central charge.

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263.2. The free fermion.

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. 303.3.

Mode expansions and the Virasoro algebra. .

. .

303.4. In- and out-states.

. .

323.5. Highest weight states.

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. 363.6.

Descendant ﬁelds. .

. .

. 393.7.

Duality and the bootstrap. .

. .

. 424.

Kac determinant and unitarity. .

. .

. 464.1.

The Hilbert space of states. .

. .

. 464.2.

Kac determinant. .

. .

. 504.3.

Sketch of non-unitarity proof. .

. .

524.4. Critical statistical mechanical models.

. .

. 564.5.

Conformal grids and null descendants. .

. .

575. Identiﬁcation of m = 3 with the critical Ising model.

. .

. 585.1.

Critical exponents. .

. .

. 595.2.

Critical correlation functions of the Ising model. .

. .

635.3. Fusion rules for c < 1 models.

. .

. 665.4.

More discrete series. .

. .

696. Free bosons and fermions.

. .

. 716.1.

Mode expansions. .

. .

. 716.2.

Twist ﬁelds. .

. .

726.3. Fermionic zero modes.

. .

777. Free fermions on a torus.

. .

. 807.1.

Back to the cylinder, on to the torus. .

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. 807.2. c = 12 representations of the Virasoro algebra.

. .

857.3. The modular group and fermionic spin structures.

. .

887.4. c = 12 Virasoro characters. .

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9117.5. Critical Ising model on the torus.

. .

. 957.6.

Recreational mathematics and ϑ-function identities. .

. .

.1018. Free bosons on a torus.

. .

.1088.1. Partition function.

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.1088.2. Fermionization.

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. .1158.3.

Orbifolds in general. .

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.1178.4. S1/Z2 orbifold.

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. .1218.5.

Orbifold comments. .

. .

.1268.6. Marginal operators.

. .

. .1288.7.

The space of c = 1 theories. .

. .

. .1309.

Aﬃne Kac-Moody algebras and coset constructions. .

. .

.1369.1. Aﬃne algebras.

. .

. .1369.2.

Enveloping Virasoro algebra. .

. .

.1389.3. Highest weight representations.

. .

.1429.4. Some free ﬁeld representations.

. .

.1479.5. Coset construction.

. .

.1529.6. Modular invariant combinations of characters.

. .

.1579.7. The A-D-E classiﬁcation of SU(2) invariants.

. .

.1609.8. Modular transformations and fusion rules.

. .

.16510. Advanced applications .

. .

.166These lectures consisted of an elementary introduction to conformal ﬁeldtheory, with some applications to statistical mechanical systems, and fewer tostring theory. They were given to a mixed audience of experts and beginners(more precisely an audience roughly 35% of which was alleged to have had noprior exposure to conformal ﬁeld theory, and a roughly equal percentage allegedto be currently working in the ﬁeld), and geared in real time to the appropriatelevel.

The division into sections corresponds to the separate (1.5 hour) lectures,except that 7 and 8 together stretched to three lectures, and I have taken theliberty of expanding some rushed comments at the end of 9.It was not my intent to be particularly creative in my presentation of thematerial, but I did try to complement some of the various introductory treat-ments that already exist. Since these lectures were given at the beginning ofthe school, they were intended to be more or less self-contained and generallyaccessible.

I tried in all cases to emphasize the simplest applications, but not toduplicate excessively the many review articles that already exist on the subject.2

More extensive applications to statistical mechanical models may be found in J.Cardy’s lectures in this volume, given concurrently, and many string theory ap-plications of conformal ﬁeld theory were covered in D. Friedan’s lectures, whichfollowed. The standard reference for the material of the ﬁrst three sections is[1].

Some of the review articles that have inﬂuenced the presentation of theearly sections are listed in [2]. A more extensive (physicist-oriented) review ofaﬃne Kac-Moody algebras, discussed here in section 9, may be found in [3].Throughout I have tried to include references to more recent papers in whichthe interested reader may ﬁnd further references to original work.Omittedreferences to relevant work are meant to indicate my prejudices rather than myignorance in the subject.I am grateful to the organizers and students at the school for insisting onthe appropriate level of pedagogy and for their informative questions, and toP.

di Francesco and especially M. Goulian for most of the answers. I thanknumerous participants at the conformal ﬁeld theory workshop at the AspenCenter for Physics (Aug., 1988) for comments on the manuscript, and thank S.Giddings, G. Moore, R. Plesser, and J. Shapiro for actually reading it.

FinallyI acknowledge the students at Harvard who patiently sat through a dry runof this material (and somewhat more) during the spring of 1988. This workwas supported in part by NSF contract PHY-82-15249, by DOE grant FG-84ER40171, and by the A. P. Sloan foundation.1.

Conformal theories in d dimensionsConformally invariant quantum ﬁeld theories describe the critical behaviorof systems at second order phase transitions. The canonical example is theIsing model in two dimensions, with spins σi = ±1 on sites of a square lat-tice.

The partition function Z = P{σ} exp(−E/T ) is deﬁned in terms of theenergy E = −ǫ P⟨ij⟩σiσj, where the summation ⟨ij⟩is over nearest neighborsites on the lattice. This model has a high temperature disordered phase (withthe expectation value ⟨σ⟩= 0) and a low temperature ordered phase (with⟨σ⟩̸= 0).

The two phases are related by a duality of the model, and there is3a 2nd order phase transition at the self-dual point. At the phase transition,typical conﬁgurations have ﬂuctuations on all length scales, so the ﬁeld theorydescribing the model at its critical point should be expected to be invariant atleast under changes of scale.

In fact, critical theories are more generally invari-ant under the full conformal group, to be introduced momentarily. In three ormore dimensions, conformal invariance does not turn out to give much moreinformation than ordinary scale invariance.

But in two dimensions, the confor-mal algebra becomes inﬁnite dimensional, leading to signiﬁcant restrictions ontwo dimensional conformally invariant theories, and perhaps ultimately givinga classiﬁcation of possible critical phenomena in two dimensions.Two dimensional conformal ﬁeld theories also provide the dynamical vari-able in string theory. In that context conformal invariance turns out to giveconstraints on the allowed spacetime (i.e.

critical) dimension and the possibleinternal degrees of freedom. A classiﬁcation of two dimensional conformal ﬁeldtheories would thus provide useful information on the classical solution spaceof string theory, and might lead to more propitious quantization schemes.1.1.

Conformal group in d dimensionsWe begin here with an introduction to the conformal group in d-dimensions.The aim is to exhibit the constraints imposed by conformal invariance in themost general context. In section 2 we shall then restrict to the case of twodimensional Euclidean space, which will be the focus of discussion for the re-mainder.We consider the space Rd with ﬂat metric gµν = ηµν of signature (p, q)and line element ds2 = gµν dxµdxν.

Under a change of coordinates, x →x′, wehave gµν →g′µν(x′) =∂xα∂x′µ ∂xβ∂x′ν gαβ(x). By deﬁnition, the conformal group isthe subgroup of coordinate transformations that leaves the metric invariant upto a scale change,gµν(x) →g′µν(x′) = Ω(x) gµν(x) .

(1.1)These are consequently the coordinate transformations that preserve the anglev · w/(v2w2)1/2 between two vectors v, w (where v · w = gµνvµwν). We note4

that the Poincar´e group, the semidirect product of translations and Lorentztransformations of ﬂat space, is always a subgroup of the conformal group sinceit leaves the metric invariant (g′µν = gµν).The inﬁnitesimal generators of the conformal group can be determined byconsidering the inﬁnitesimal coordinate transformation xµ →xµ + ǫµ, underwhichds2 →ds2 + (∂µǫν + ∂νǫµ)dxµdxν .To satisfy (1.1) we must require that ∂µǫν + ∂νǫµ be proportional to ηµν,∂µǫν + ∂νǫµ = 2d(∂· ǫ)ηµν ,(1.2)where the constant of proportionality is ﬁxed by tracing both sides with ηµν.Comparing with (1.1) we ﬁnd Ω(x) = 1 + (2/d)(∂· ǫ). It also follows from (1.2)thatηµν+ (d −2)∂µ∂ν∂· ǫ = 0 .

(1.3)For d > 2, (1.2) and (1.3) require that the third derivatives of ǫ mustvanish, so that ǫ is at most quadratic in x. For ǫ zeroth order in x, we havea) ǫµ = aµ, i.e.

ordinary translations independent of x.There are two cases for which ǫ is linear in x:b) ǫµ = ωµν xν (ω antisymmetric) are rotations,andc) ǫµ = λ xµ are scale transformations.Finally, when ǫ is quadratic in x we haved) ǫµ = bµ x2 −2xµ b · x, the so-called special conformal transformations. (these last may also be expressed as x′µ/x′2 = xµ/x2 + bµ, i.e.

as an inversionplus translation). Locally, we can conﬁrm that the algebra generated by aµ∂µ,ωµνǫν∂µ, λx · ∂, and bµ(x2∂µ −2xµx · ∂) (a total of p + q + 12(p + q)(p +q −1) + 1 + (p + q) =12(p + q + 1)(p + q + 2) generators) is isomorphic toSO(p + 1, q + 1) (Indeed the conformal group admits a nice realization actingon Rp,q, stereographically projected to Sp,q, and embedded in the light-cone ofRp+1,q+1.

).5Integrating to ﬁnite conformal transformations, we ﬁnd ﬁrst of all, as ex-pected, the Poincar´e groupx →x′ = x + ax →x′ = Λ x(Λµν ∈SO(p, q))(Ω= 1) . (1.4a)Adjoined to it, we have the dilatationsx →x′ = λx(Ω= λ−2) ,(1.4b)and also the special conformal transformationsx →x′ =x + bx21 + 2b · x + b2x2Ω(x) = (1 + 2b · x + b2x2)2.

(1.4c)Note that under (1.4c) we have x′2 = x2/(1+2b·x+b2x2), so that points on thesurface 1 = 1 +2b·x+b2x2 have their distance to the origin preserved, whereaspoints on the exterior of this surface are sent to the interior and vice-versa. (Under the ﬁnite transformation (1.4c) we also continue to have x′µ/x′2 =xµ/x2 + bµ.)1.2.

Conformal algebra in 2 dimensionsFor d = 2 and gµν = δµν, (1.2) becomes the Cauchy-Riemann equation∂1ǫ1 = ∂2ǫ2 ,∂1ǫ2 = −∂2ǫ1 .It is then natural to write ǫ(z) = ǫ1 + iǫ2 and ǫ(z) = ǫ1 −iǫ2, in the complexcoordinates z, z = x1 ± ix2. Two dimensional conformal transformations thuscoincide with the analytic coordinate transformationsz →f(z) ,z →f(z) ,(1.5)the local algebra of which is inﬁnite dimensional.

In complex coordinates wewriteds2 = dz dz →∂f∂z2dz dz ,(1.6)and have Ω= |∂f/∂z|2.6

To calculate the commutation relations of the generators of the conformalalgebra, i.e. inﬁnitesimal transformations of the form (1.5), we take for basisz →z′ = z + ǫn(z)z →z′ = z + ǫn(z)(n ∈Z) ,whereǫn(z) = −zn+1ǫn(z) = −zm+1 .The corresponding inﬁnitesimal generators areℓn = −zn+1∂zℓn = −zn+1∂z(n ∈Z) .

(1.7)The ℓ’s and ℓ’s are easily veriﬁed to satisfy the algebrasℓm, ℓn= (m −n)ℓm+nℓm, ℓn= (m −n)ℓm+n ,(1.8)and [ℓm, ℓn] = 0. In the quantum case, the algebras (1.8) will be corrected toinclude an extra term proportional to a central charge.

Since the ℓn’s commutewith the ℓm’s, the local conformal algebra is the direct sum A ⊕A of twoisomorphic subalgebras with the commutation relations (1.8).Since two independent algebras naturally arise, it is frequently useful toregard z and z as independent coordinates. (More formally, we would saythat since the action of the conformal group in two dimensions factorizes intoindependent actions on z and z, Green functions of a 2d conformal ﬁeld theorymay be continued to a larger domain in which z and z are treated as independentvariables.) In terms of the original coordinates (x1, x2) ∈R2, this amounts totaking instead (x1, x2) ∈C2, and then the transformation to z, z coordinatesis just a change of variables.

In C2, the surface deﬁned by z = z∗is the ‘real’surface on which we recover (x, y) ∈R2. This procedure allows the algebraA ⊕A to act naturally on C2, and the ‘physical’ condition z = z∗is left tobe imposed at our convenience.

The real surface speciﬁed by this condition ispreserved by the subalgebra of A⊕A generated by ℓn+ℓn and i(ℓn−ℓn). In thesections that follow, we shall frequently use the independence of the algebrasA and A to justify ignoring anti-holomorphic dependence for simplicity, thenreconstruct it afterwards by adding terms with bars where appropriate.7We have been careful thus far to call the algebra (1.8) the local conformalalgebra.

The reason is that the generators are not all well-deﬁned globally onthe Riemann sphere S2 = C ∪∞. Holomorphic conformal transformations aregenerated by vector ﬁeldsv(z) = −Xnanℓn =Xnan zn+1∂z .Non-singularity of v(z) as z →0 allows an ̸= 0 only for n ≥−1.

To investigatethe behavior of v(z) as z →∞, we perform the transformation z = −1/w,v(z) =Xnan−1wn+1 dzdw−1∂w =Xnan−1wn−1∂w .Non-singularity as w →0 allows an ̸= 0 only for n ≤1. We see that only theconformal transformations generated by anℓn for n = 0, ±1 are globally deﬁned.The same considerations apply to anti-holomorphic transformations.In two dimensions the global conformal group is deﬁned to be the group ofconformal transformations that are well-deﬁned and invertible on the Riemannsphere.It is thus generated by the globally deﬁned inﬁnitesimal generators{ℓ−1, ℓ0, ℓ1} ∪{ℓ−1, ℓ0, ℓ1}.

From (1.7) and (1.4) we identify ℓ−1 and ℓ−1 asgenerators of translations, ℓ0 + ℓ0 and i(ℓ0 −ℓ0) respectively as generators ofdilatations and rotations (i.e. generators of translations of r and θ in z = reiθ),and ℓ1, ℓ1 as generators of special conformal transformations.

The ﬁnite formof these transformations isz →az + bcz + dz →a z + bc z + d ,(1.9)where a, b, c, d ∈C and ad−bc = 1). This is the group SL(2, C)/Z2 ≈SO(3, 1),also known as the group of projective conformal transformations.

(The quotientby Z2 is due to the fact that (1.9) is unaﬀected by taking all of a, b, c, d to minusthemselves.) In SL(2, C) language, the transformations (1.4) are given bytranslations :1B01dilatations :λ00λ−1rotations :eiθ/200e−iθ/2special conformal :10C1,8

where B = a1 + ia2 and C = b1 −ib2.The distinction encountered here between global and local conformalgroups is unique to two dimensions (in higher dimensions there exists onlya global conformal group). Strictly speaking the only true conformal group intwo dimensions is the projective (global) conformal group, since the remainingconformal transformations of (1.5) do not have global inverses on C ∪∞.

Thisis the reason the word algebra rather than the word group appears in the titleof this subsection.The global conformal algebra generated by {ℓ−1, ℓ0, ℓ1}∪{ℓ−1, ℓ0, ℓ1} is alsouseful for characterizing properties of physical states. Suppose we work in abasis of eigenstates of the two operators ℓ0 and ℓ0, and denote their eigenvaluesby h and h respectively.

Here h and h are meant to indicate independent (real)quantities, not complex conjugates of one another. h and h are known as theconformal weights of the state.

Since ℓ0 +ℓ0 and i(ℓ0 −ℓ0) generates dilatationsand rotations respectively, the scaling dimension ∆and the spin s of the stateare given by ∆= h + h and s = h −h. In later sections, we shall generalizethese ideas to the full quantum realization of the algebra (1.8).1.3.

Constraints of conformal invariance in d dimensionsWe shall now return to the case of an arbitrary number of dimensionsd = p + q and consider the constraints imposed by conformal invariance onthe N-point functions of a quantum theory. In what follows we shall prefer toemploy the jacobian,∂x′∂x =1pdet g′µν= Ω−d/2 ,(1.10)to describe conformal transformations, rather than directly the scale factor Ωof (1.1).

For dilatations (1.4b) and special conformal transformations (1.4c),this jacobian is given respectively by∂x′∂x = λdand∂x′∂x =1(1 + 2b · x + b2x2)d . (1.11)9We deﬁne a theory with conformal invariance to satisfy some straightfor-ward properties:1) There is a set of ﬁelds {Ai}, where the index i speciﬁes the diﬀerentﬁelds.This set of ﬁelds in general is inﬁnite and contains in particular thederivatives of all the ﬁelds Ai(x).2) There is a subset of ﬁelds {φj} ⊂{Ai}, called “quasi-primary”, thatunder global conformal transformations, x →x′ (i.e.

elements of O(p+1, q+1)),transform according toφj(x) →∂x′∂x∆j/dφj(x′) ,(1.12)where ∆j is the dimension of φj (the 1/d compensates the exponent of d in(1.10)). The theory is then covariant under the transformation (1.12), in thesense that the correlation functions satisfyφ1(x1) .

. .

φν(xn)=∂x′∂x∆1/dx=x1· · ·∂x′∂x∆n/dx=xnφ1(x′1) . .

. φn(x′n).

(1.13)3) The rest of the {Ai}’s can be expressed as linear combinations of thequasi-primary ﬁelds and their derivatives.4) There is a vacuum |0⟩invariant under the global conformal group.The covariance property (1.13) under the conformal group imposes severerestrictions on 2- and 3-point functions of quasi-primary ﬁelds. To identify in-dependent invariants on which N-point functions might depend, we constructsome invariants of the conformal group in d dimensions.

Ordinary translationinvariance tells us that an N-point function depends not on N independentcoordinates xi, but rather only on the diﬀerences xi −xj (d(N −1) independentquantities). If we consider for simplicity spinless objects, then rotational invari-ance furthermore tells us that for d large enough, there is only dependence onthe N(N −1)/2 distances rij ≡|xi −xj|.

(As we shall see, for a given N-pointfunction in low enough dimension, there will automatically be linear relationsamong coordinates that reduce the number of independent quantities.) Next,10

imposing scale invariance (1.4b) allows dependence only on the ratios rij/rkl.Finally, since under the special conformal transformation (1.4c), we have|x′1 −x′2|2 =|x1 −x2|2(1 + 2b · x1 + b2x21)(1 + 2b · x2 + b2x22) ,(1.14)only so-called cross-ratios of the formrij rklrik rjl(1.15)are invariant under the full conformal group. The number of independent cross-ratios of the form (1.15), formed from N coordinates, is N(N −3)/2 [4].

(To seethis, use translational and rotational invariance to describe the N coordinatesas N −1 points in a particular N −1 dimensional subspace, thus characterizedby (N −1)2 independent quantities. Then use rotational, scale, and specialconformal transformations of the N −1 dimensional conformal group, a total of(N −1)(N −2)/2+1+(N−1) parameters, to reduce the number of independentquantities to N(N −3)/2.

)According to (1.13), the 2-point function of two quasi-primary ﬁelds φ1,2in a conformal ﬁeld theory must satisfyφ1(x1) φ2(x2)=∂x′∂x∆1/dx=x1∂x′∂x∆2/dx=x2φ1(x′1) φ2(x′2). (1.16)Invariance under translations and rotations (1.4a) (for which the jacobian isunity) forces the left hand side to depend only on r12 ≡|x1 −x2|.

Invarianceunder the dilatations x →λx then implies thatφ1(x1) φ2(x2)=C12r∆1+∆212,where C12 is a constant determined by the normalization of the ﬁelds. Finally,using the special conformal transformation (1.14) for r12 and (1.11) for itsjacobian, we ﬁnd that (1.16) requires that ∆1 = ∆2 if c12 ̸= 0, and henceφ1(x1) φ2(x2)=c12r2∆12∆1 = ∆2 = ∆0∆1 ̸= ∆2 .

(1.17)11The 3-point function is similarly restricted. Invariance under translations,rotations, and dilatations requiresφ1(x1) φ2(x2) φ3(x3)=Xa,b,cCabcra12 rb23 rc13,where the summation (in principle this could be an integration over a continuousrange) over a, b, c is restricted such that a + b + c = ∆1 + ∆2 + ∆3.

Thencovariance under the special conformal transformations (1.4c) in the form (1.14)requires a = ∆1 + ∆2 −∆3, b = ∆2 + ∆3 −∆1, and c = ∆3 + ∆1 −∆2. Thusthe 3-point function depends only on a single constant C123,φ1(x1) φ2(x2) φ3(x3)=C123r∆1+∆2−∆312r∆2+∆3−∆123r∆3+∆1−∆213.

(1.18)It might seem at this point that conformal invariant theories are rathertrivial since the Green functions thus far considered are entirely determined upto some constants. The N-point functions for N ≥4, however, are not so fullydetermined since they begin to have in general a dependence on the cross-ratios(1.15).

The 4-point function, for example, may take the more general formG(4)(x1, x2, x3, x4) = Fr12 r34r13 r24, r12 r34r23 r41 Yi

In two dimensions,however, the local conformal group provides additional constraints that we shallstudy in the next section.2. Conformal theories in 2 dimensions2.1.

Correlation functions of primary ﬁeldsWe now apply the general formalism of section 1 to the special case of twodimensions, as introduced in subsection 1.2. Recall from (1.6) that the lineelement ds2 = dz dz transforms under z →f(z) asds2 →∂f∂z ∂f∂zds2 .12

We shall generalize this transformation law to the formΦ(z, z) →∂f∂zh ∂f∂zhΦf(z), f(z),(2.1)where h and h are real-valued (and h again does not indicate the complexconjugate of h). (2.1) is equivalent to the statement that Φ(z, z)dzhdzh isinvariant.

It is similar in form to the tensor transformation propertyAµ...ν(x) →∂x′α∂xµ · · · ∂x′β∂xν Aα···β(x′) ,under x →x′. In two dimensional complex coordinates, a tensor Φzzz...z z(z, z),with m lower z indices and n lower z indices, would transform as (2.1) withh = m, h = n.The transformation property (2.1) deﬁnes what is known as a primary ﬁeldΦ of conformal weight (h, h).

Not all ﬁelds in conformal ﬁeld theory will turnout to have this transformation property — the rest of the ﬁelds are known assecondary ﬁelds. A primary ﬁeld is automatically quasi-primary, i.e.

satisﬁes(1.12) under global conformal transformations. (A secondary ﬁeld, on the otherhand, may or may not be quasi-primary.

Quasi-primary ﬁelds are sometimesalso termed SL(2, C) primaries.). Inﬁnitesimally, under z →z + ǫ(z), z →z + ǫ(z), we have from (2.1)δǫ,ǫ Φ(z, z) =h∂ǫ + ǫ∂+h ∂ǫ + ǫ∂Φ(z, z) ,(2.2)where ∂≡∂z.Now the 2-point function G(2)(zi, zi) =Φ1(z1, z1)Φ2(z2, z2)is supposedto satisfy the inﬁnitesimal form of (1.13),δǫ,ǫ G(2)(zi, zi) =δǫ,ǫ Φ1, Φ2+Φ1, δǫ,ǫ Φ2= 0 ,giving the partial diﬀerential equationǫ(z1)∂z1 + h1∂ǫ(z1)+ǫ(z2)∂z2 + h2∂ǫ(z2)+ǫ(z1)∂z1 + h1∂ǫ(z1) + ǫ(z2)∂z2 + h2∂ǫ(z2)G(2)(zi, zi) = 0 .

(2.3)13Then paralleling the arguments that led to (1.17), we use ǫ(z) = 1 andǫ(z) = 1 to show that G(2) depends only on z12 = z1 −z2, z12 = z1 −z2; thenuse ǫ(z) = z and ǫ(z) = z to require G(2) = C12/(zh1+h212zh1+h212); and ﬁnallyǫ(z) = z2 and ǫ(z) = z2 to require h1 = h2 = h, h1 = h2 = h. The result isthat the 2-point function is constrained to take the formG(2)(zi, zi) =C12z2h12 z2h12. (2.4)To make contact with (1.17), we consider bosonic ﬁelds with spin s = h−h = 0.In terms of the scaling weight ∆= h + h, we see that (2.4) is equivalent toG(2)(zi, zi) =C12|z12|2∆.The 3-point function G(3) = ⟨Φ1Φ2Φ3⟩is similarly determined, by argu-ments parallel to those leading to (1.18), to take the formG(3)(zi, zi) = C1231zh1+h2−h312zh2+h3−h123zh3+h1−h213·1zh1+h2−h312zh2+h3−h123zh3+h1−h213,(2.5)where zij = zi −zj.

As in (1.18), the 3-point function depends only on a singleconstant. This is because three points z1, z2, z3 can always be mapped by aconformal transformation to three reference points, say ∞, 1, 0, where we havelimz1→∞z2h11z2h11G(3) = C123.

The coordinate dependence for general z1, z2, z3can be reconstructed by conformal invariance. For all ﬁelds taken to be spinless,so that si = hi −hi = 0, (2.5) correctly reduces to (1.18) with ∆i = hi +hi andrij = |zij|.As in (1.19), the 4-point function, on the other hand, is not so fully deter-mined just by conformal invariance.

Global conformal invariance allows it totake the formG(4)(zi, zi) = f(x, x)Yi

where h = P4i=1 hi, h = P4i=1 hi.In (2.6) the cross-ratio x is deﬁned asx = z12z34/z13z24. (We note that this cross-ratio is annihilated by the dif-ferential operator P4i=1 ǫ(zi)∂zi so the analog of (2.3) leaves the function fundetermined.) In two dimensions, the two cross-ratios of (1.19) are linearlyrelated (because 4 points constrained to be coplanar must satisfy an additionallinear relation).

The six possible cross ratios of the form (1.15), constructedfrom four zi’s, are given byx = z12z34z13z24,1 −x = z14z23z13z24,x1 −x = z12z34z14z23,and their inverses. With respect to the argument that ﬁxed the form of the3-point function (2.5), we can understand the residual x dependence of (2.6)by recalling that global conformal transformations only allow us to ﬁx threecoordinates, so the best we can do is to take say z1, z2, z3, z4 = ∞, 1, x, 0.In (2.4)–(2.6), the hi’s and hi’s are in principle arbitrary.

Later on weshall see how they may be constrained by unitarity. We shall also formulatediﬀerential equations which, together with monodromy conditions, allow one inprinciple to determine all the unknown functions (generalizing the f of (2.6))for arbitrary N-point functions in a given theory.2.2.

Radial quantization and conserved chargesTo probe more carefully the consequences of conformal invariance in atwo dimensional quantum ﬁeld theory, we enter into some of the details of thequantization procedure. We begin with ﬂat Euclidean “space” and “time” co-ordinates σ1 and σ0.

In Minkowski space, the standard light-cone coordinateswould be σ0 ± σ1.In Euclidean space the analogs are instead the complexcoordinates ζ, ζ = σ0 ± iσ1. The two dimensional Minkowski space notions ofleft- and right-moving massless ﬁelds become Euclidean ﬁelds that have purelyholomorphic or anti-holomorphic dependence on the coordinates.

For this rea-son we shall occasionally call the holomorphic and anti-holomorphic ﬁelds left-and right-movers respectively. To eliminate any infrared divergences, we com-pactify the space coordinate, σ1 ≡σ1 +2π.

This deﬁnes a cylinder in the σ1, σ0coordinates.15Next we consider the conformal map ζ →z = exp ζ = exp(σ0 + iσ1)that maps the cylinder to the complex plane coordinatized by z (see ﬁg. 1.

)Then inﬁnite past and future on the cylinder, σ0 = ∓∞, are mapped to thepoints z = 0, ∞on the plane. Equal time surfaces, σ0=const, become circlesof constant radius on the z-plane, and time reversal, σ0 →−σ0, becomes z →1/z∗.

To build up a quantum theory of conformal ﬁelds on the z-plane, wewill need to realize the operators that implement conformal mappings of theplane. For example dilatations, z →eaz, on the cylinder are just the timetranslations σ0 →σ0 + a.

So the dilatation generator on the conformal planecan be regarded as the Hamiltonian for the system, and the Hilbert space isbuilt up on surfaces of constant radius. This procedure for deﬁning a quantumtheory on the plane is known as radial quantization[5].

It is particularly usefulfor two dimensional conformal ﬁeld theory in the Euclidean regime since itfacilitates use of the full power of contour integrals and complex analysis toanalyze short distance expansions, conserved charges, etc. Our intuition formanipulations in this scheme will frequently come from referring things back tothe cylinder.σ1σ0zFig.

1. Map of the cylinder to the plane16

Symmetry generators in general can be constructed via the Noether pre-scription.A d + 1 dimensional quantum theory with an exact symmetryhas an associated conserved current jµ, satisfying ∂µjµ = 0. The conservedcharge Q = R ddx j0(x), constructed by integrating over a ﬁxed-time slice, gen-erates, according to δǫA = ǫ[Q, A], the inﬁnitesimal symmetry variation inany ﬁeld A.

In particular, local coordinate transformations are generated bycharges constructed from the stress-energy tensor Tµν, in general a symmetricdivergence-free tensor.In conformally invariant theories, Tµν is also trace-less.This follows from requiring the conservation 0 = ∂· j = T µµ of thedilatation current jµ = Tµν xν (associated to the ordinary scale transforma-tions xµ →xµ + λxµ). The current associated to other inﬁnitesimal conformaltransformations is jµ = Tµν ǫν, where ǫµ satisﬁes (1.2).

This current as wellhas an automatically vanishing divergence, ∂· j = 12T µµ(∂· ǫ) = 0, due to thetraceless condition on Tµν.To implement the conserved charges on the conformal z-plane, we introducethe necessary complex tensor analysis. The ﬂat Euclidean plane (gµν = δµν)in complex coordinates z = x + iy has line element ds2 = gµν dxµdxν = dx2 +dy2 = dz dz.

The components of the metric referred to complex coordinateframes are thus gzz = gz z = 0 and gzz = gzz = 12, and the components of thestress-energy tensor referred to these frames are Tzz = 14T00 −2iT10 −T11,Tz z =14T00 + 2iT10 −T11, and Tzz = Tzz =14T00 + T11=14T µµ. Theconservation law gαµ∂αTµν = 0 gives two relations, ∂zTzz + ∂zTzz = 0 and∂zTz z + ∂zTzz = 0.

Using the traceless condition Tzz = Tzz = 0, these imply∂zTzz = 0and∂zTz z = 0 .The two non-vanishing components of the stress-energy tensorT (z) ≡Tzz(z)andT(z) = Tz z(z)thus have only holomorphic and anti-holomorphic dependences. We shall ﬁndnumerous properties of conformal theories on the z-plane to factorize similarlyinto independent left and right pieces.17It is natural to expect T and T, the remnants of the stress-energy tensorin complex coordinates, to generate local conformal transformations on the z-plane.In radial quantization, the integral of the component of the currentorthogonal to an “equal-time” (constant radius) surface becomesRj0(x) dx →R jr(θ) dθ.

Thus we should takeQ =12πiI dz T (z)ǫ(z) + dz T(z)ǫ(z)(2.7)as the conserved charge.The line integral is performed over some circle ofﬁxed radius and our sign conventions are such that both the dz and the dzintegrations are taken in the counter-clockwise sense. Note that (2.7) is a formalexpression that cannot be evaluated until we specify what other ﬁelds lie insidethe contour.The variation of any ﬁeld is given by the “equal-time” commutator withthe charge (2.7),δǫ,ǫΦ(w, w) =12πiI hdz T (z) ǫ(z) , Φ(w, w)i+hdz T(z) ǫ(z) , Φ(w, w)i.

(2.8)Now products of operators A(z)B(w) in Euclidean space radial quantization areonly deﬁned for |z| > |w|. (In general, recall that to continue any Minkowskispace Green functionA1(x1, t1) .

. .

An(xn, tn)to Euclidean space, we let A(x, t) →eHτA(x, 0)e−Hτ, where t = iτ. In a theorywith energy bounded from below, the Euclidean space Green functionA1(x1, 0)e−H(τ1−τ2)A2(x2, 0) .

. .

e−H(τn−1−τn)A(xn, 0)is guaranteed to converge only for operators that are time-ordered, i.e. for whichτj > τj+1.

The analytic continuation of time-ordered Euclidean Green functionsthen gives the desired solution to the Minkowski space equations of motionon the cylinder. In a Euclidean space functional integral formulation, Greenfunctionsφ1 .

. .

φn=Zϕexp−S[ϕ]ϕ1 . .

. ϕn.

Zϕexp−S[ϕ]18

are computed in terms of dummy integration variables ϕ, which automaticallycalculate the time-ordered (convergent) result.) Thus we deﬁne the radial or-dering operation R asRA(z)B(w)=A(z)B(w)|z| > |w|B(w)A(z)|z| < |w|(2.9)(or with a minus sign for fermionic operators).

This allows us to deﬁne themeaning of the commutators in (2.8). The equal-time commutator of a localoperator A with the spatial integral of an operator B will become the contourintegral of the radially ordered product,Rdx B, AE.T.

→Hdz RB(z)A(w).In ﬁg. 2 we have represented the contour integrations that we need toperform in order to evaluate the commutator in (2.8).

We see that the diﬀerencecombines into a single integration about a contour drawn tightly around thepoint w. (The reader might derive further insight into the map (ﬁg. 1) fromthe cylinder to the plane by pulling back ﬁg.

2 to the cylinder and seeing whatit looks like in terms of equal time σ0 contours.) We may thus rewrite (2.8) inthe formδǫ,ǫΦ(w,w) =12πi I|z|>|w|−I|z|<|w|!

dz ǫ(z)RT (z)Φ(w, w)+ dz ǫ(z)RT(z)Φ(w, w)=12πiI dz ǫ(z)RT (z)Φ(w, w)+ dz ǫ(z)RT (z)Φ(w, w)= h ∂ǫ(w)Φ(w, w) + ǫ(w)∂Φ(w, w)+ h ∂ǫ(w) Φ(w, w) + ǫ(w) ∂Φ(w, w) ,where in the last line we have substituted the desired result, i.e. the result of thetransformation (2.1) in the case of inﬁnitesimal f(z) = z + ǫ(z).

In order thatthe charge (2.7) induce the correct inﬁnitesimal conformal transformations, weinfer that the short distance singularities of T and T with Φ should beRT (z)Φ(w, ¯w)=h(z −w)2 Φ(w, w) +1z −w ∂wΦ(w, w) + . .

.RT(z)Φ(w, w)=h(z −w)2 Φ(w, w) +1z −w ∂wΦ(w, w) + . .

. .19These short distance properties can be taken to deﬁne the quantum stress-energy tensor.

They are naturally realized by standard canonical deﬁnitionsof the stress-energy tensor in two dimensions (since they ordinarily result ingenerators of coordinate transformations). In a moment, we shall conﬁrm howall of this works in some speciﬁc examples.zw=−zwzwFig.

2. Evaluation of “equal-time” commutator on the conformal plane.We see that the transformation law (2.1) for primary ﬁelds leads to a shortdistance operator product expansion for the holomorphic and anti-holomorphicstress-energy tensors, T and T, with a primary ﬁeld.

From now on we shall dropthe R symbol and consider the operator product expansion itself as a shorthandfor radially ordered products. The operator product expansion that deﬁnes thenotion of a primary ﬁeld is abbreviated asT (z)Φ(w, ¯w) =h(z −w)2 Φ(w, w) +1z −w ∂wΦ(w, w) + .

. .T(z)Φ(w, w) =h(z −w)2 Φ(w, w) +1z −w ∂wΦ(w, w) + .

. .

,(2.10)and encodes the conformal transformation properties of Φ. In the next section,we shall see how operator product expansions are also equivalent to canonicalcommutators of the modes of the ﬁelds.We pause at this point to recall some of the standard lore concerningoperator product expansions[6].

In general, the singularities that occur when20

operators approach one another are encoded in operator product expansions ofthe formA(x)B(y) ∼XiCi(x −y)Oi(y) ,(2.11)where the Oi’s are a complete set of local operators and the Ci’s are (singular)numerical coeﬃcients. Ordinarily (2.11) is an asymptotic expansion, but in aconformal theory it has been argued to converge (since e−ℓ/|z−w| type termsthat would be expected if the series did not converge require a dimensionalparameter ℓ, absent in a conformal ﬁeld theory).

For operators of ﬁxed scalingdimension d in (2.11), we can determine the coordinate dependence of the Ci’sby dimensional analysis to be Ci ∼1/|x −y|dA+dB−dOi.In two dimensional conformal ﬁeld theories, we can always take a basis ofoperators φi with ﬁxed conformal weight. If we normalize their 2-point functions(2.4) asφi(z, z) φj(w, w)= δij1(z −w)2hi1(z −w)2hi ,(2.12)then the operator product coeﬃcients Cijk deﬁned byφi(z, z) φj(w, w) ∼XkCijk (z −w)hk−hi−hj (z −w)hk−hi−hj φk(w, w) (2.13)are symmetric in i, j, k. By taking the limit as any two of the zi’s in the 3-pointfunction ⟨φiφjφk⟩approach one another, and using (2.12), it is easy to showthat the Cijk’s of (2.13) coincide precisely with the numerical factors in the3-point functions (2.5).2.3.

Free boson, the exampleWe shall now illustrate the formalism developed thus far in the case of asingle massless free boson, also known as the gaussian model. We use the stringtheory normalization for the action,S =ZL = 12πZ∂X ∂X ,(2.14)so that X(z, z) has propagatorX(z, z)X(w, w)= −12 log |z −w|.

(This iscalculated using z = 12(σ1 + iσ0), and integration measure 2idz∧dz = dσ1∧dσ021in (2.14), although ultimately only the normalization of the propagator itselfis important in what follows.) The standard statistical mechanical convention(see e.g.

section 4.2 of Cardy’s lectures) uses instead a factor of g/4π in frontin the action (2.14). For solutions of the equations of motion, we ﬁnd thatX(z, z) = 12x(z)+x(z)splits into two pieces with only holomorphic and anti-holomorphic dependence respectively.

(These are the massless left-movers andright-movers. To avoid any ambiguity we could write xL(z) and xR(z), but themeaning is usually clear from context.) These pieces have propagatorsx(z)x(w)= −log(z −w) ,x(z) x(w)= −log(z −w) .

(2.15)Note that the ﬁeld x(z) is not itself a conformal ﬁeld, but its derivative, ∂x(z),has leading short distance expansion∂x(z) ∂x(w) = −1(z −w)2 + . .

. ,(2.16)inferred by taking two derivatives of (2.15).

We see from the scaling propertiesof the right hand side of (2.16) that ∂x(z) has a chance to be a (1,0) conformalﬁeld.Concentrating for the moment on the holomorphic dependence of the the-ory, we deﬁne the stress-energy tensor T (w) via the normal-ordering prescrip-tionT (w) = −12 : ∂x(z)∂x(w):≡−12 limz→w∂x(z)∂x(w) +1(z −w)2. (2.17)Using the Wick rules and Taylor expanding, we can compute the singular partofT (z) ∂x(w) = −12 : ∂x(z)∂x(z): ∂x(w)= −12∂x(z)∂x(z)∂x(w)· 2 + .

. .= ∂x(z)1(z −w)2 + .

. .=∂x(w) + (z −w)∂2x(w)1(z −w)2 + .

. .

,22

in the limit z →w. We ﬁndT (z)∂x(w) ∼∂x(w)(z −w)2 +1z −w ∂2x(w) + .

. .

,in accord with (2.10) for a (1,0) primary ﬁeld. Moreover substituting in (2.8),we see thatIdz2πi T (z)ǫ(z) , ∂x(w)=Idz2πiǫ(z) ∂x(w)(z −w)2 + ∂2x(w)z −w + .

. .= ∂ǫ(w)∂x(w) + ǫ(w)∂2x(w) .This is all as expected since under z →z + ǫ, we have x(z) →x(z + ǫ) =x(z) + ǫ∂x(z), and consequently ∂x(z) →∂x(z) + ∂ǫ∂x(z) + ǫ∂2x(z).Theabove result is just the statement that ∂x transforms as in (2.1) as a tensor ofmass dimension h = 1.As another illustration of (2.10), we consider the operator : exp iαx(w): .The normal ordering symbol is meant to remind us not to contract the x(w)’sin the expansion of the exponent with one another.

(This prescription is equiv-alent to a multiplicative wave function renormalization, and for convenience wewill frequently drop the normal ordering symbol in the following). Taking theoperator product expansion with T (z) as z →w, we ﬁnd the leading singularbehavior−12∂x(z)2eiαx(w) = −12∂x(z)iαx(w)2eiαx(w)−12 2 ∂x(z)∂x(z)iαx(w)eiαx(w)=α2/2(z −w)2 eiαx(w) + iα∂x(z)z −weiαx(w)=α2/2(z −w)2 eiαx(w) +1z −w ∂eiαx(w) .

(2.18)exp(iαx) is thus a primary ﬁeld of conformal dimension h = α2/2.This result could also be inferred from the 2-point functionDeiαx(z)e−iαx(w)E= eα2⟨x(z)x(w)⟩=1(z −w)α2 ,(2.19)23where the ﬁrst equality is a general property of free ﬁeld theory, and the sec-ond equality follows from the speciﬁcally two dimensional logarithmic behavior(2.15) (recall that the propagator in d > 2 spacetime dimensions goes insteadasRddp exp(ipx)/p2 ∼1/xd−2).We see that the logarithmic divergence ofthe scalar propagator leads to operators with continuously variable anomalousdimensions in two dimensions, even in free ﬁeld theory.Identical considerations apply equally to anti-holomorphic operators, suchas ∂x(z) and exp(iαx(z). Their operator products with T(z) = 12 : ∂x(z)∂x(z):shows them to have conformal dimensions (0, 1) and (0, α2/2).

More generallyif we took an action S =12πR∂Xµ∂Xµ with a vector of ﬁelds Xµ(z, z) =12(xµ(z) + xµ(z), thenxµ(z)xν(w)= −δµν log(z −w) and exp±iαµxµ(z)for example has conformal dimension (α · α/2, 0).Before closing this introduction to massless scalars in two dimensions, weshould dispel an occasional unwarranted confusion concerning the result of [7],which states that the Goldstone phenomenon does not occur in two dimensions.In the present context this does not mean that there is anything particularlypeculiar about massless scalar ﬁelds, only that they are not Goldstone bosons.Although it appears that (2.14) has a translation symmetry X →X + a thatcan be spontaneously broken, this symmetry is an illusion at the quantum level.That is because the ﬁeld X is itself ill-deﬁned due to the incurable infraredlogarithmic divergence of its propagator. ∂µX is of course well deﬁned but isnot sensitive to the putative symmetry breaking.

Exponentials of X as in (2.19)can also be deﬁned by appropriate extraction of wave function normalization,but their non-vanishing correlation functions all have simple power law falloﬀ,and again show no signal of symmetry breakdown. This is all consistent withthe result of [7].2.4.

Conformal Ward identitiesWe complete our discussion of conformal formalities by writing down theconformal Ward identities satisﬁed by correlations functions of primary ﬁeldsφi. Ward identities are generally identities satisﬁed by correlation functions asa reﬂection of symmetries possessed by a theory.

They are easily derived in the24

functional integral formulation of correlation functions for example by requiringthat they be independent of a change of dummy integration variables. TheWard identities for conformal symmetry can thus be derived by considering thebehavior of n-point functions under a conformal transformation.

This shouldbe considered to take place in some localized region containing all the operatorsin question, and can then be related to a surface integral about the boundaryof the region.For the two dimensional conformal theories of interest here, we shall insteadimplement this procedure in the operator form of the correlation functions. Byglobal conformal invariance, these correlation functions satisfy (compare with(1.13))φ1(z1, z1) .

. .

φn(zn, zn)=Yj∂f(zj)hj∂f(zj)hj φ1(w1, w1) . .

. φn(wn, wn),(2.20)with w = f(z) and w = f(z) of the form (1.9).

To gain additional informationfrom the local conformal algebra, we consider an assemblage of operators atpoints wi as in ﬁg. 3, and perform a conformal transformation in the interiorof the region bounded by the z contour by line integrating ǫ(z)T (z) aroundit.

By analyticity, the contour can be deformed to a sum over small contoursencircling each of the points wi, as depicted in the ﬁgure. The result of thecontour integration is thusDIdz2πi ǫ(z)T (z) φ1(w1, w1) .

. .

φn(wn, wn)E=nXj=1Dφ1(w1, w1) . .

.Idz2πi ǫ(z)T (z)φj(wj, wj). .

. φn(wn, wn)E=nXj=1φ(w1, w1) .

. .

δǫφj(wj, wj) . .

. φn(wn, wn).

(2.21)In the last line we have used the inﬁnitesimal transformation propertyδǫφ(w, w) =Idz2πi ǫ(z)T (z)φ(w, w) =ǫ(w)∂+ h∂ǫ(w)φ(w, w) ,encoded in the operator product expansion (2.10).25z=w1w2w3w4w5w1w2w3w4w5Fig. 3.

Another deformed contourSince (2.21) is true for arbitrary ǫ(z) andHdzT (z) = 0, we can write anunintegrated form of the conformal Ward identities,T (z)φ1(w1, w1) . .

. φn(wn, wn)=nXj=1hj(z −wj)2 +1z −wj∂∂wj φ1(w1, w1) .

. .

φn(wn, wn). (2.22)This states that the correlation functions are meromorphic functions of z withsingularities at the positions of inserted operators.

The residues at these sin-gularities are simply determined by the conformal properties of the operators.Later on we shall use (2.22) to show that 4-point correlation functions involvingso-called degenerate ﬁelds satisfy hypergeometric diﬀerential equations.3. The central charge and the Virasoro algebra3.1.

The central chargeNot all ﬁelds satisfy the simple transformation property (2.1) under con-formal transformations. Derivatives of ﬁelds, for example, in general have morecomplicated transformation properties.

A secondary ﬁeld is any ﬁeld that hashigher than the double pole singularity (2.10) in its operator product expansion26

with T or T. In general, the ﬁelds in a conformal ﬁeld theory can be groupedinto families [φn] each of which contains a single primary ﬁeld φn and an inﬁniteset of secondary ﬁelds (including its derivative), called its descendants. Thesecomprise the irreducible representations of the conformal group, and the pri-mary ﬁeld can be regarded as the highest weight of the representation.

The setof all ﬁelds in a conformal theory {Ai} = Pn[φn] may be composed of either aﬁnite or inﬁnite number of conformal families.An example of a ﬁeld that does not obey (2.1) or (2.10) is the stress-energy tensor. By performing two conformal transformations in succession, wecan determine its operator product with itself to take the formT (z)T (w) =c/2(z −w)4 +2(z −w)2 T (w) +1z −w ∂T (w) .

(3.1)The (z −w)−4 term on the right hand side, with coeﬃcient c a constant, isallowed by analyticity, Bose symmetry, and scale invariance.Its coeﬃcientcannot be determined by the requirement that T generate conformal transfor-mations, since that only involves the commutator of T with other operators.Apart from this term, (3.1) is just the statement that T (z) is a conformal ﬁeldof weight (2,0). The constant c is known as the central charge and its valuein general will depend on the particular theory under consideration.SinceT (z)T (0)= (c/2)/z4, we expect at least that c ≥0 in a theory with a posi-tive semi-deﬁnite Hilbert space.Identical considerations apply to T, so thatT(z) T(w) =c/2(z −w)4 +2(z −w)2 T(w) +1z −w ∂T(w) ,(3.2)where c is in principle an independent constant.

(Later on we shall see thatmodular invariance constrains c −c = 0 mod 24.) A theory with a Lorentz-invariant, conserved 2-point functionTµν(p) Tαβ(−p)requires c = c. This isequivalent to requiring cancellation of local gravitational anomalies[8], allowingthe system to be consistently coupled to two dimensional gravity.

In heteroticstring theory, for example, this is achieved by adding ghosts to the system sothat c = c = 0.27In general, the inﬁnitesimal transformation law for T (z) induced by (3.1)isδǫT (z) = ǫ(z) ∂T (z) + 2∂ǫ(z) T (z) + c12∂3ǫ(z) .It can be integrated to giveT (z) →(∂f)2 Tf(z)+ c12 S(f, z)(3.3)under z →f(z), where the quantityS(f, z) = ∂zf ∂3zf −32(∂2zf)2(∂zf)2is known as the Schwartzian derivative.It is the unique weight two objectthat vanishes when restricted to the global SL(2, R) subgroup of the two di-mensional conformal group. (It also satisﬁes the composition law S(w, z) =(∂zf)2S(w, f) + S(f, z).) The stress-energy tensor is thus an example of a ﬁeldthat is quasi-primary, i.e.

SL(2, C) primary, but not (Virasoro) primary.We can readily calculate (3.1) for the free boson stress-energy tensor (2.17),T (z) = −12 : ∂x(z)∂x(z): . The result isT (z)T (w)=−1222∂x(z)∂x(w)2+ 4∂x(z)∂x(w)∂x(z)∂x(w)+ .

. .=1/2(z −w)4 +2(z −w)2−12∂x(w)2+1z −w ∂−12∂x(w)2,and thus the leading term in (3.1) is normalized so that a single free boson hasc = 1.A variation on (2.17) is to take insteadT (w) = −12 : ∂x(z)∂x(w): + i√2α0 ∂2x(z) .

(3.4)The extra term is a total derivative of a well-deﬁned ﬁeld and does not aﬀectthe status of T (z) as a generator of conformal transformations. Using (2.16)28

and proceeding as above, we can show that the T (z) of (3.4) satisﬁes (3.1) withcentral chargec = 1 −24α20 .We see that the eﬀect of the extra term in (3.4) is to shift c < 1 for α0 real.Since the stress-energy tensor in (3.4) has an imaginary part, the theory itdeﬁnes is not unitary for arbitrary α0. For particular values of α0, it turns outto contain a consistent unitary subspace.

(In section 4, we will discuss the roleplayed by unitarity in ﬁeld theory and statistical mechanical models and alsoimplicitly identify the relevant values of α0. )The modiﬁcation of T (z) in (3.4) is interpreted as the presence ofa ‘background charge’ −2α0 at inﬁnity.This is created by the operator: exp−i2√2α0x(∞): , so we take as out-state(−2α0) =⟨0|V−2α0(∞)⟨0|V−2α0(∞) V2α0(0)|0⟩,where Vβ(z) ≡: expi√2βx(z): .

Thus the only non-vanishing correlation func-tions of strings of operators Vβj(z) are those with Pj βj = 2α0. n-point correla-tion functions may be derived by sending a V−2α0(z) to inﬁnity in an n+1-pointfunction.

For example, the result (2.19) for the 2-point function is modiﬁed toVβ(z) V2α0−β(w)=1(z −w)2β(β−2α0) .The operators in this 2-point function are regarded as adjoints of one anotherin the presence of the background charge, and each thus has conformal weighth = β(β −2α0). We arrive at the same result (rather than simply h = β2)by calculating the conformal weight of the operator Vβ(z) as in (2.18), onlyusing the modiﬁed deﬁnition (3.4) of T (z).

This formalism was anticipated inancient times[9] and has more recently been used to great eﬀect[10] to calculatecorrelation functions of the c < 1 theories to be discussed in the next section.These and other applications are described in more detail in Zuber’s lectures.293.2. The free fermionAnother free system that will play a major role later on here is that of afree massless fermion.

With both chiralities, we write the actionS = 18πZ ψ∂ψ + ψ∂ψ. (3.5)The equations of motion determine that ψ(z) and ψ(z) are respectively theleft- and right-moving “chiralities”.

(Recall that in 2 Euclidean dimensions theDirac operator can be represented as/∂= σx∂x + σy∂y =∂x −i∂y∂x + i∂y∼∂∂,so that the operators ∂, ∂are picked out by the chirality projectors12(1 ±σz).) The normalization of (3.5) is chosen so that the leading short distancesingularities areψ(z)ψ(w) = −1z −w ,ψ(z)ψ(w) = −1z −w .This system has holomorphic and anti-holomorphic stress-energy tensorsT (z) = 12 : ψ(z)∂ψ(z): ,T(z) = 12 : ψ(z) ∂ψ(z):that satisfy (3.1) with c = c = 12.

From the T (z)ψ(w) and T(z)ψ(w) operatorproducts we verify that ψ and ψ are primary ﬁelds of conformal weight ( 12, 0)and (0, 12).3.3. Mode expansions and the Virasoro algebraIt is convenient to deﬁne a Laurent expansion of the stress-energy tensor,T (z) =Xn∈Zz−n−2Ln ,T(z) =Xn∈Zz−n−2Ln ,(3.6)in terms of modes Ln (which are themselves operators).

The exponent −n −2in (3.6) is chosen so that for the scale change z →z/λ, under which T (z) →30

λ2 T (z/λ), we have L−n →λn L−n. L−n and L−n thus have scaling dimensionn.

(3.6) is formally inverted by the relationsLn =Idz2πi zn+1 T (z) ,Ln =Idz2πi zn+1 T(z) . (3.7)To compute the algebra of commutators satisﬁed by the modes Ln and Ln,we employ a procedure for making contact between local operator products andcommutators of operator modes that will repeatedly prove useful.

The commu-tator of two contour integrationsHdz,Hdwis evaluated by ﬁrst ﬁxing w anddeforming the diﬀerence between the two z integrations into a single z contourdrawn tightly around the point w, as in ﬁg. 2.

In evaluating the z contourintegration, we may perform operator product expansions to identify the lead-ing behavior as z approaches w. The w integration is then performed withoutfurther subtlety. For the modes of the stress-energy tensor, this procedure givesLn, Lm=Idz2πiIdw2πi −Idw2πiIdz2πizn+1 T (z)wm+1T (w)=Idz2πiIdw2πi zn+1 wm+1c/2(z −w)4 +2T (w)(z −w)2 + ∂T (w)z −w + .

. .=Idw2πi c12(n + 1)n(n −1)wn−2wm+1+ 2(n + 1)wnwm+1T (w) + wn+1wm+1∂T (w).

(where the residue of the ﬁrst term results from 13!∂3zzn+1|z=w = 16(n + 1)n(n −1)wn−2). Integrating the last term by parts and combining with the secondterm gives (n −m)wn+m+1T (w), so performing the w integration givesLn, Lm= (n −m)Ln+m + c12(n3 −n)δn+m,0 .

(3.8a)The identical calculation for T results inLn, Lm= (n −m)Ln+m + c12(n3 −n)δn+m,0 . (3.8b)Since T (z) and T(z) have no power law singularities in their operator product,on the other hand, we have the commutationLn, Lm= 0 .

(3.8c)31In (3.8a–c) we ﬁnd two copies of an inﬁnite dimensional algebra, calledthe Virasoro algebra, originally discovered in the context of string theory [11].Every conformally invariant quantum ﬁeld theory determines a representationof this algebra with some value of c and c. For c = c = 0, (3.8a, b) reducesto the classical algebra (1.8). The form of the algebra may be altered a bitby shifting the Ln’s by constants.

In (3.8a) this freedom is exhausted by therequirement that the subalgebra L−1, L0, L1 satisfy[L∓1, L0] = ∓L∓1[L1, L−1] = 2L0 ,with no anomaly term.The global conformal group SL(2, C) generated byL−1,0,1 and L−1,0,1 thus remains an exact symmetry group despite the centralcharge in (3.8).3.4. In- and out-statesTo analyze further the properties of the modes, it is useful to introduce thenotion of adjoint,A(z, z)† = A1z , 1z 1z2h1z2h ,(3.9)(on the real surface z = z∗), for Euclidean-space ﬁelds that correspond to real(Hermitian) ﬁelds in Minkowski space.

Although (3.9) might look strange, itis ultimately justiﬁed by considering the continuation back to the Minkowskispace cylinder, as described in section 2.2. The missing factors of i in Euc-lidean-space time evolution, A(x, τ) = eHτA(x, 0)e−Hτ, must be compensatedin the deﬁnition of the adjoint by an explicit Euclidean-space time reversal,τ →−τ.

As discussed earlier, this is implemented on the plane by z →1/z∗.The additional z, z dependent factors on the right hand side of (3.9) are requiredto give the adjoint the proper tensorial properties under the conformal group.We derive further intuition by considering in- and out-states in confor-mal ﬁeld theory. In Euclidean ﬁeld theory we ordinarily associate states withoperators via the identiﬁcation|Ain⟩=limσ0→−∞A(σ0, σ1)|0⟩=limσ0→−∞eHσ0A(σ1)|0⟩.32

Since time σ0 →−∞on the cylinder corresponds to the origin of the z-plane,it is natural to deﬁne in-states as|Ain⟩≡limz,z→0 A(z, z)|0⟩.To deﬁne ⟨Aout| we need to construct the analogous object for z →∞. Confor-mal invariance, however, allows us relate a parametrization of a neighborhoodabout the point at ∞on the Riemann sphere to that of a neighborhood aboutthe origin via the map z = 1/w.

If we call eA(w, w) the operator in the co-ordinates for which w →0 corresponds to the point at ∞, then the naturaldeﬁnition is⟨Aout| ≡limw,w→0⟨0| eA(w, w) . (3.10a)Now we need to relate eA(w, w) to A(z, z).

Recall that for primary ﬁelds wehave under w →f(w)eA(w, w) = Af(w), f(w) ∂f(w)h∂f(w)h ,so that in particular under f(w) = 1/w we haveeA(w, w) = A 1w, 1w −w−2h −w −2h .The deﬁnition (3.9) of adjoint then gives the natural relation between ⟨Aout|and |Ain⟩(up to a spin dependent phase ignored here for convenience),⟨Aout| =limw,w→0⟨0| eA(w, w)deﬁnition= limz,z→0⟨0|A1z , 1z 1z2h1z2hconformal invariance= limz,z→0⟨0|A(z, z)†adjoint=hlimz,z→0 A(z, z)|0⟩i†= |Ain⟩† . (3.11)33Occasionally we shall be sloppy and write the out-state in the form ⟨Aout| ≡limz,z→∞⟨0|A(z, z) — this should be recognized as shorthand for⟨Aout| ≡limz,z→∞⟨0|A(z, z) z2hz2h ,(3.10b)as follows from the deﬁnition (3.10a) and the second line of (3.11).

(Eqns. (3.10a, b)are actually correct for any quasi-primary ﬁeld, since we only make use of theSL(2, C) transformation w →1/w to deﬁne the out-state.

For general sec-ondary ﬁelds, on the other hand, the slightly more complicated expression maybe found for example in [12]. )(We point out that in deﬁning our in- and out-states by means of ﬁeldsof well-deﬁned scaling dimension, we are proceeding somewhat diﬀerently thanin ordinary perturbative ﬁeld theory calculations.

The procedure here deﬁnesasymptotic states that are eigenstates of the exact Hamiltonian of the system,rather than eigenstates of some ﬁctitious asymptotically non-interacting Hamil-tonian. Our ability to do this in conformal ﬁeld theories in two dimensions stemsfrom their providing non-trivial examples of solvable quantum ﬁeld theories.

Ifwe could implement such a prescription in non-trivial 3+1 dimensional ﬁeld the-ories, we of course would. We also point out that the correspondence betweenoperators and states in ﬁeld theory is not ordinarily one-to-one — in massiveﬁeld theories, for example, more than one operator typically creates the samestate as σ0 →−∞.

In conformal ﬁeld theory, the number of ﬁelds and stateswith any ﬁxed conformal weight is ordinarily ﬁnite so by orthogonalization wecan associate a unique ﬁeld with each state. )Note that for the stress-energy tensor, equality ofT †(z) =X L†mzm+2andT1z 1z4 =XLmz−m−21z4results inL†m = L−m .

(3.12)(3.12) should be regarded as the condition that T (z) is hermitian. Hermiticityof T(z) equivalently results in L†m = L−m.34

Other important conditions on the Ln’s can be derived by requiring theregularity ofT (z)|0⟩=Xm∈ZLm z−m−2|0⟩at z = 0. Evidently only terms with m ≤−2 are allowed, so we learn thatLm|0⟩= 0 ,m ≥−1 .

(3.13a)From (3.11) we have also that ⟨0|L†m = 0, m ≥−1.L0,±1|0⟩= 0 is thestatement that the vacuum is SL(2, R) invariant, and we see that this followsdirectly just from the requirement that z = 0 be a regular point (the rest ofthe vanishing Lm|0⟩= 0, m ≥1, come along for free). From (3.12) we ﬁndL†m|0⟩= 0, m ≤1, and thus from (3.11) that⟨0|Lm = 0 ,m ≤1 .

(3.13b)The states L−n|0⟩for n ≥2, on the other hand, are in principle non-trivialHilbert space states that transform as part of some representation of the Vira-soro algebra.The only generators in common between (3.13a, b), annihilating both ⟨0|and |0⟩, are L±1,0. It is easy to show, using the commutation relations (3.8a),that this is the only ﬁnite subalgebra of the Virasoro algebra for which thisis possible.

Identical results apply as well for the Ln’s, and we shall call thevacuum state |0⟩, annihilated by both L±1,0 and L±1,0, the SL(2, C) invariantvacuum. (Strictly speaking we could denote this as the tensor product |0⟩⊗|0⟩of two SL(2, R) invariant vacuums, but any ambiguity in the symbol |0⟩isordinarily resolved by context.

)The conditions (3.13) together with the commutation rules (3.8a) can beused to verify thatT (z) T (w)= ⟨0|Xn∈ZLn z−n−2 Xm∈ZLm w−m−2|0⟩=c/2(z −w)4 ,(3.14)35giving an easy way to calculate c in some theories. Similarly, we can computeall higher point correlation functions of the formT (w1) · · · T (wn) T(z1) · · · T(zm)=T (z1) · · · T (zn) T (z1) · · · T(zm),(3.15)by substituting the mode expansions (3.6) and commuting the Ln’s with npositive (negative) to the right (left).

We can also see the condition c > 0 toresult from the algebra (3.8a), and the relations (3.13a) and (3.12):c2 = ⟨0|L2, L−2|0⟩= ⟨0|L2 L†2|0⟩≥0 ,since the norm satisﬁes ∥L†2|0⟩∥2 ≥0 in a positive Hilbert space.3.5. Highest weight statesLet us now consider the state|h⟩= φ(0)|0⟩(3.16)created by a holomorphic ﬁeld φ(z) of weight h. From the operator productexpansion (2.10) between the stress-energy T and a primary ﬁeld φ we ﬁndLn, φ(w)=Idz2πi zn+1 T (z)φ(w) = h(n + 1)wnφ(w) + wn+1∂φ(w) , (3.17)so thatLn, φ(0)= 0, n > 0.

The state |h⟩thus satisﬁesL0|h⟩= h|h⟩Ln|h⟩= 0, n > 0 . (3.18a)More generally, an in-state |h, h⟩created by a primary ﬁeld φ(z, z) of conformalweight (h, h) will also satisfy (3.18a) with the replacements L →L, h →h.Since L0 ± L0 are the generators of dilatations and rotations, we identify h ± has the scaling dimension and Euclidean spin of the state.Any state satisfying (3.18a) is known as a highest weight state.

Statesof the form L−n1 · · · L−nk|h⟩(ni > 0) are known as descendant states. Theout-state ⟨h|, deﬁned as in (3.10), evidently satisﬁes⟨h|L0 = h⟨h|⟨h|Ln = 0, n < 0 .

(3.18b)36

The states ⟨h|Ln1 · · · Lnk (ni > 0) are the descendants of the out-state. Using(3.12), (3.18), and (3.8a), we evaluate⟨h|L†−n L−n|h⟩= ⟨h|Ln, L−n|h⟩= 2n⟨h|L0|h⟩+ c12(n3 −n)⟨h|h⟩=2nh + c12(n3 −n)⟨h|h⟩.

(3.19)Again, this quantity must be positive if the Hilbert space has a positive norm.For n large this tells us that we must have c > 0, and for n = 1 this requiresthat h ≥0. In the latter case we also see that h = 0 only if L−1|h⟩= 0, i.e.only if |h⟩is identically the SL(2, R) invariant vacuum |0⟩.We can also show for c = 0 that the Virasoro algebra has no interestingunitary representations.

From (3.19), we see that all states L−n|0⟩would havezero norm and hence should be set equal to zero. Moreover for arbitrary h if weconsider[13] the matrix of inner products in the 2×2 basis L−2n|h⟩, L2−n|h⟩, weﬁnd a determinant equal to 4n3h2(4h −5n).

For h ̸= 0 this quantity is alwaysnegative for large enough n. Thus for c = 0 the only unitary representation ofthe Virasoro algebra is completely trivial: it has h = 0 and all the Ln = 0.It follows from (3.17) that a ﬁeld φ with conformal weight (h, 0) is purelyholomorphic. We ﬁrst note from (3.17) adapted to the anti-holomorphic casethatL−1, φ= ∂φ, then argue as in (3.19) to show that the norm of the stateL−1φ|0⟩= 0, and hence that ∂φ = 0.To see what (3.16) means in termsof modes, we generalize the mode expansions (3.6) to arbitrary holomorphicprimary ﬁelds φ(z) of weight (h, 0),φ(z) =Xn∈Z−hφn z−n−h ,again chosen so that φ−n has scaling weight n. The modes satisfyφn =Idz2πi zh+n−1φ(z) .Regularity of φ(z)|0⟩at z = 0 requires φn|0⟩= 0 for n ≥−h + 1, generalizingthe case h = 2 in (3.13a).

From (3.16) we see that the state |h⟩is created by37the mode φ−h: |h⟩= φ−h|0⟩. To check that the states φn|0⟩have the correctL0 values, we use (3.17) to calculate the commutatorLn, φm=Idw2πiwh+m−1h(n + 1)wnφ(w) + wn+1∂φ(w)=Idw2πiwh+m+n−1h(n + 1) −(h + m + n)φ(w)=n(h −1) −mφm+n .

(3.20)SoL0, φm= −mφm, consistent for example with L0|h⟩= L0φ−h|0⟩= h|h⟩.Before turning to a detailed consideration of descendant ﬁelds, we showhow the formalism of this subsection may be used to derive the generalization of(2.3) to n-point functions. We ﬁrst use the SL(2, C) invariance of the vacuum,U|0⟩= |0⟩for U ∈SL(2, C), to derive (1.13) (or rather (2.20)) in the form⟨0|U−1φ1U · · · U−1φnU|0⟩= ⟨0|φ1 · · · φn|0⟩,(3.21)where the φi’s are quasi-primary ﬁelds (i.e.

satisfyU−1φ(z, z)U =∂f(z)h∂f(z)hφf(z), f(z),for f of the form (1.9)). Inﬁnitesimally, (3.21) takes the obvious form0 = ⟨0|Lk, φ1(z1).

. .

φn(zn)|0⟩+ · · · + ⟨0|φ1(z1) . .

.Lk, φn(z1)|0⟩,for k = 0, ±1. Using (3.17) we write this equivalently asnXi=1∂i⟨0|φ1(z1) .

. .

φn(zn)|0⟩= 0nXi=1(zi∂i + hi)⟨0|φ1(z1) . .

. φn(zn)|0⟩= 0nXi=1(z2i ∂i + 2zihi)⟨0|φ1(z1) .

. .

φn(zn)|0⟩= 0 ,(3.22)implying respectively invariance under translations, dilatations, and special con-formal transformations. We also point out that (3.21) applies as well to the38

correlation functions (3.15) even though T is not a primary ﬁeld. Recall thatthe Schwartzian derivative S(f, z) of (3.3) vanishes for the global transforma-tions (1.9), implying that T is quasi-primary, and that suﬃces to show that itscorrelation functions transform covariantly under SL(2, C).3.6.

Descendant ﬁeldsAs mentioned at the beginning of this section, representations of the Vira-soro algebra start with a single primary ﬁeld. Remaining ﬁelds in the represen-tation are given by successive operator products with the stress-energy tensor.Together all these ﬁelds comprise a representation [φn].

(In terms of modes, thedescendant ﬁelds are obtained by commuting L−n’s with primary ﬁelds.) Act-ing on the vacuum, the descendant ﬁelds create descendant states.

We shall seethat the conformal ward identities give diﬀerential equations that determinethe correlation functions of descendant ﬁelds in terms of those of primaries.The utility of organizing a two dimensional conformal ﬁeld theory in terms ofconformal families, i.e. irreducible representations of the Virasoro algebra, isthat the theory may then be completely speciﬁed by the Green functions of theprimary ﬁelds.We extract the descendant ﬁelds bL−nφ, n > 0, from the less singular partsof the operator product expansion of T (z) with a primary ﬁeld,T (z) φ(w, w) ≡Xn≥0(z −w)n−2 bL−nφ(w, w)=1(z −w)2 bL0φ +1z −wbL−1φ + bL−2φ + (z −w)bL−3φ + .

. .

. (3.23)The ﬁeldsbL−nφ(w, w) =Idz2πi1(z −w)n−1 T (z)φ(w, w)(3.24)are sometimes also denoted as φ(−n) (and in the presence of larger algebraicstructures are called Virasoro descendants to avoid ambiguity).

The conformalweight of the descendant ﬁeld bL−nφ is (h+n, h). Note from (2.10) that the ﬁrsttwo descendant ﬁelds are given by φ(0) = bL0φ = hφ and φ(−1) = bL−1φ = ∂φ.39A simple example of a descendant ﬁeld isbL−21(w) =Idz2πi1z −wT (z)1 = T (w) .Thus 1(−2)(w) =bL−21(w) = T (w), and we see that the stress-energy tensoris always a level 2 descendant of the identity operator.

This explains why theoperator product (3.1) of the stress-energy tensor with itself does not take thecanonical form (2.10) of that for a primary ﬁeld.For n > 0, primary ﬁelds satisfy bLnφ = 0. The ﬁrst few descendant ﬁelds,ordered according to their conformal weight, areleveldimension ﬁeld0hφ1h + 1bL−1φ2h + 2bL−2φ, bL2−1φ3h + 3bL−3φ, bL−1bL−2φ, bL3−1φ· · ·Nh + NP(N) ﬁelds ,(3.25)where the number at level N is given by P(N), the number of partitions of Ninto positive integer parts.

P(N) is given in terms of the generating function1Q∞n=1(1 −qn) =∞XN=0P(N) qN ,(3.26)where P(0) ≡1. The ﬁelds in (3.25) arise from repeated short distance ex-pansion of the primary ﬁeld φ with T (z), and constitute the conformal family[φ] based on φ.

Since bL−1ψ = ∂ψ for any ﬁeld ψ, [φ] naturally contains inparticular all derivatives of each of its ﬁelds.All the correlation functions of the secondary ﬁelds are given by diﬀerentialoperators acting on those of primary ﬁelds. For example if we let z →wn in(2.22), expand in powers of z −wn, and use the deﬁnition (3.23) of secondaryﬁelds, we ﬁndφ1(w1, w1) .

. .

φn−1(wn−1, wn−1)bL−kφ(z, z)= L−kφ1(w1, w1) . .

. φn−1(wn−1, wn−1) φ(z, z),(3.27a)40

where the diﬀerential operator (for k ≥2) is deﬁned byL−k = −n−1Xj=1 (1 −k)hj(wj −z)k +1(wj −z)k−1∂∂wj. (3.27b)The L’s provide a diﬀerential realization of (3.8a) with c = 0.

With z = z = 0,we see from (3.24) and (3.7) that bL−kφ(0) →L−kφ(0). Thus (3.27) can alsobe derived at z = 0 by using (3.17) to commute L−k to the left, and thenusing the highest weight property (3.13b) of the out vacuum.

(Although (2.22)was derived for |z| greater than all the |wi|’s, it is easy to show either bycontour integral methods or by substituting the mode expansion for T andcommuting L’s that it remains true for any ordering of the arguments). By thesame methods, the generalization of (3.27) to correlation functions involvingone arbitrary secondary ﬁeld is⟨0|φ1(w1, w1) .

. .

φn−1(wn−1, wn−1)bL−k1 . .

. bL−kℓφ(z, z)|0⟩= L−k1 .

. .

L−kℓ⟨0|φ1(w1, w1) . .

. φn−1(wn−1, wn−1)φ(z, z)|0⟩.

(3.28)In principle one can write down expressions for correlation functions of arbitrarysecondary ﬁelds in terms of those for primaries, but there is no convenientclosed form expression in the most general case. A particular case of interest isthe 2-point function.

If we take orthogonal primary ﬁelds as in (2.12), then itfollows directly from (2.22) that the 2-point functions of descendants of diﬀerentprimary ﬁelds must vanish.A problem related to calculating correlation functions of secondary ﬁeldsis to write the operator product coeﬃcients (2.13) for descendants in terms ofthose for primaries. Let us consider (2.13) with φi and φj primary ﬁelds, andgroup together all the secondary ﬁelds belonging to the conformal family [φp]in the summation to writeφi(z, z)φj(w, w) =Xp{kk}C{kk}ijpz(hp−hi−hj+Σℓkℓ) z(hp−hi−hj+Σℓkℓ) φ{k k}p(w, w) .

(3.29)41Here we have labeled the descendantsbL−k1 · · · bL−kn bL−k1 · · · bL−km φpof a primary ﬁeld φp by φ{kk}p, and we assume the normalization (2.12). Theoperator product coeﬃcients in this normalization are symmetric and from (2.5)coincide with the numerical factor in the 3-point function⟨φi|φj(z, z)|φp⟩=φi(∞)φj(z, z)φp(0)= Cijp zhi−hj−hp zhi−hj−hp ,where these ﬁelds are either primary or secondary.

Using (3.28) in the caseof the 3-point function for ﬁelds as in (3.29) (or by performing a conformaltransformation on both sides of (3.29) and comparing terms), one can show[1]thatC{kk}ijp= Cijp βp{k}ijβ p{k}ij,(3.30)where the Cijp’s are the operator product coeﬃcients for primary ﬁelds, andβp{k}ij( β p{k}ij) is a function of the four parameters hi, hj, hp, and c (hi,hj, hp,and c) determined entirely by conformal invariance (and can in principle becomputed mechanically). Moreover the 3-point function for any three descen-dant ﬁelds can be determined from that of their associated primaries (althoughas noted after (3.28), the explicit form of the relation is awkward to write downin all generality).

The primary Cijp’s thus determine the allowed non-vanishing3-point functions for any members of the families [φi], [φj], and [φp].We see that the complete information to specify a two dimensional confor-mal ﬁeld theory is provided by the conformal weights (hi, hi) of the Virasorohighest weight states, and the operator product coeﬃcients Cijk between theprimary ﬁelds that create them.Everything else follows from the values ofthese parameters, which themselves cannot be determined solely on the basisof conformal symmetry.3.7. Duality and the bootstrapTo determine the Cijk’s and h’s, we need to apply some dynamical principleto obtain additional information.Up to now, we have considered only the42

local constraints imposed by the inﬁnite conformal algebra. Associativity ofthe operator algebra (2.13), on the other hand, imposes global constraints oncorrelation functions.

To see how this works, we consider evaluating the 4-pointfunctionφi(z1, z1)φj(z2, z2)φℓ(z3z3)φm(z4, z4)(3.31)in two ways. First we take z1 →z2, z3 →z4, and ﬁnd the schematic resultdepicted in the left hand side of ﬁg.

4, where the sum over p is over both primaryand secondary ﬁelds. (3.31) can alternatively be evaluated by taking z1 →z3,z2 →z4, and we have represented this result diagrammatically in the righthand side of ﬁg.

4. Associativity of the operator algebra implies that these twomethods of calculating the 4-point function should give the same result.

Theirequality is a necessary consistency requirement, known as crossing symmetryof the 4-point function.XpCijp Cℓmpijpℓm=XqCiℓq CjmqijqℓmFig. 4.

Crossing symmetryIn ﬁg. 4, we thus have an inﬁnite number of equations that the Cijk’s mustsatisfy.

The sum over all the descendant states can be performed in principle,and the relations in ﬁg. 4 become algebraic equations for the Cijk’s.

These verystrong constraints were originally suggested to give a means of characterizingall conformally invariant systems in d dimensions (the procedure of solving therelations of ﬁg. 4 to ﬁnd conformal ﬁeld theories is known as ‘the conformalbootstrap’).

This program however proved too diﬃcult to implement in prac-tice. In two dimensions the problem becomes somewhat more tractable, since43one need only consider the primary ﬁelds, vastly reducing the number of inde-pendent quantities in the problem.

There remains however the possibility ofencountering an unmanageable number of primary ﬁelds, and as well one muststill evaluate the objects represented diagrammatically in ﬁg. 4.

In [1], it wasshown that there are certain special c, h values where things simplify dramati-cally (such values were also noted in [14]), as we shall discuss momentarily.First we need to convert ﬁg. 4 to an analytic expression.

We can write thecontribution to the 4-point function from ‘intermediate states’ belonging only tothe conformal family [φp] as Fℓmij (p|x) Fℓmij (p|x). This amplitude is representedin ﬁg.

5, and we are for simplicity taking z1, z2, z3, z4 = 0, x, 1, ∞in the 4-pointfunction (3.31). The amplitude projected onto a single conformal family takes afactorized form because the sums over descendants in the holomorphic and anti-holomorphic families [φp] and [φp] (generated by T and T) are independent.

TheFℓmij (p|x) depend on the parameters hi, hj, hℓ, hm, hp, and c, and are knownas “conformal blocks” since any correlation function can be built from them.Fℓmij (p|x) F ℓmij (p|x)=0xijpℓm1∞Fig. 5.

Single channel amplitudeIn terms of the conformal blocks, we can write an analytic form of thediagrammatic equations ﬁg. 4 asXpCijp Cℓmp Fℓmij (p|x) Fℓmij (p|x)=XqCiℓq Cjmq Fjmiℓ(q|1 −x) Fjmiℓ(q|1 −x) .

(3.32)44

If we know the conformal blocks F, then (3.32) yields a system of equationsthat determine the Cijk’s and h, h’s. This has not been carried out in generalbut at the special values of c, h mentioned earlier, the F’s can be determinedas solutions of linear diﬀerential equations (that result from the presence ofso-called null states).In section 5, we shall see some examples of how thisworks.The particular values of c for which things simplify, as mentioned above,take the formc = 1 −6(m′ −m)2mm′,where m and m′ are two coprime positive integers.

In [1], these models werecalled ‘minimal models’, and it was shown that they possessed a closed operatoralgebra with only a ﬁnite number of primary ﬁelds.For these models thebootstrap equation (3.32) can be solved completely, and everything about theseconformal ﬁeld theories can be determined in principle.These models thusrealize an old hope[15] that the most singular part of the operator productexpansion should deﬁne a closed, ﬁnite-dimensional algebra of primary ﬁelds ina theory. We shall see in the next section that imposing as well the criterionof unitary selects an even smaller subset of these models (with m′ = m + 1),known as the unitary discrete series.

In section 9, we shall see how the fusionrules for their closed operator algebras can be calculated.The relation represented in ﬁg. 4 is also known as ‘duality of the 4-pointfunction’ (not to be confused with various other forms of duality that appearin these notes).

This notion of duality generalizes to the n-point correlationfunctionsφ1(z1, z1) . .

. φn(zn, zn)of sensible conformal ﬁeld theories on arbitrary genus Riemann surfaces.

Therequirement of duality states that any such correlation function should 1) be asingle-valued real analytic function of the zi’s and the moduli of the Riemannsurface, and 2) be independent of the basis of conformal blocks used to computeit. Requirement 2) generalizes (3.32) and insures that the correlation functionis not sensitive to the particular decomposition of the Riemann surface into45thrice-punctured spheres (and also that it be independent of the order of theφi’s).

Pictorially this generalizes ﬁg. 4 to n-point functions, and is discussedfurther in the contribution of Dijkgraaf to these proceedings.4.

Kac determinant and unitarity4.1. The Hilbert space of statesWe now return to consider more carefully the Hilbert space of states ofa conformal ﬁeld theory.

For the time being it will be suﬃcient to consideronly the holomorphic half of the theory. We recall that a highest weight state|h⟩= φ(0)|0⟩, satisfying L0|h⟩= h|h⟩, is created by acting with a primary ﬁeldφ of conformal weight h on the SL(2, R) invariant vacuum |0⟩, which satisﬁesLn|0⟩= 0, n ≥−1.

We have seen from (3.19) that a positive Hilbert spacerequires h ≥0. Descendant states are created by acting on |h⟩with a string ofL−ni’s, ni > 0.

These states can also be regarded to result from the action ofa descendant ﬁeld acting on the vacuum, e.g.L−n|h⟩= L−nφ(0)|0⟩=bL−nφ(0)|0⟩= φ(−n)(0)|0⟩.We wish to verify that every sensible representation of the Virasoro algebrais characterized by such a highest weight state. Generally we are interested inscaling operators, i.e.

operators of ﬁxed conformal weight, whose associatedstates diagonalize the action of L0. Thus we focus on eigenstates |ψ⟩of L0,say with L0|ψ⟩= hψ|ψ⟩.

Then since [L0, Ln] = −nLn, we have L0Ln|ψ⟩=(hψ −n)Ln|ψ⟩and Ln lowers the eigenvalue of L0 for n > 0. But dilatation inz on the plane, generated by L0 + L0, corresponds to translation in σ0 on thecylinder, generated by the energy H. L0 + L0 should thus be bounded belowin any sensible quantum ﬁeld theory.

Since L0 and L0 reside in independentholomorphic and anti-holomorphic algebras, they must be separately boundedfrom below. By acting with Ln’s, we must therefore ultimately reach a stateannihilated by Ln, n > 0 (and similarly by Ln).

This state is the highest weight,or primary, state, that we have been calling |h⟩. We see that we can regard the46

Ln’s, n > 0, as an inﬁnite number of harmonic oscillator annihilation operatorsand the L†n = L−n’s as creation operators. The representation theory of theVirasoro algebra thus resembles that of SU(2), with L0 playing the role of J3and the L±n’s playing the roles of an inﬁnite number of J∓’s.We also wish to show that every state in a positive Hilbert space can beexpressed as a linear combination of primary and descendant states.

Supposenot, i.e. suppose that there exists a state |λ⟩that is not a descendant of a highestweight state.

Then in a positive metric theory, we can decompose |λ⟩= |δ⟩+|ψ⟩,where |ψ⟩is orthogonal to all descendants |δ⟩. If |ψ⟩has L0 eigenvalue hψ, letK = [hψ] (the greatest integer part).

Now consider some order K combinationof the Lni’s (such that P ni = K for any term), symbolically denoted LK. Then|h⟩= LK|ψ⟩is a highest weight state with h = hψ−K (it must be annihilated byall the Ln’s, n > 0, since otherwise they would create a state with h < 0).

Butwe also have ⟨h|h⟩= ⟨ψ|L†K|h⟩= 0, since ⟨ψ| is orthogonal to all descendants.It follows that |h⟩= 0. We next consider the state L(K−1)|ψ⟩= |h + 1⟩, whereL(K−1) is order (K −1) in the Ln’s.

The same argument as above shows that|h + 1⟩too must be highest weight but have zero norm, and consequently mustvanish. By induction we ﬁnd that |ψ⟩itself is a highest weight state, concludingthe argument.With this characterization of the Hilbert space of states in hand, we turnto a more detailed consideration of the state representations of the Virasoroalgebra.

(Via the correspondence between states and ﬁelds, we could equallyproceed in terms of the ﬁelds (3.25), but framing the discussion in terms ofstates turns out to be slightly more convenient for our purposes.) Starting froma highest weight state |h⟩, we build the set of statesleveldimension state0h|h⟩1h + 1L−1|h⟩2h + 2L−2|h⟩, L2−1|h⟩3h + 3L−3|h⟩, L−1L−2|h⟩, L3−1|h⟩· · ·Nh + NP(N) states ,(4.1)47known as a Verma module.

We are not guaranteed however that all the abovestates are linearly independent. That depends on the structure of the Virasoroalgebra (3.8a) for given values of h and c.A linear combination of statesthat vanishes is known as a null state, and the representation of the Virasoroalgebra with highest weight |h⟩is constructed from the above Verma moduleby removing all null states (and their descendants).

(It is useful at this point to contrast the situation in two dimensions withthat of higher dimensions, where the conformal algebra is ﬁnite dimensional.The ﬁnite dimensional analog in two dimensions is the closed SL(2, C) subalge-bra generated by L0,±1, L0,±1. Its irreducible representations are much smallerthan those of the full inﬁnite dimensional Virasoro algebra.

In general an ir-reducible representation of the Virasoro algebra contains an inﬁnite number ofSL(2, C) representations, whose behavior is thereby tied together. It is this ad-ditional structure that enables a more extensive analysis of conformal theoriesin two dimensions.

)Let us now consider the consequences of a linear combination of states thatvanishes. At level 1, the only possibility is thatL−1|h⟩= 0 ,but this just implies that h = 0, i.e.

|h⟩= |0⟩. At level 2, on the other hand, itmay happen thatL−2|h⟩+ aL2−1|h⟩= 0for some value of a.

so that the central charge must satisfy c = 2(−6ah−4h) = 2h(5−8h)/(2h+1).We conclude that a highest weight state |h⟩of the Virasoro algebra at this valueof c satisﬁesL−2 −32(2h + 1)L2−1|h⟩= 0 . (4.2)Such a state |h⟩, with a null descendant at level 2, is also called degenerate atlevel 2.For a degenerate primary ﬁeld, the analogous statement isbL−2 −32(2h + 1)bL2−1φ = 0 .By (3.27), correlation functions of such a ﬁeld are annihilated by the diﬀerentialoperator L−2 −32(2h+1)L2−1.

To express this diﬀerential equation in a form thatwill prove useful later, we write bL−2φ = −abL2−1φ = −a ∂2∂z2 φ for a ﬁeld φdegenerate at level 2. From the deﬁnition (3.23), as z →w,bL−2φ(w, w) = T (z)φ(w, w) −hφ(w, w)(z −w)2 −∂φ(w, w)z −w−.

. .

,together with (2.22) in the limit z →w1, we derive−a ∂2∂w21φ1(w1, w1) . .

. φn(wn, wn)=DT (z)φ1(w1, w1) −hφ1(w1, w1)(z −w1)2−∂φ1(w1, w1)z −w1· φ2(w2, w2) .

. .

φn(wn, wn)Ez→w1=Xj̸=1hj(w1 −wj)2 +1w1 −wj∂∂wj φ1(w1, w1) . .

. φn(wn, wn).

(4.3)This is a second order diﬀerential equation for any n-point function involving aprimary ﬁeld φ1 with a null descendant at level 2. In the case of 4-point func-tions, the solutions to (4.3) are expressible in terms of standard hypergeometricfunctions.

In section 5, we shall show how monodromy conditions can be usedto select particular solutions that are physically relevant.494.2. Kac determinantAt any given level, the quantity to calculate to determine more generallywhether there are any non-trivial linear relations among the states is the matrixof inner products at that level.

A zero eigenvector of this matrix gives a linearcombination with zero norm, which must vanish in a positive deﬁnite Hilbertspace. At level 2, for example, we work in the 2×2 basis L−2|h⟩, L2−1|h⟩, andcalculate⟨h|L2 L−2|h⟩⟨h|L21 L−2|h⟩⟨h|L2 L2−1|h⟩⟨h|L21 L2−1|h⟩=4h + c/26h6h4h(1 + 2h).

(4.4a)We can write the determinant of this matrix as2(16h3 −10h2 + 2h2c + hc) = 32h −h1,1(c)h −h1,2(c)h −h2,1(c), (4.4b)where h1,1(c) = 0 and h1,2, h2,1 =116(5 −c) ∓116p(1 −c)(25 −c). The h = 0root is actually due to the null state at level 1, L−1|0⟩= 0, which impliesalso the vanishing L−1L−1|0⟩= 0.

This is a general feature: if a null state|h + n⟩= 0 occurs at level n, then at level N there are P(N −n) null statesL−n1 · · · L−nk|h + n⟩= 0 (with Pi ni = N −n). Thus a null state for somevalue of h that ﬁrst appears at level n implies that the determinant at level Nwill have aP(N −n)th order zero for that value of h (and the ﬁrst term inthe product (4.4b) can be reexpressed ash −h1,1(c)P (1) to reﬂect its origin).At level N, the Hilbert space consists of all states of the formX{ni}an1···nk L−n1 · · · L−nk|h⟩,where Pi ni = N. We can pick P(N) basis states as in (4.1), and the levelN analog of (4.4a, b) is to take the determinant of the P(N)×P(N) matrixMN(c, h) of inner products of the form⟨h|Lmℓ· · · Lm1 L−n1 · · · L−nk|h⟩(where Pℓi=1 mi = Pkj=1 nj = N).

If det MN(c, h) vanishes, then there existsa linear combination of states with zero norm for that c, h. If negative, then50

the determinant has an odd number of negative eigenvalues (i.e. at least one).The representation of the Virasoro algebra at those values of c and h includesstates of negative norm, and is consequently not unitary.The formula generalizing (4.4b),det MN(c, h) = αNYpq≤N(h −hp,q(c))P (N−pq) ,(4.5a)is due to Kac and was proven in [16].

The product in (4.5a) is over all positiveintegers p, q whose product is less than or equal to N, and αN is a constant inde-pendent of c and h. The hp,q(c)’s are most easily expressed by reparametrizingc in terms of the (in general complex) quantitym = −12 ± 12r25 −c1 −c .Then the hp,q’s of (4.5) are given byhp,q(m) =(m + 1)p −mq2 −14m(m + 1). (4.5b)(For c < 1 we conventionally choose the branch m ∈(0, ∞) — in any eventthe determinant (4.5a) is independent of the choice of branch since it can becompensated by the interchange p ↔q in (4.5b).) We easily verify that (4.5)reduces to (4.4b) for N = 2.

We also note that c is given in terms of m byc = 1 −6/m(m + 1). Finally we point out that the values of the hp,q’s in (4.5b)possess the symmetry p →m −p, q →m + 1 −q.Although (4.5) can be proven by relatively straightforward methods, weshall not undertake to reproduce a complete proof since only the result itself willbe needed in what follows.

Here we brieﬂy indicate how the proof goes[16][17].To begin with one writes down an explicit set of states parametrized by in-tegers p, q, shows that they are null, and calculates their eigenvalue h. Sincedet MN(c, h) is a polynomial in h, it can be determined up to a constant by itszeros in h and their multiplicities. Making use of the observation after (4.4b)that a zero of det Mn leads to a multiplicity P(N −n) zero of det MN, the51explicit enumeration of states shows that det MN has at least all the zeros ap-pearing on the right hand side of (4.5a).

To show that this is indeed the fullpolynomial, i.e. that there are no other zeroes, it suﬃces to show that the orderof the r.h.s.

of (4.5a) coincides with the order νN of det MN(c, h) as a poly-nomial in h. This latter order can be determined by noting that the highestpower of h in det MN(c, h) comes from the product of the diagonal elementsof the matrix MN(c, h) (these elements result in the maximum number of L0’sgenerated by commuting Lk’s through an identical set of L−k’s). The diagonalelement for a state L−n1 · · · L−nk|h⟩gives a contribution proportional to hk.The order of det MN(c, h) is thus given byνN =X{n1+...+nk=N}k =Xpq≤NP(N −pq) ,where the summation on the left is over all {ni > 0} with Pki=1 ni = N, andthe right hand side follows from a standard number theoretic identity.

We seethat the order of the polynomial on the right hand side of (4.5a) coincides withthat of det MN(h, c), showing that the states explicitly exhibited in [16],[17]exhaust all the zeros and hence determine the determinant up to a constant.4.3. Sketch of non-unitarity proofNow we are ready to investigate the values of c and h for which the Vi-rasoro algebra has unitary representations[18].

In ﬁeld theory, unitarity is thestatement of conservation of probability and is fundamental. In statistical me-chanical systems, it does not necessarily play as central a role.

There unitarityis expressed as the property of reﬂection positivity, and consequently the exis-tence of a hermitian transfer matrix. Statistical mechanical systems that canbe described near a second order phase transition by an eﬀective ﬁeld theoryof a local order parameter, however, are always expected to be described by aunitary theory.

Higher derivative interactions which might spoil unitarity of aLagrangian theory are generically irrelevant operators, and do not survive tothe long distance eﬀective theory. For the remainder here, we will thus restrictattention to unitary theories.

(That is not to say, however, that unitary theories52

necessarily exhaust all cases of interest. The Q →0 limit of the Q-state Pottsmodel, for example, useful in studying percolation, is not described by a localorder parameter and is not a unitary theory.

The Yang-Lee edge singularityalso appears in a non-unitary theory, in this case due to the presence of animaginary ﬁeld. )The analysis of unitary representations of the Virasoro algebra proceedsfrom a study of the Kac determinant (4.5).As mentioned in the previoussubsection, if the determinant is negative at any given level it means that thereare negative norm states at that level and the representation is not unitary.If the determinant is greater than or equal to zero, further investigation candetermine whether or not the representation at that level is unitary.In the region c > 1, h ≥0, it is easy to see that there are no zeroes ofthe Kac determinant (4.5) at any level.

For 1 < c < 25, m is not real, and thehp,q’s of (4.5b) either have an imaginary part or (for p = q) are negative. Forc ≥25 we can choose the branch −1 < m < 0 and ﬁnd that all the hp,q’s arenegative.

Now we can show that the non-vanishing of det MN in this regionimplies that all the eigenvalues of MN are positive. This is because for h large,the matrix becomes dominated by its diagonal elements (as shown at the endof the previous subsection, these are highest order in h).

Since these matrixelements are all positive, the matrix has all positive eigenvalues for large h. Butsince the determinant never vanishes for c > 1, h ≥0, all of the eigenvaluesmust stay positive in the entire region.On the boundary c = 1, the determinant vanishes at the points h = n2/4but does not become negative. Thus the Kac determinant (4.5) poses no ob-stacle in principle to having unitary representations of the Virasoro algebra forany c ≥1, h ≥0.Only the region 0 < c < 1, h > 0 is delicate to treat, although all steps inthe argument are elementary.

First we draw the vanishing curves h = hp,q(c)in the h, c plane (see ﬁg. 6), by reexpressing (4.5b) in the formhp,q(c) = 1 −c96 (p + q) ± (p −q)r25 −c1 −c!2−4.

(4.5b′)53(In this form it is evident that the convention for which branch in m is chosen iscompensated by the interchange p ↔q). The behavior near c = 1 is determinedby taking c = 1 −6ǫ which gives, to leading order in ǫ,hp,qc = 1 −6ǫ= 14(p −q)2 + 14(p2 −q2)√ǫ(p ̸= q)hp,pc = 1 −6ǫ= 14(p2 −1)ǫ .By analyzing the curves (4.5b′), it is easy to show that one may connect anypoint in the region 0 < c < 1, h > 0 to the c > 1 region by a path that crossesa single vanishing curve of the Kac determinant at some level.

The vanishing isdue to a single eigenvalue crossing through zero, so the determinant reverses signpassing through the vanishing curve and there must be a negative norm state atthat level. This excludes unitary representations of the Virasoro algebra at allpoints in this region, except those on the vanishing curves themselves where thedeterminant vanishes.

A more careful analysis[18] of the determinant shows thatthere is an additional negative norm state everywhere on the vanishing curvesexcept at certain points where they intersect, as indicated in ﬁg. 6.This discrete set of points, where unitary representations of the Virasoroalgebra are not excluded, occur at values of the central chargec = 1 −6m(m + 1)m = 3, 4, .

. .

(4.6a)(m = 2 is the trivial theory c = 0). To each such value of c there are m(m−1)/2allowed values of h given byhp,q(m) =(m + 1)p −mq2 −14m(m + 1)(4.6b)where p, q are integers satisfying 1 ≤p ≤m −1, 1 ≤q ≤p.Thus we see that the necessary conditions for unitary highest weight repre-sentations of the Virasoro algebra are (c ≥1, h ≥0) or (4.6a, b).

That the latterof these two conditions is also suﬃcient, i.e. that there indeed exist unitary rep-resentations of the Virasoro algebras for these discrete values of c, h, was shownin [19] via a coset space construction (to be discussed in section 9).

The overall54

ch111_21_27_104_51_41_16h 3,1h 2,2h 3,3h 1,2h 2,3h 2,1h 3,2h 1,3h 1,1h 2,4h 4,20Fig. 6.

First few vanishing curves h = hp,q(c) in the h, c plane.status of conformal ﬁeld theories with c ≥1 is not as yet well understood, andmuch eﬀort is currently being expended to develop more powerful techniques55to investigate them.4.4. Critical statistical mechanical modelsWe pause here to emphasize the import of (4.6a, b).

The representationtheory of the Virasoro algebra in principle allows us to describe the possiblescaling dimensions of ﬁelds of two dimensional conformal ﬁeld theories, andthereby the possible critical indices of two dimensional systems at their secondorder phase transitions. In the case of unitary systems with c ≤1, this hasturned out to give a complete classiﬁcation of possible two dimensional criticalbehavior.

We shall later see how to identify the particular representations ofthe Virasoro algebra which occur in the description of a given two dimensionalsystem at its critical point. (In Cardy’s lectures (section 3.2), we have alreadyseen how to calculate the central charge of the Q-state Potts model.

)While the c < 1 discrete series distinguishes a set of representations of theVirasoro algebra, it is not obvious that these should be realized by readily con-structed statistical mechanical model at their critical points. The ﬁrst few mem-bers of the series (4.6a) with m = 3, 4, 5, 6, i.e.

central charge c = 12, 710, 45, 67,were associated in [18] respectively with the critical points of the Ising model,tricritical Ising model, 3-state Potts model, and tricritical 3-state Potts model,by comparing the allowed conformal weights (4.6b) with known scaling dimen-sions of operators in these models. The ﬁrst of these, m = 3, we will treatin great detail in the next section.

In general, there may exist more than onemodel at a given discrete value of c < 1, corresponding to diﬀerent consistentsubsets of the full unitarity-allowed operator content (4.6b).By coincidence, at roughly the same time as the unitarity analysis, theauthors of [20] had constructed a new series of exactly solvable models of RSOS(restricted solid-on-solid) type. The critical points of these models models werequickly identiﬁed[21] to provide particular realizations of all members of thediscrete series (4.6a).

The RSOS models of [20] are deﬁned in terms of heightvariables ℓi that live at the sites of a square lattice. The heights are subject tothe restriction ℓi = 1, .

. .

, m, and nearest neighbor heights are also constrainedto satisfy ℓi = ℓj ±1. m is here an integer parameter that characterizes diﬀerent56

models. The Boltzmann weights for the models are given in terms of four-heightinteractions around each plaquette of the lattice (known as ‘IRF’ interactionsfor ‘interactions round a face’).

These weights are deﬁned so that each modelhas a second order phase transition at a self-dual point. The continuum limittheory of the RSOS model with heights restricted to take values from 1 tom turns out to give a realization of the Virasoro algebra with central chargec = 1 −6/m(m + 1).

(The nearest neighbor constraint in the case m = 3, forexample, causes the lattice to decompose to an even sublattice on which ℓi = 2for all sites, and an odd sublattice on which ℓi = 1, 3. The even sublatticedecouples, and the remaining 2-state model on the odd sublattice is the Isingmodel.) Other models of RSOS type were later constructed[22] and have criticalpoints also described by unitary representations of the Virasoro algebra withc < 1, but have a diﬀerent operator content than the models of [20].Forexample, the model of [20] with m = 5 (c = 4/5) is in the universality classof the tetracritical Ising model, whereas a model of [22] with the same valueof c is in the universality class of the 3-state Potts model (these two may beassociated respectively to the Dynkin diagrams of A5 and D4).

We shall returnto say a bit more about these models in section 9.4.5. Conformal grids and null descendantsTo prepare for our discussion of the operator content in later sections, weneed a convenient way of organizing the allowed highest weights hp,q of (4.6b).As noted, the hp,q are invariant under p →m −p, q →m + 1 −q.

Thus ifwe extend the range of q to 1 ≤q ≤m, we will have a total of m(m −1)values of hp,q with each appearing exactly twice. It is frequently convenient toarrange this extended range in an (m −1) × m “conformal grid” with columnslabeled by p and rows by q.

For the cases m = 3 (Ising model, c = 1/2), m = 4(tricritical Ising model, c = 7/10), and m = 5 (3-state Potts model, c = 4/5),we ﬁnd the conformal weights tabulated in ﬁg. 7.

Note that the symmetry in pand q mentioned above means that the diagram is left invariant by a rotationby π about its center. The singly-counted set of operators with q ≤p are thosebelow the q = p diagonal in ﬁg.

7.Another way of eliminating the doublecounting is to restrict to operators with p + q even — this selects operators ina checkerboard pattern starting from the identity operator at lower left.57↑qp →120116116012327160353801101103803507163237525013182140140182311511523181402140138025753Fig. 7.

Conformal grids for the cases m = 3, 4, 5 (c = 12, 710, 45).In general we have seen from the Kac determinant formula that the primarystate with L0 eigenvalue hp,q has a null descendant at level pq. For the threeallowed values h1,1 = 0, h2,1 = 12, and h1,2 =116 at m = 3, the associated nullstates at levels one and two were determined to beL−1|0⟩= 0(4.7a)and (from (4.2))L−2 −32(2h2,1 + 1)L2−1 12=L−2 −34L2−1 12= 0L−2 −32(2h1,2 + 1)L2−1 116=L−2 −43L2−1 116= 0 .

(4.7b)For higher values of m, null states begin to occur at higher levels pq. For m = 4,for example, the state |h3,1⟩= 32has a null descendant at level three, and isthus annihilated by a linear combination of L−3, L−2L−1, and L3−1, as easilydetermined by applying the commutation rules of the Virasoro generators withc = 7/10.5.

Identiﬁcation of m = 3 with the critical Ising modelThe unitary representation theory of the Virasoro algebra plays the samerole in studying two dimensional critical phenomena as representation theoryof ﬁnite and Lie groups plays in other branches of physics. Once the relevant58

symmetry group of a physical system has been identiﬁed, the analysis of itsspectrum and interactions is frequently reduced to a straightforward exercise ingroup representation theory and branching rules. For a given critical statisticalmechanical model, the 2-point correlation functions allow an identiﬁcation ofthe scaling weights of the operators in the theory and in many cases that issuﬃcient to identify the relevant representation of the Virasoro algebra.

Wehave already mentioned that the discrete unitary series with c < 1, for example,provides a set of possibilities for 2d critical behavior that can be matched upwith that of known statistical mechanical systems.We shall now make explicit the identiﬁcation of the ﬁrst member of thediscrete unitary series, i.e. the case m = 3 with c = 1/2, with the Ising modelat its critical point.

Up to now we have concentrated on the analytic dependenceT (z) of the stress-energy tensor. The physical systems we shall consider herealso have a non-trivial T(z) with central charge c = c. The primary ﬁelds in ourtheory are thus described by the two scaling weights h and h (the eigenvalues ofthe associated highest weight state under L0 and L0).

The simplest possibilityis to consider the left-right symmetric ﬁelds Φp,q(z, z) = φp,q(z) φp,q(z) withconformal weights (h, h)Φ1,1 : (0, 0)Φ2,1 : ( 12, 12)Φ1,2 : ( 116, 116)(5.1)(we shall later infer that this is the only possibility allowed by modular invari-ance for the theory on a torus).5.1. Critical exponentsThe (0, 0) ﬁeld above is present in every theory and is identiﬁed as theidentity operator.

To compare the remaining ﬁelds in (5.1) with those presentin the conventional description of the Ising model on a lattice, we need to makea brief digression into some of the standard lore of critical phenomena. (Fora review of the material needed here, see [23].

)Suppose we have a systemwith an order parameter σ (such as the spin (σ = ±1) in the Ising model.Suppose further that the system has a 2nd order transition separating a high59temperature (disordered) phase with ⟨σ⟩= 0 from a low temperature (ordered)phase with ⟨σ⟩̸= 0. In the high temperature phase the 2-point function of theorder parameter will fall oﬀexponentially ⟨σn σ0⟩∼exp(−|n|/ξ), where thecorrelation length ξ depends on the temperature (we see ξ−1 can be regardedas a mass for the theory).

At the critical point the correlation length diverges(theory becomes massless) and the 2-point function instead falls oﬀas a powerlaw⟨σn σ0⟩∼1|n|d−2+η ,where d is the dimension of the system and this expression deﬁnes the criti-cal exponent η. Another exponent, ν, can be deﬁned in terms of the 4-pointfunction at criticality⟨εn ε0⟩∼⟨σnσn+1σ0σ1⟩∼1|n|2(d−1/ν)(5.2)(more precisely εn should be deﬁned by averaging over all nearest neighbor sitesto n, but for our purposes here any one nearest neighbor, which we denote n+1,suﬃces).The critical exponents calculated for the two dimensional Ising model areη = 1/4, ν = 1.

Therefore the 2-point function behaves as⟨σnσ0⟩∼1|n|1/4 ∼1r2∆σ ,where the r dependence is appropriate for the 2-point function of a conformalﬁeld of scaling dimension ∆σ = hσ + hσ and spin sσ = hσ −hσ = 0. We seethat ∆σ = 2hσ = 2hσ = 1/8 and hence the ( 116, 116) ﬁeld in (5.1) should beidentiﬁed with the spin σ of the Ising model.

The energy operator, on the otherhand, satisﬁes⟨εnε0⟩∼1|n|2∆ε .Its scaling weight, then, can be identiﬁed from (5.2) with ν = 1 as d −1/ν =1 = ∆ε = hε + hε. Thus the ( 12, 12) ﬁeld in (5.1) should be identiﬁed with theenergy operator of the Ising model.

This completes the identiﬁcation of the60

primary ﬁelds in the Ising model, which turns out to have a total of only threeconformal families. (Although we have chosen to introduce the exponents η and ν in terms ofcritical correlation functions, we mention that many exponents are also deﬁnedin terms of oﬀ-critical correlation functions.

Diﬀerent deﬁnitions of the sameexponent are related by the scaling hypothesis. The critical exponent ν, forexample, is deﬁned alternatively in terms of the divergence of the correlationlength close to criticality,ξ ∼t−ν ,where t = (T −Tc)/Tc parametrizes the deviation of temperature from thecritical temperature Tc.

Another common critical exponent is deﬁned similarlyin terms of the divergence of the speciﬁc heat,C ∼t−α ,near the critical point.Now according to the scaling hypothesis, the divergence of all thermody-namic quantities at the critical point is due to their dependence on the correla-tion length ξ. Dimensional analysis thus allows us to ﬁnd relations between crit-ical exponents. For example the free energy density has dimension (length)−din d-dimensions so we ﬁndf ∼ξ−d ∼tνd .The speciﬁc heat, on the other hand, is given byC ∼∂2f∂t2 ∼tνd−2 ,so the scaling hypothesis implies the relation α = 2 −νd.

Finally the energydensity itself satisﬁesε ∼∂f∂t ∼tνd−1 ∼ξ−(νd−1)/ν ,(5.3)and comparing with (5.2) we see that the scaling hypothesis implies coincidenceof the two deﬁnitions of ν.61To make the relationship more precise, we consider the continuum limit ofthe correlation functionε(r)ε(0)= 1rp g(r/ξ)close to criticality. Then the speciﬁc heat satisﬁesC ∼∂2f∂t2 ∼Zddrε(r)ε(0)∼ξd−p ∼t−ν(d−p) ∼t−α ,so that p = d−α/ν = 2(d−1/ν).

At the critical point, ξ →∞, andε(r) ε(0)=g(0)/rp = g(0)/r2(d−1/ν), in accord with the deﬁnition (5.2).We note from (5.3) that in two dimensions the scaling weight of a spinlessenergy operator is hε = hε = (1 −α)/(2 −α). For other magnetization typeoperators, one can deﬁne exponents β by m ∼tβ, and proceeding as above weﬁndm ∼tβ ∼ξ−β/ν ∼ξ−dβ/(2−α) .For spinless magnetization type operators in two dimensions, we thus havehm = hm = β/(2 −α).

The reader might beneﬁt from repeating the argumentof the preceding paragraph to see how the exponent β may be alternativelydeﬁned via a 2-point function at the critical point. )In (3.5), we introduced another c = c = 12 system consisting of free fermionsψ(z) and ψ(z).

In [24], it is shown that the Ising model can generally be writtenas a theory of a free lattice fermion. At the critical point the fermion becomesmassless and renormalizes onto a massless continuum fermion.

The free fermionsystem (3.5) thus turns out to be equivalent to the critical Ising model ﬁeldtheory. From the standpoint of the free fermion description of the Ising criticalpoint, we see that the energy operator corresponds to the ( 12, 12) ﬁeld ψ(z)ψ(z).Moving away from criticality by adding a perturbation proportional to theenergy operator thus corresponds to adding a mass term δm ψ(z)ψ(z).

Theemergence of the ( 116, 116) ﬁeld σ in the fermionic language, on the other hand,is not as immediately obvious. In section 6 we shall see why a ﬁeld of thatweight should naturally occur.

In section 7 we shall further exploit the freefermion representation of the Ising model to investigate its spectrum.62

As described in Cardy’s lectures, the Ising model also possesses a disorderoperator µ, dual to the spin σ. Since the critical point occurs at the self-dualpoint of the model, at the critical point the ﬁeld µ(z, z) will have the sameconformal weights and operator algebra as the spin ﬁeld σ(z, z).

Thus the fulloperator content of the Ising model includes two ( 116, 116) ﬁelds, although thetwo are not mutually local (and neither is local with respect to the fermions ψ,ψ ). Both σ and µ are each individually local, on the other hand, with respectto the energy operator ε.5.2.

Critical correlation functions of the Ising modelSince, as noted after (3.30), the non-vanishing operator products for anymembers of conformal families are determined by those of the primaries, it ispossible to write “fusion rules” [φi][φj] = Pk[φk] for conformal families. Theydetermine which conformal families [φk] may have their members occurring inthe operator product between any members of conformal families [φi] and [φj].In the case of the Ising model, we write the three conformal families associatedto the primary ﬁelds of (5.1) as 1, [ǫ], and [σ].

The fusion rules allowed by thespin reversal (σ →−σ) and duality (ε →−ε) symmetries of the critical Isingmodel are[σ][σ] = 1 + [ε][σ][ε] = [σ][ε][ε] = 1 . (5.4)We shall shortly conﬁrm that 4-point correlation functions in the critical Isingmodel are consistent with the non-vanishing operator products represented by(5.4).In the conformal ﬁeld theory description of the critical point, both the en-ergy and spin (order/disorder) operators of (5.1) have null descendants at level2.

That means that any correlation function of these operators will satisfy asecond order diﬀerential equation. Speciﬁcally from (4.7b) we see that corre-lation functions involving either µ or σ will be annihilated by the diﬀerential63operator (L−2 −43L2−1).

From (4.3), we ﬁnd furthermore that any correlationfunction of σ’s and µ’s,G(2M,2N) =σ(z1, z1) · · · σ(z2M, z2M)µ(z2M+1, z2M+1) · · · µ(z2M+2N, z2M+2N),will satisfy the diﬀerential equations (i = 1, . .

. , 2M + 2N)43∂2∂z2i−2M+2NXj̸=i1/16(zi −zj)2 +1zi −zj∂∂zjG(2M,2N) = 0 ,(5.5)and similarly for zi →zi.Here we shall illustrate (following Appendix E of [1]) how these diﬀerentialequations can be used to determine the 4-point function G(4) of four σ’s at thecritical point of the Ising model.

The constraints of global conformal invariancediscussed in section 2 ﬁrst of all require thatG(4) =σ(z1, z1)σ(z2, z2)σ(z3, z3)σ(z4, z4)=z13z24z12z23z34z411/8 z13z24z12z23z34z411/8F(x, x)(5.6)where x = z12z34/z13z24 is the conformally invariant cross-ratio and zij =zi −zj. (To facilitate comparison with the conventional Ising model result Ihave absorbed some additional x dependence in the prefactor to F in (5.6) withrespect to the canonical form of 4-point functions given in (2.6).

The result isalso frequently cited in terms of the prefactor in (5.6) written in the equivalentformz13 z24 x(1 −x)−1/4. )(5.5) then yields the second order ordinary diﬀerential equationx(1 −x) ∂2∂x2 + 12 −x ∂∂x + 116F(x, x) = 0(5.7)satisﬁed by F (and a similar equation with x →x).

(5.7) has regular singu-lar points at x = 0, 1, ∞and the exponents at these singular points can beobtained by standard asymptotic analysis. The two independent solutions areexpressible as hypergeometric functions which in the case at hand reduce to the64

elementary functions f1,2(x) =1 ± √1 −x1/2. Taking also into account thez dependence, G(4) takes the formG(4) =z13z24z12z23z34z411/42Xi,j=1aij fi(x)fj(x) .

(5.8)But when x is the complex conjugate of x, single-valuedness of G(4) allows onlythe linear combination af1(x)2 +f2(x)2. The resulting expression agreeswith that derived directly in the critical Ising model[25].Now that we have determined the 4-point function, it is possible to identifythe coeﬃcient Cσσε in the operator product expansionσ(z1, z1)σ(z2, z2) ∼1z1/812 z1/812+ Cσσε z3/812 z3/812 ε(z2, z2) + .

. .

,(5.9)where the ﬁrst term ﬁxes the normalization conventions for the σ’s. (5.9) impliesthat (5.6) must behave in the x →0 limit asG(4) ∼1|z12|1/41|z34|1/4 + C2σσε|z12|3/4|z34|3/4|z24|2+ .

. .

. (5.10)Comparison of the ﬁrst term above with the leading small x behavior of (5.8)determines that a = a11 = a22 = 12, i.e.G(4) = 12z13z24z12z23z34z411/4 1 +√1 −x +1 −√1 −x.

(5.11)Comparing the next leading terms of (5.10) and (5.11) as x →0 we ﬁnd Cσσε =12.The non-vanishing operator product coeﬃcients considered thus far areconsistent with the fusion rules (5.4).Similar methods may be used to obtain the other 4-point functions. Insteadof (5.6), we can calculateG(2,2) =σ(z1, z1)µ(z2, z2)σ(z3, z3)µ(z4, z4)=z13z24z12z23z34z411/4F(x, x) .

(5.12)65G(2,2) satisﬁes the same diﬀerential equation (5.7), only now we require the solu-tion to be double-valued as z1 is taken around z2 (x taken around 0). This allowsanother solution with a21 = −a12, a11 = a22 = 0.

In the limit x →∞(z1 →z3, z2 →z4), we have G(2,2) ∼(σ(z1, z1)σ(z3, z3)(µ(z2, z2)µ(z4, z4)=|z13 z24|−1/4, the same leading behavior as in (5.10). This determines a21 =−a12 = i2, i.e.G(2,2) = i2z13z24z12z23z34z411/4 h1 −√1 −x1/21 +√1 −x1/2−1 +√1 −x1/21 −√1 −x1/2i.

(5.13)In the next section we will use the non-leading terms in (5.13) to determinesome of the operator product coeﬃcients involving σ and µ.In principle one can use the (p, q) →(m−p, m+1−q) symmetry of (4.5b) togenerate both an order pq and an order (m−p)(m+1 −q) diﬀerential equationfor correlation functions involving a φp,q operator. In some cases[26], combiningthe two equations allows one to derive a lower order diﬀerential equation forcorrelation functions involving the ﬁeld in question.For the (m = 3) Isingmodel, for example, this procedure gives both second and third order diﬀerentialequations for correlation functions involving the operator ε = Φ2,1.

These canbe combined to give readily solved ﬁrst-order partial diﬀerential equations forthe 4-point functions ⟨εεεε⟩and ⟨εεσσ⟩.5.3. Fusion rules for c < 1 modelsAlthough rather cumbersome in general, the above diﬀerential equationmethod in principle gives the correlation functions of any set of degenerate op-erators and can be used to determine the operator product coeﬃcients Cijk (forthe 3-state Potts model this has been carried out in [27]).

A diﬀerent method,based on the background charge ideas described after (3.4), gives instead in-tegral representations for the correlation functions which have been studiedextensively in [10]. Again the results for the 4-point functions can be used toinfer the Cijk’s.66

Applied directly to the 3-point functions, the above diﬀerential equationmethod does not determine the Cijk’s, but does give useful selection rules thatdetermine which are allowed to be non-vanishing. For example, the 3-pointfunctionφ2,1(z1)φp,q(z2)φp′,q′(z3)is annihilated by the second order diﬀer-ential operator L−2 −32(2h2,1+1)L2−1.If we substitute the operator productexpansion for φ2,1(z1) and φp,q(z2) into this diﬀerential equation and considerthe most singular term as z1 →z2, the characteristic equation gives a quadraticrelation between hp,q and hp′,q′ which is satisﬁed only for p′ = p±1 and q′ = q.For 3-point functions involving φ1,2, we ﬁnd similar the selection rule p′ = pand q′ = q ± 1.By considering multiple insertions of φ1,2 and φ2,1 and using associativityof the operator product expansion, it is possible to derive the general selectionrules for non-vanishingφp1,q1φp2,q2φp3,q3.

If we choose the φp,q’s of ﬁg. 7 withp = 1, .

. .

, m −1, q = 1, . .

. , m, and p + q even, these selection rules may beexpressed asφp1,q1 × φp2,q2 =min(p1+p2−1,2m−1−(p1+p2) )Xp3=|p1−p2|+1min(q1+q2−1,2m+1−(q1+q2) )Xq3=|q1−q2|+1φp3,q3 .

(5.14)The selection rules take a more intuitive form reexpressed in terms of ‘spins’pi = 2ji + 1, qi = 2j′i + 1. They then resemble SU(2) branching rules, i.e.allowed j3 are those that appear in the decomposition of j1 × j2 consideredas representations of SU(2) (and cyclic permutations).

The same conditionsmust be satisﬁed by the j′’s. These conditions allow, among other things, non-vanishing Cijk’s only for all p’s odd (all vector-like) or two even, one odd (twospinor-like, one vector-like).

The selection rules are not quite those of SU(2)because of the upper restriction involving m on the summations. In fact theyare the selection rules instead for what is known as aﬃne SU(2) (at levelsk = m−2 and m−1 respectively for p and q).

We will derive the selection rules(5.14) from this point of view when we discuss aﬃne algebras and the cosetconstruction of these models in section 9.67We have deliberately written (5.14) in a notation slightly diﬀerent from(5.4). (5.14) involves only the holomorphic parts of the ﬁelds and determines acommutative associative algebra.

In general we write such fusion rules as[28]φi × φj =XkNijkφk ,(5.15)where the φi’s denote a set of primary ﬁelds. In the event that the chiral al-gebra is larger than the Virasoro algebra, they should be taken as the ﬁeldsprimary with respect to the larger algebra (later on we shall encounter exam-ples of extended chiral algebras).

The Nijk’s on the right hand side of (5.15)are integers that can be interpreted as the number of independent fusion pathsfrom φi and φj to φk (the k index is distinguished to allow for the possibility ofnon-self-conjugate ﬁelds). (5.4), on the other hand, symbolically indicates theconformal families that may occur in operator products of conformal families ofoperators with combined z, z dependence, but has no natural integral normal-ization.

The algebra (5.15) together with its anti-holomorphic counterpart canalways be used in any given theory to reconstruct less precise structures suchas (5.4).The Nijk’s are automatically symmetric in i and j and satisfy a quadraticcondition due to associativity of (5.15).They can be analyzed extensivelyin a class of theories known as ‘rational conformal ﬁeld theories’. These aretheories[29] that involve only a ﬁnite number of primary ﬁelds with respect tothe (extended) chiral algebra.

The c < 1 theories of section 4 are particularexamples (in which there are a ﬁnite number of primaries with respect to theVirasoro algebra itself). The rationality condition means that the indices ofthe Nijk’s run only over a ﬁnite set of values, and summations over them arewell-deﬁned.

If we use a matrix notation (Ni)jk = Nijk, then the ij symmetrycan be used to write the associativity condition either asNiNℓ= NℓNi ,or asNiNj =XkNijkNk .The Ni’s themselves thus form a commutative associative matrix representationof the fusion rules (5.15). They can be simultaneously diagonalized and their68

eigenvalues λ(n)iform one dimensional representations of the fusion rules. Thealgebra (5.15) is an algebra much like algebras that occur in ﬁnite group theory,such as for the multiplication of conjugacy classes or for the branching rules forrepresentations.

It is a generalization that turns out to embody these algebrasin the orbifold models to be discussed in section 8. We shall see how the Nijk’sthemselves may be determined[28][30] in section 9.5.4.

More discrete seriesSince we have mentioned the idea of extended chiral algebras, we pausehere to exhibit some speciﬁc examples of algebras larger than the Virasoroalgebra. Supersymmetric extensions of the Virasoro algebra are obtained bygeneralizing conformal transformations to superconformal transformations ofsupercoordinates z = (z, θ), where θ is an anticommuting coordinate (θ2 =0).

Superconformal transformations are generated by the moments of a superstress-energy tensor. If there is only a single anti-commuting coordinate (N=1supersymmetry), then the super stress-energy tensor T(z) = TF (z)+θT (z) hascomponents that satisfy the operator products[31][32]T (z1) T (z2) ∼3ˆc/4(z1 −z2)4 +2(z1 −z2)2 T (z2) +1z1 −z2∂T (z2) ,T (z1) TF (z2) ∼3/2(z1 −z2)2 TF (z2) +1z1 −z2∂TF(z2) ,TF (z1) TF (z2) ∼ˆc/4(z1 −z2)3 +1/2z1 −z2T (z2) ,(5.16)where ˆc = 23c.

The conventional normalization is such that a single free super-ﬁeld x(z) + θψ(z) has central charge ˆc = 1 in (5.16), just as the stress-energytensor for a single bosonic ﬁeld x(z) had central charge c = 1 in (3.1). The sec-ond equation in (5.16) is the statement that TF is a primary ﬁeld of dimension3/2.In terms of the moments Ln of T , and the momentsGn =Idz2πi zn+1/2 2 TF(z)(5.17)69of TF , the operator product expansions (5.16) are equivalent to the (anti-)commutation relations[Lm, Ln] = (m −n)Lm+n + ˆc8(m3 −m)δm+n,0[Lm, Gn] =m2 −nGm+n{Gm, Gn} = 2Lm+n + ˆc2m2 −14δm+n,0 .

(5.18)The algebra (5.16) has a Z2 symmetry, TF →−TF, so there are two possiblemodings for the Gn’s. For integer moding (n ∈Z) of Gn, the supersymmetricextension of the Virasoro algebra is termed the Ramond (R) algebra; for half-integer moding (n ∈Z + 12), it is termed the Neveu-Schwarz (NS) algebra.Primary ﬁelds are again associated with highest weight states |h⟩, satisfyingLn|h⟩= Gn|h⟩= 0, n > 0, and L0|h⟩= h|h⟩.

Note that (5.18) requires thata highest weight state in the Ramond sector have eigenvalue h −ˆc/16 underG20. For ˆc > 1, the only restrictions imposed by unitarity are h ≥0 (NS), andh ≥ˆc/16 (R), and the Verma modules again provide irreducible representations(no null states) except when the latter inequalities are saturated.For ˆc < 1 (c < 32), on the other hand, unitary representations of (5.16) canoccur only at the discrete valuesc = 321 −8m(m + 2)(5.19)(m = 3, 4, .

. .

), and discrete values of h from a formula analogous to (4.6b).Notice that the ﬁrst value is c = 7/10, and coincides with the second memberof the discrete series (4.6a), identiﬁed as the tricritical Ising model. Furtherdiscussion of the supersymmetry in this model may be found in [32][33].There are also generalizations of (5.16) with more than one supersymmetrygenerator.

In the case N = 2 [34], there is a discrete series [35]c = 31 −2m(5.20)(m = 3, 4, . .

.) of allowed values for c < 3, and a continuum of allowed valuesfor c ≥3.The boundary value c = 3 can be realized in terms of a single70

free complex superﬁeld. The ﬁrst value, c = 1, coincides with the second non-trivial member of the series (5.19).

The N = 2 superconformal algebra containsa U(1) current algebra, under which the supersymmetry generators transformwith non-zero charge. For N = 3 supersymmetry, unitary representations occur[36] only at the discrete set of values c = 32k (k = 1, 2, .

. .

); and for N = 4supersymmetry, only at the values c = 6k (k = 1, 2, . .

. ).

In these last two casesunitarity allows no continuum of values for the central charge. This is relatedto the fact that the N = 3, 4 algebras contain an SU(2) current algebra underwhich the supersymmetry generators transform non-trivially (we shall discussaﬃne SU(2) in some detail in section 9).6.

Free bosons and fermionsUseful properties of conformal ﬁeld theories can frequently be illustratedby means of free ﬁeld realizations. In this section, we shall apply the general for-malism of sections 1–3 to the cases of free bosons and free fermions, introducedin subsections 2.3 and 3.2.

These will prove most useful in our applications ofconformal ﬁeld theory in succeeding sections.6.1. Mode expansionsIn section 3, we introduced mode expansions for general primary ﬁelds.

Inparticular, for free bosons and fermions we havei∂zx(z) =Xnαn z−n−1iψ(z) =Xψn z−n−1/2 . (6.1)In what follows we shall take n to run over either integers or half-integers,depending on the boundary conditions chosen for the ﬁelds.

(The factors of ihave been inserted in (6.1) to give more familiar commutation relations for themodes. They compensate the choice of sign in (2.16).) The expansions (6.1)are easily inverted to giveαn =Idz2πizn i∂zx(z)ψn =Idz2πizn−1/2 iψ(z) .

(6.2)71In section 3 we also saw how the operator product expansion (3.1) of thestress-energy tensor T (z) implied commutation relations for the modes Ln ofthe Virasoro algebra. In the case of the bosonic modes, we ﬁnd that the shortdistance expansion (2.16) implies the commutation rules[αn, αm] = i2Idz2πi,Idw2πizn∂zx(z) wm∂wx(w)= i2Idw2πiwmIdz2πizn−1(z −w)2 =Idw2πinwmwn−1= nδn+m,0 ,(6.3)where we have evaluated the commutator of integrals by ﬁrst performing thez-integral with the contour drawn tightly around w, and then performing thew-integral.Similarly, we ﬁnd{ψn, ψm} = i2Idz2πi,Idw2πizn−1/2wm−1/2ψ(z)ψ(w)= i2Idw2πiwm−1/2Idz2πizn−1/2 −1z −w=Idw2πiwm−1/2wn−1/2 = δn+m,0 ,(6.4)although in this case we obtain an anti-commutator due to the fermionic natureof ψ which gives an extra minus sign when we change the order of ψ(z) andψ(w).6.2.

Twist ﬁeldsWe shall choose to consider periodic (P) and anti-periodic (A) bound-ary conditions on the fermion ψ(z) as z rotates by 2π about the origin,ψ(e2πiz) = ±ψ(z). Ultimately consideration of the two boundary conditionsis dictated by the fact that spinors naturally live on a double cover of thepunctured plane, and only bilinears in spinors, i.e.

vectors, need transform assingle-valued representations of the 2d Euclidean group. (On higher genus Rie-mann surfaces, spinors generally live in the spin bundle, i.e.

the double cover72

of the principle frame bundle of the surface.) In the course of our discussionwe shall also encounter other ways in which the twisted structure naturallyemerges.

From (6.1) we see that the two boundary conditions select respec-tively half-integer and integer modingsψ(e2πiz) = +ψ(z)n ∈Z + 12(P)ψ(e2πiz) = −ψ(z)n ∈Z(A) . (6.5)In preparation for the anti-periodic case, we ﬁrst consider the calculation ofthe 2-point function in the periodic case ψ(e2πiz) = ψ(z).

Then with n ∈Z+ 12,we ﬁnd the expected result,−ψ(z)ψ(w)=∞Xn=1/2ψn z−n−1/2−∞Xm=−1/2ψm w−m−1/2=∞Xn=1/2z−n−1/2wn−1/2 = 1z∞Xn=0wzn=1z −w . (6.6)For the anti-periodic case, it is useful to introduce the twist operator σ(w)whose operator product with ψ(z),ψ(z)σ(w) ∼(z −w)−1/2 µ(w) + .

. .

,(6.7)is deﬁned to have a square-root branch cut. The ﬁeld µ appearing in (6.7)is another twist ﬁeld which by dimensional analysis has the same conformalweight as the ﬁeld σ.

Our immediate object is to infer the dimension of σ bycalculating the 2-point function of ψ. Due to the square-root in (6.7), when theﬁeld ψ is transported around σ it changes sign and the twist ﬁeld σ can be usedto change the boundary conditions on ψ.

We can thus view the combinationσ(0) and σ(∞) to create a cut (the precise location of which is unimportant)from the origin to inﬁnity passing through which the fermion ψ(z) ﬂips sign. (The similarity with the Ising disorder operator described in Cardy’s lectures,sec.

5.2, is not accidental.) Equivalently, we can view the state σ(0)|0⟩as a newincoming vacuum, and the operator product (6.7) allows only fermions with73anti-periodic boundary conditions (half-integral modes) to be applied to thisvacuum, resulting in overall single-valued states.In either interpretation, the 2-point function of the fermion with anti-periodic boundary conditions is given byψ(z) ψ(w)A ≡⟨0|σ(∞) ψ(z) ψ(w) σ(0)|0⟩(6.8)(see (3.10b) for what we mean by ⟨0|σ(∞) here).

The evaluation of this quan-tity proceeds as in (6.6) except that now for anti-periodic fermions ψ(e2πiz) =−ψ(z), we take n ∈Z.That means the fermion mode algebra now has azero mode ψ0 that by (6.4) formally satisﬁes {ψ0, ψ0} = 1. We shall discussthe fermion zero mode algebra in some detail a bit later, but for the momentsubstituting ψ20 = 12 gives−ψ(z)ψ(w)A = ∞Xn=0ψn z−n−1/2−∞Xm=0ψm w−m−1/2A=∞Xn=1z−n−1/2wn−1/2 + 121√zw=1√zwwz −w + 12=12p zw + p wzz −w.

(6.9)This result has the property that it agrees with the result (6.6) in the z →w limit (the short distance behavior is independent of the global boundaryconditions), and also changes sign as either z or w makes a loop around 0 or∞. It could alternatively have been derived as the unique function with theseproperties.We now wish to show how (6.9) may be used to infer the conformal weighthσ of the ﬁeld σ(w).This is extracted from the operator product with thestress-energy tensorT (z)σ(0)|0⟩∼hσ σ(0)z2|0⟩+ .

. .

,(6.10)where the stress-energy tensor is deﬁned as the limitT (z) = 12ψ(z)∂wψ(w) +1(z −w)2z→w.74

The expectation value of the stress-energy tensor in the state σ(0)|0⟩may beevaluated from (6.9) by taking the derivative with respect to w and then settingz = w + ǫ in the limit ǫ →0,ψ(z)∂wψ(w)A = −12p zw + p wz(z −w)2+ 141w3/2z1/2 = −1ǫ2 + 181w2 ,so thatT (z)A = 1161z2 .If we now take the limit z →0 and compare with (6.10) we ﬁnd that hσ =116.Before turning to the promised treatment of the fermion zero modes, weoutline an analogous treatment for a bosonic twist ﬁeld. As in (6.7), we write∂x(z)σ(w) ∼(z −w)−1/2 τ(w) + .

. .

,(6.11)where now by dimensional analysis the “excited twist ﬁeld” τ has hτ = hσ + 12.A twist ﬁeld σ(w, w) (with hσ = hσ) that twists both x(z) and x(z) can then beconstructed as a product of separate holomorphic and anti-holomorphic pieces.We deﬁne the 2-point function for the boson with anti-periodic boundaryconditions as in (6.8),∂x(z) ∂x(w)A ≡⟨0|σ(∞) ∂x(z) ∂x(w) σ(0)|0⟩,(6.12)and again evaluate using the mode expansion (6.1). Now the boson with anti-periodic boundary conditions requires n ∈Z + 12, so that−∂x(z)∂x(w)A = ∞Xn= 12z−n−1αn−∞Xm=−12w−m−1αmA=∞Xn= 12n z−n−1wn−1 =1(zw)1/21z∞Xn=0n + 12 wzn=1(zw)1/2w(z −w)2 + 121z −w=12p zw + p wz(z −w)2.

(6.13)This result could equally have been derived by requiring the correct short dis-tance behavior (2.16) as z →w, together with the correct sign change for z orw taken around 0 or ∞.75We may now use (6.13) to evaluate the expectation value of the stress-energy tensor in the twisted sectorT (z)A = −12 limz→w∂x(z)∂x(w) +1(z −w)2A=116z2 .Taking z →0 we again infer fromT (z)σ(0)|0⟩∼hσ σ(0)z2|0⟩+ . .

.that the twist ﬁeld for a single holomorphic boson has hσ =116.At ﬁrst this result may seem strange, since a single c = 1 boson is nominallycomposed of two c = 12 fermions. The correspondence is given byψ±(z) =: e±ix(z): ,(6.14)where by (2.19), ψ±(z) are seen to have conformal weight h = 12 appropriateto fermions.

Under the twist x →−x we see that ψ± →ψ∓. In terms of realfermions ψ1,2 deﬁned by ψ± =i√2(ψ1±iψ2), we have ψ1 →ψ1, ψ2 →−ψ2.

Thebosonic twist x →−x thus corresponds to taking only one of the two fermionsto minus itself, and it is natural that the twist operator for a boson have thesame conformal weight as the twist operator for a single fermion. We can alsounderstand this result by considering the currentψ1(z)ψ2(z) = limz→w1i√2(ψ+(z) + ψ−(z)−1√2ψ+(w) −ψ−(w)= ∂x(z)(here we have used: e±ix(z): : e∓ix(w): ∼: e±ix(z)∓ix(w):z −w∼: e±i(z−w)∂x(w):z −w∼±i∂x(w) ,following from (2.19), and pulled out the leading term as z →w).

Again wesee that twisting the (1,0) current ∂x →−∂x requires twisting only one of thetwo fermions ψ1 or ψ2.There is a nice intuitive picture for calculating correlation functions in-volving twist ﬁelds (see e.g. [37]).

A cut along which two fermions change sign76

is equivalent to an SO(2) gauge ﬁeld concentrated along the cut whose ﬁeldstrength, non-zero only at the endpoints of the cut, is adjusted to give a phasechange of π for parallel transport around them. In this language, the twistﬁeld looks like a point magnetic vortex, and changing the position of the cutjust corresponds to a gauge transformation of its gauge potential.

The physi-cal spectrum of the model should consist only of operators that do not see thestring of the vortex, so that the theory is local. If we bosonize the fermions, thencorrelations of twist ﬁelds can be calculated as ratios of partition functions ofa free scalar ﬁeld with and without these point sources of ﬁeld strength.

Theseratios in turn are readily calculated correlation functions of exponentials of freescalars, and result in power law dependences for the correlators of twist ﬁelds.For their 2-point function, this reproduces in particular the conformal weightcalculated earlier.6.3. Fermionic zero modesNow we return to a more careful treatment of the fermionic zero modementioned before (6.9).

We begin by introducing an operator (−1)F, deﬁned toanticommute with the fermion ﬁeld, (−1)F ψ(z) = −ψ(z)(−1)F, and to satisfy(−1)F 2 = 1. In terms of modes, this means that{(−1)F, ψn} = 0for all n,(6.15)so (−1)F will have eigenvalue ±1 acting on states with even or odd numbers offermion creation operators.From (6.4) and (6.15) we thus have for n ∈Z the anti-commutators{ψ0, ψn̸=0} = 0,{(−1)F, ψ0} = 0,andψ20 = 12(6.16)with the zero mode ψ0.

Since the mode ψ0 acting on a state does not change theeigenvalue of L0, in particular the ground state must provide a representationof the 2d cliﬀord algebra consisting of (−1)F and ψ0. The smallest irreduciblerepresentation of this algebra consists of two states that we label 116±.

The77action of operators on these states can be represented in terms of Pauli matrices,deﬁned to act asσz 116± = ± 116±σx 116± = 116∓.Then(−1)F = σz(−1)Pψ−nψnandψ0 =1√2σx(−1)Pψ−nψn(6.17)provide a representation of (6.16) in a (−1)F diagonal basis. Since ψ20 116± =12 116±, if we identify the state σ(0)|0⟩in (6.9) with 116+, the remaining stepsin (6.9) are now justiﬁed.

The state 116−, on the other hand, can be identiﬁedwith µ(0)|0⟩, where µ(z) is the conjugate twist ﬁeld appearing in the right handside of (6.7). (If we are willing to give up having a well-deﬁned (−1)F , we could alsouse either of1√2 116+ ± 116−as our ground state in (6.9).In terms ofﬁelds, this would mean trading the two ﬁelds σ and µ for a single ﬁeld eσ, takenas either of1√2(σ ± µ).Instead of the fusion rule [ψ][σ] = [µ] of (6.7), wewould have [ψ][eσ] = [eσ].

The theories we consider later on here, however, willgenerally require a realization of (−1)F on the Hilbert space, so we have chosento incorporate it into the formalism from the outset. )An additional subtlety occurs when we consider both holomorphic fermionsψ(z) and their anti-holomorphic partners ψ(z).

Then the ψ’s satisfy the analogof (6.4), and as wellψn, ψm= 0∀n, m . (6.18)If we wish to realize separate operators (−1)FL, (−1)FR, satisfying(−1)FL, ψ(z)=0,(−1)FR, ψ(z)= 0, then we simply duplicate the structure (6.17) for theψ’s and ψ’s to give fourh =116, h =116ground states of the form 116L± ⊗ 116R± .

(6.19)But in general we need not require the existence of both chiral (−1)FLand (−1)FR, but rather only the non-chiral combination (−1)F = (−1)FL+FR.78

In fact (6.18) implies that ψ0 and ψ0 already form a two dimensional Cliﬀordalgebra, so the combination ψ0ψ0 automatically serves to represent the non-chiral (−1)F restricted now to a two-dimensional ground state representationh =116, h =116±. If we write the action of Pauli matrices on this basis asσx 116, 116± = 116, 116∓σy 116, 116± = ∓i 116, 116∓σz 116, 116± = ± 116, 116± ,(6.20a)then it is easily veriﬁed that the zero mode representationψ0 = σx + σy2(−1)Pn>0 ψ−nψn+ψ−nψnψ0 = σx −σy2(−1)Pn>0 ψ−nψn+ψ−nψn(−1)F = σz(−1)Pn>0 ψ−nψn+ψ−nψn(6.20b)satisﬁes the algebra (6.16),(6.18).

In (6.20b) we have chosen to represent theCliﬀord algebra in a rotated basis,1√2(σx ± σy) =e∓iπ/4e±iπ/4,since this is the representation we shall ﬁnd induced by our choice of phaseconventions (choice of gauge) for operator product expansions. (The four dimensional representation (6.19), irreducible under the full chi-ral algebra including both (−1)FL and (−1)FR, is reducible under the subalge-bra that includes only the non-chiral (−1)F .

Explicitly the two two-dimensionalirreducible representations of the non-chiral subalgebra are given by 116, 116± = 116L+ ⊗ 116R±+ 116L−⊗ 116R∓ 116, 116′± = 116L+ ⊗ 116R±− 116L−⊗ 116R∓.We see that only the operators (−1)FL and (−1)FR act to connect the orthogonalHilbert spaces built on 116, 116± and 116, 116′±. Had we begun with the fourdimensional representation (6.19), but required only the existence of the non-chiral (−1)FL+FR, then we could consistently throw out all the states builtsay on 116, 116′± and be left with the minimal two-dimensional representation(6.20) of the zero mode algebra.

Similar considerations apply in the case ofrealizations of N = 1 superconformal algebras without chiral (−1)F [38].)797. Free fermions on a torusIn this section we shall consider conformal theory not on the conformalplane, but rather on a torus, i.e., on a Riemann surface of genus one.

Ourmotivation for doing this is both statistical mechanical and ﬁeld theoretical.From the statistical mechanical point of view, it turns out that the fact thata given model admits a consistent formulation on the torus acts to constrainits operator content already on the plane. From the ﬁeld theoretical point ofview, conformal ﬁeld theory achieves its full glamour when formulated on anarbitrary genus Riemann surface.

Higher genus is also the natural arena forapplications of conformal ﬁeld theory to perturbative string theory. The torusis the ﬁrst non-trivial step in this direction, and turns out to probe all of theessential consistency requirements for conformal ﬁeld theory formulated on anarbitrary genus Riemann surface.

We refer the reader to Friedan’s lectures formore on the higher genus extension.7.1. Back to the cylinder, on to the torusOur strategy for constructing conformal ﬁeld theory on the torus is to makeuse of the local properties of operators already constructed on the conformalplane, map them to the cylinder via the exponential map, and then arriveat a torus via discrete identiﬁcation.

While this procedure preserves all localproperties of operators in a theory, it does not necessarily preserve all of theirglobal properties.For example since the torus maps to an annulus on theplane, only the generators of dilatations and rotations, i.e. L0 and L0, surviveas global symmetry generators.

On the torus, L±1 and L±1 are reduced toplaying the role of local symmetry generators, as played by the remaining Ln,Ln (n ̸= 0, ±1) on the plane, and the global symmetry group is reduced toU(1) × U(1).Another global property aﬀected by the passage from the plane to thecylinder (or torus) is boundary conditions on ﬁelds. Let us consider the map80

w →z = ew, mapping the cylinder, coordinatized by w, to the plane, coordi-natized by z. Since ϕ(z, z)dzhdzh is invariant under this map, we ﬁndϕcyl(w, w) = dzdwh dzdwhϕ(z, z) = zh zhϕ(z, z) .

(7.1)This means that a ﬁeld ϕ(z, z) on the plane that is invariant under z →e2πiz,z →e−2πiz corresponds to a ﬁeld ϕcyl(w, w) that picks up a phase e2πi(h−h)under w →w + 2πi, w →w −2πi. Fields with integer spin s = h −h thushave the same boundary conditions on the plane and cylinder.

Fields with half-integer spin having periodic boundary conditions become anti-periodic, andvice-versa, when passing from the plane to the cylinder.We can see the same eﬀect in terms of the mode expansion ϕ(z) =Pn ϕnz−n−h of a holomorphic ﬁeld. The mode expansion induced on the cylin-der,ϕcyl(w) = dzdwhϕ(z) = zh Xnϕnz−n−h =Xnϕn e−nw ,(7.2)becomes an ordinary Fourier series.

Again however a ﬁeld moded as n ∈Z −hso that it is non-singular at the origin of the conformal plane is no longer single-valued under w →w + 2πi on the cylinder.For a fermion, with h =12, h = 0, we have from (7.1) that ψcyl(w) =z1/2ψ(z) so A boundary conditions on the plane become P on the cylinder, andvice-versa. In terms of the mode expansion (7.2), we haveψcyl(w) =Xnψn e−nw,n ∈ Z(P)Z + 12(A),(7.3)opposite to the case (6.5) on the plane where the same modes ψn give A forn ∈Z and P for n ∈Z + 12.

On the cylinder it is thus the P sector that hasground state L0 eigenvalue larger by116. We point out that even if we thoughtonly one of the A or P boundary conditions the more natural, we would beforced to consider the other anyway in moving back and forth from plane tocylinder (giving a possible motivation for considering both on equal footingfrom the outset).

(For superpartners ψµ of spacetime bosonic coordinates in81string theory, the sectors corresponding to P and A on the cylinder, i.e. n ∈Zand n ∈Z + 12 respectively, are ordinarily termed the Ramond (R) and Neveu-Schwarz (NS) sectors.) Since the modes ψn in our mode expansion (7.3) on thecylinder are identically those on the plane (6.1) (local operator products arenot aﬀected by conformal mapping), they satisfy the same anti-commutationrules (6.4),{ψn, ψm} = δn+m,0 .ψ−n and ψn (n > 0) thus continue to be regarded as fermionic creation andannihilation operators acting on a vacuum state |0⟩, deﬁned to satisfy ψn|0⟩= 0(n > 0), and the Hilbert space of states ψ−n1 .

. .

ψ−nk|0⟩is built up by applyingcreation operators ψ−n to |0⟩.For a ﬁeld such as the stress-energy tensor T (z) that does not transformtensorially under conformal transformations, an additional subtlety arises inthe transfer to the cylinder. Under conformal transformations w →z, T (z) ingeneral picks up an anomalous piece proportional to the Schwartzian derivativeS(z, w) =∂wz ∂3wz −32(∂2wz)2/(∂wz)2 as in (3.3).

For the exponential mapw →z = ew, we have S(ew, w) = −1/2, soTcyl(w) = ∂z∂w2T (z) + c12S(z, w) = z2 T (z) −c24 .Substituting the mode expansion T (z) = P Ln z−n−2, we ﬁndTcyl(w) =Xn∈ZLn z−n −c24 =Xn∈ZLn −c24 δn0e−nw . (7.4)The translation generator (L0)cyl on the cylinder is thus given in terms of thedilatation generator L0 on the plane as(L0)cyl = L0 −c24 .Ordinarily one can always shift the energy of the vacuum by a constant (equiva-lently change the normalization of a functional integral), but in conformal ﬁeldtheory, scale and rotational invariance of the SL(2, C) invariant vacuum on the82

plane naturally ﬁxes L0 and L0 to have eigenvalue zero on the vacuum, therebyﬁxing the zero of energy once and for all.Conformal ﬁeld ﬁeld theory on a cylinder coordinatized by w can nowbe transferred to a torus as follows. We let H and P denote the energy andmomentum operators, i.e.

the operators that eﬀect translations in the “space”and “time” directions Re w and Im w respectively. On the plane we saw thatL0 ± L0 respectively generated dilatations and rotations, so according to thediscussion of radial quantization at the beginning of subsection 2.2, we haveH = (L0)cyl + (L0)cyl and P = (L0)cyl −(L0)cyl.

To deﬁne a torus we need toidentify two periods in w. It is convenient to redeﬁne w →iw, so that one of theperiods is w ≡w+2π. The remaining period we take to be w ≡w+2πτ, whereτ = τ1 + iτ2 and τ1 and τ2 are real parameters.

This means that the surfacesIm w = 2πτ2 and Im w = 0 are identiﬁed after a shift by Re w →Re w + 2πτ1(see ﬁg. 8).

The complex parameter τ parametrizing this family of distinct toriis known as the modular parameter.ττ + 101Re wIm wFig. 8.

Torus with modular parameter τ.Since we are deﬁning (imaginary) time translation of Im w by its period2πτ2 to be accompanied by a spatial translation of Re w by 2πτ1, the operatorimplementation for the partition function of a theory with action S on a torus83with modular parameter τ isZe−S = tr e2πiτ1P e−2πτ2H= tr e2πiτ1(L0)cyl −(L0)cyle−2πτ2(L0)cyl + (L0)cyl= tr e2πiτ(L0)cyl e−2πiτ(L0)cyl = tr q(L0)cylq(L0)cyl= tr qL0 −c24 qL0 −c24 = q−c24 q−c24 tr qL0 qL0 ,(7.5)where q ≡exp(2πiτ). For the c = c = 12 theory of a single holomorphic fermionψ(w) and a single anti-holomorphic fermion ψ(w) on the torus, we would thusﬁndZe−S = (qq)−c24 tr qL0qL0 = (qq)−148 tr qL0qL0 .

(7.6)Before turning to a treatment of free fermions in terms of the representationtheory of the Virasoro algebra, we pause here to mention that the vacuumenergies derived in section 6 can be alternatively interpreted to result from avacuum normal ordering prescription on the cylinder. We ﬁnd for example(L0)cyl = 12Xnn : ψ−nψn: =Xn>0nψ−nψn −12Xn>0n=Xn>0nψ−nψn +(−12ζ(−1) =124n ∈Z−12(−12ζ(−1)) = −148n ∈Z + 12,where we have used ζ-function regularization to evaluate the inﬁnite sums.

Wesee that the result for n ∈Z + 12 agrees with the result given earlier in thissubsection for the A sector on the cylinder. For n ∈Z we as well ﬁnd correctlythat the vacuum energy is shifted up by124 −(−148) =116.

The justiﬁcation forthis ζ-function regularization procedure ultimately resides in its compatibilitywith conformal and modular invariance. For a boson on the cylinder we wouldinstead ﬁnd(L0)cyl = 12Xn: α−nαn: =Xn>0α−nαn + 12Xn>0n=Xn>0α−nαn +(12ζ(−1) = −124n ∈Z12(−12ζ(−1)) =148n ∈Z + 12.84

For n ∈Z the result correctly gives −c24, now with c = 1. For n ∈Z + 12 we seethat the vacuum energy is increased by116, again correctly giving the conformalweight of the bosonic twist ﬁeld calculated in the previous section.

(Note thatthe vacuum normal ordering constants for a single boson on the cylinder aresimply opposite in sign from those for the fermion.) The anti-periodic bosonparametrizes what is known as a Z2 orbifold, and will be treated in detail inthe next section.More generally this vacuum normal ordering prescription can be used tocalculate the vacuum energy for a complex holomorphic fermion (i.e.

two c = 12holomorphic fermions) with boundary condition twisted by a complex phaseψcyl(w + 2πi) = exp(2πiη) ψcyl(w), 0 ≤η ≤1. The resulting vacuum normalordering constant calculated as above is f(η) =112 −12η(1−η).

(As a consistencycheck, a single real fermion has one-half of f as vacuum energy, and consequentlywe conﬁrm that 12f( 12) = −148 and 12f(0) =124 for vacuum energy in the A andP sectors respectively on the cylinder).7.2. c = 12 representations of the Virasoro algebraHaving introduced all of the necessary formalism for treating free fermionson the torus, we are now prepared to make contact with the general repre-sentation theory of the Virasoro algebra introduced in section 4.Since thestress-energy tensor for a single free fermion has c = 12, we should expect toﬁnd free fermionic realizations of the three unitary irreducible representationsallowed for this value of c, namely h = {h1,1, h2,1, h2,2} = {0, 12, 116}.We begin by considering the states built in the A sector of the fermion onthe torus. In this case states take the form ψ−n1 .

. .

ψ−nk|0⟩, with ni ∈Z + 12.The ﬁrst few such states, ordered according to their eigenvalue under L0 =85Pn>0 nψ−nψn, areL0 eigenvaluestate0|0⟩1/2ψ−1/2|0⟩3/2ψ−3/2|0⟩2ψ−3/2ψ−1/2|0⟩5/2ψ−5/2|0⟩3ψ−5/2ψ−1/2|0⟩7/2ψ−7/2|0⟩4ψ−7/2ψ−1/2|0⟩,ψ−5/2ψ−3/2|0⟩. .

..(7.7)Denoting the trace in this sector by trA, we calculatetrA qL0 = 1 + q1/2 + q3/2 + q2 + q5/2 + q3 + q7/2 + 2q4 + . .

. .In general traces of the form tr qL0 = Pn Nn qn characterize the number ofstates Nn that occur at a given level n (eigenvalue of L0).q may thus beregarded as a formal parameter analogous to the Cartan angles that appear incharacter formulae for Lie groups.

q (= e2πiτ) obtains additional signiﬁcance interms of the modular parameter τ when these traces are regarded as the resultof calculating functional integrals (7.5) for ﬁeld theories on a torus.The states (7.7) form a (not necessarily irreducible) representation of theVirasoro algebra with c =12.From the eigenvalues of L0, we immediatelyidentify the representation as the direct sum [0] ⊕[ 12] of the highest weightrepresentations with h = 0 and h =12.Since there is only a single statewith h = 0 and only a single state with h = 12 we see that each of these tworepresentations appears with unit multiplicity. Moreover since states createdby applying L−n’s to a single highest weight state all have integrally spacedL0 eigenvalues, we see that the states of the representation [0] are identicallythose with even fermion number, and hence L0 ∈Z; the states of [ 12] are those86

with odd fermion number and hence L0 ∈Z + 12. These two sets of states areprecisely distinguished by their opposite eigenvalues under the operator (−1)F ,i.e.trA (−1)F qL0 = 1 −q1/2 −q3/2 + q2 −q5/2 + q3 −q7/2 + 2q4 + .

. .

.The projection operators 12(1 ± (−1)F ) may therefore be used to disentanglethe two representations, givingq−1/48 trA121 + (−1)F qL0 = q−1/48(1 + q2 + q3 + 2q4 + . .

. )= q−1/48 trh=0 qL0 ≡χ0q−1/48 trA121 −(−1)F qL0 = q−1/48(q1/2 + q3/2 + q5/2 + q7/2 + .

. .

)= q−1/48 trh=1/2 qL0 ≡χ1/2 ,(7.8)where χ0,1/2 are the characters of the h = 0, 12 representations of the c = 12Virasoro algebra (deﬁned to include the oﬀset of L0 by −c/24).In the periodic sector of the fermion on the torus, on the other hand, wehave L0 = Pn>0 ψ−nψn +116 with n ∈Z.As seen in (6.17), the fermionzero mode algebra together with (−1)F requires two ground states 116±, witheigenvalues ±1 under (−1)F, that satisfyψ0 116± =1√2 116∓.The states of the Hilbert space in this sector thus take the formL0 eigenvaluestate116 + 0 116±116 + 1ψ−1 116±116 + 2ψ−2 116±116 + 3ψ−3 116± ,ψ−2ψ−1 116±. .

..(7.9)87We ﬁnd two irreducible representations of the c =12 Virasoro algebra withhighest weight h =116. Again they can be disentangled by projecting onto ±1eigenstates of (−1)F ,q−1/48 trP121 ± (−1)F qL0 = q1/24(1 + q + q2 + 2q3 + .

. .

)= q−1/48 trh=1/16 qL0 ≡χ1/16 . (7.10)Although it happens that trP(−1)F qL0 = 0 in this sector, due to a cancellationbetween equal numbers of states at each level with opposite (−1)F , its insertionin (7.10) has the formal eﬀect of assigning states with even numbers of fermionsbuilt on 116+, or odd numbers on 116−, to one representation 116+ with(−1)F = 1, and vice-versa to the other representation 116−with (−1)F = −1.7.3.

The modular group and fermionic spin structuresWe shall now introduce some essentials of the Lagrangian functional inte-gral formalism for fermions ψ(w) that live on a torus. (For the remainder ofthis section, ψ will always mean ψcyl.) This formalism will facilitate writingdown and manipulating explicit forms for the characters of the h = 0, 12, 116representations of the c = 12 Virasoro algebra.

In general a torus is speciﬁedby two periods which by rescaling coordinates we take as 1 and τ, where τ isthe modular parameter introduced in the previous subsection. Symbolically wewrite w ≡w + 1 ≡w + τ, which means that ﬁelds that live on the torus mustsatisfy ϕ(w + 1) = ϕ(w + τ) = ϕ(w).

It is convenient to write the coordinate win terms of real coordinates σ0,1 ∈[0, 1) as w = σ1 + τσ0.To specify a fermionic theory, we now need to generalize the considerationsof section 6 from a choice of P or A boundary conditions around the one non-trivial cycle on the cylinder, or punctured plane, to two such choices around thetwo non-trivial cycles of the torus. (This is known as choosing a spin structurefor the fermion on a genus one Riemann surface.) In the coordinates σ0, σ1,this amounts to choosing signs ψ(σ0 + 1, σ1) = ±ψ(σ0, σ1), ψ(σ0, σ1 + 1) =±ψ(σ0, σ1).

As in section 6, we can view this sign ambiguity to result fromspinors actually living on a double cover of the frame bundle, so that only88

bilinears, corresponding to two dimensional vector-like representations, need beinvariant under parallel transport around a closed cycle.We shall denote the result of performing the functional integralRexp(−Rψ∂ψ)over fermions with a given ﬁxed spin structure by the symbol xy. The resultfor the spin structure with periodic (P) boundary condition in the σ0 (time)and anti-periodic boundary condition (A) in the σ1 (space) direction, for ex-ample, we denote by PA.

The result of the functional integral can also beregarded as calculating the square root of the determinant of the operator ∂for the various choices of boundary conditions. Due to the zero mode (i.e.

theconstant function) allowed by PP boundary conditions, we see for example thatPP=detPP ∂1/2 = 0.In ordinary two-dimensional ﬁeld theory on a torus, it would suﬃce tochoose any particular spin structure and that would be the end of the story.But there is an additional invariance, modular invariance, that we shall imposeon “good” conformal ﬁeld theories on a torus that forces consideration of non-trivial combinations of spin structures. (In general a “really good” conformaltheory is required to be sensible on an arbitrary Riemann surface, i.e.

be mod-ular invariant to all orders. This turns out to be guaranteed by duality of the4-point functions on the sphere together with modular invariance of all 1-pointfunctions on the torus[30][39].

Intuitively this results from the possibility ofconstructing all correlation functions on arbitrary genus surfaces by “sewing”together objects of the above form. Thus all the useful information about con-formal ﬁeld theories can be obtained by studying them on the plane and on thetorus.

)The group of modular transformations is the group of disconnected dif-feomorphisms of the torus, generated by cutting along either of the non-trivialcycles, then regluing after a twist by 2π. Cutting along a line of constant σ0,then regluing, gives the transformation T : τ →τ + 1, while cutting then reglu-ing along a line of constant σ1 gives the transformation U : τ →τ/(τ +1).

(Thisis the new ratio of periods (see ﬁg. 9), and hence the new modular parameter89after the coordinate rescaling w →w/(τ + 1).

)These two transformationsgenerate a group of transformationsτ →aτ + bcτ + dabcd∈SL(2, Z)(7.11)(i.e. a, b, c, d ∈Z, ad−bc = 1), known as the modular group.

Since reversing thesign of all of a, b, c, d in (7.11) leaves the action on τ unchanged, the modulargroup is actually PSL(2, Z) = SL(2, Z)/Z2. By a modular transformation onecan always take τ to lie in the fundamental region −12 < Re τ ≤12, |τ| ≥1 (Re τ ≥0), |τ| > 1 (Re τ < 0).Usually one uses T : τ →τ + 1 andS = T−1UT−1 : τ →−1/τ to generate the modular group.

They satisfy therelations S2 = (ST )3 = 1.ττ + 110Fig. 9.

The modular transformation U : τ →τ/(τ + 1).Now we consider the transformation properties of fermionic spin structuresunder the modular group. Under T , we have for exampleτ →τ + 1 :AA↔PA.

(7.12a)We can see this starting from AAsince shifting the upper edge of the boxone unit to the right means that the new “time” direction, from lower left to90

upper right, sees both the formerly anti-periodic boundary conditions, to givean overall periodic boundary condition. (see ﬁg.

10) From PAthe oppositeoccurs. The spin structures APand PP, on the other hand, transform intothemselves under T .ττ + 110APAFig.

10. The modular transformation T : τ →τ + 1.The action of U : τ →τ/(τ + 1) on any spin structure can be determinedsimilarly, and thence the action of S = T−1UT−1.We ﬁnd that S acts tointerchange the boundary conditions in “time” and “space” directions, so thatτ →−1/τ :PA↔AP,(7.12b)while AAand PPtransform into themselves.

Since S and T generate themodular group, (7.12a, b) determine the transformation properties under ar-bitrary modular transformations (7.11). It is evident, for example, that thefunctional integral for the spin structure PPis invariant under all modulartransformations (and in fact, as noted earlier, vanishes identically due to thezero mode).

For the moment, (7.12a, b) are intended as symbolic representationsof modular transformation properties of diﬀerent fermionic spin structures. Weshall shortly evaluate the functional integrals and ﬁnd that (7.12a, b) becomecorrect as equations, up to phases.7.4.

c = 12 Virasoro charactersThe c = 12 Virasoro characters (7.8) and (7.10) introduced in the previoussubsection may be written explicitly in terms of fermionic functional integrals91over diﬀerent spin structures. For example the result of the functional integralfor a single holomorphic fermion with spin structure AA, according to (7.5), issimply the trace in the anti-periodic sector q−1/48trA qL0 (where the prefactorq−1/48 results from the vacuum energy discussed earlier).

The spin structurePAin Hamiltonian language corresponds to taking the trace of the insertion ofan operator that anticommutes with the fermion (thereby ﬂipping the boundaryconditions in the time direction). Since (−1)F ψ = −ψ(−1)F , (−1)F is justsuch an operator and PA= q−1/48 trA (−1)F qL0.For the periodic sector,we deﬁne AP=1√2 q−1/48 trP qL0 and PP=1√2 q−1/48 trP(−1)F qL0 (=0).

(The factor1√2 is included ultimately to simplify the behavior under modulartransformations).The calculation of these traces is elementary. In the 2×2 basis|0⟩, ψ−n|0⟩for the nth fermionic mode, we haveqnψ−nψn =1qn,and thus tr qnψ−nψn = 1+qn, and similarly tr(−1)F qnψ−nψn = 1−qn.

It followsthatqL0 = qPn>0 nψ−nψn =Yn>0qnψ−nψn =Yn>01qn.Since the trace of a direct product of matrices ⊗iMi satisﬁes tr⊗iMi = Qi trMi,we ﬁnd trA qL0 = Q∞n=0(1 + qn+1/2), trA(−1)F qL0 = Q∞n=0(1 −qn+1/2), andtrP qL0 = q1/16 Q∞n=0(1 + qn).Expanding out the ﬁrst few terms, we cancompare with (7.8) and (7.10) and see how these inﬁnite products enumerateall possible occupations of modes satisfying Fermi-Dirac statistics. In the casewith (−1)F inserted, each state is in addition signed according to whether it iscreated by an even or odd number of fermionic creation operators.From (7.5), we may thus summarize the partition functions for a singlec = 12 holomorphic fermion asAA= q−1/48 trA qL0 = q−1/48∞Yn=0(1 + qn+1/2) =sϑ3η ,(7.13a)92

PA= q−1/48 trA(−1)F qL0 = q−1/48∞Yn=0(1 −qn+1/2) =sϑ4η ,(7.13b)AP=1√2 q−1/48 trP qL0 =1√2 q1/24∞Yn=0(1 + qn) =sϑ2η ,(7.13c)PP=1√2 q−1/48 trP (−1)F qL0 =1√2 q1/24∞Yn=0(1 −qn) = 0“ = ”sϑ1iη(7.13d)(where trA,P continues to denote the trace in the anti-periodic and periodicsectors). In (7.13a–d) we have also indicated that these partition functions maybe expressed directly in terms of standard Jacobi theta functions ϑi ≡ϑi(0, τ)[40] and the Dedekind eta function η(q) = q1/24 Q∞n=1(1 −qn).It might seem strange that Jacobi and his friends managed to deﬁne func-tions including identically even the correct factor of q−c/24 that we derivedhere physically as a vacuum energy on the torus.

Their motivation, as we shallconﬁrm a bit later, is that these functions have nice properties under modulartransformations. (The connection between conformal invariance and modulartransformations in this context is presumably due to the rescaling of coordinatesinvolved in the transformation τ →−1/τ.) With the explicit results (7.13) inhand, we can now reconsider the exact meaning of equations (7.12a, b).

Byinspection of (7.13) we ﬁnd ﬁrst of all under τ →τ + 1 thatAA→e−iπ24 PAPA→e−iπ24 AAAP→eiπ12 AP. (7.14a)The derivation of the transformation properties under τ →−1/τ uses the Pois-son resummation formula, which we shall introduce at the end of this section.The even simpler (phase-free) result in this case isAA→AAAP→PAPA→AP.

(7.14b)93We also defer to the end of this section some other identities satisﬁed by theseobjects.For the time being, we point out that the deﬁnitions implicit in(7.13a–c) may be used to derive immediately one of the standard ϑ-functionidentities,sϑ2ϑ3ϑ4η3=√2∞Yn=1(1 −q2n−1)(1 + qn)=√2∞Yn=1 1 −qn1 −q2n(1 + qn) =√2 ,usually written in the formϑ2ϑ3ϑ4 = 2η3 . (7.15)Equations (7.13a–d) can be regarded as basic building blocks for a varietyof theories.They also provide a useful heuristic for thinking about Jacobielliptic functions in terms of free fermions.

This representation can be used togive a free fermionic realization of certain integrable models, where away fromcriticality q acquires signiﬁcance as a function of Boltzmann weights instead ofas the modular parameter on a continuum torus.Equations (7.13a–d) also have an interpretation asdet ∂1/2 for the dif-ferent fermionic spin structures, and indeed can be calculated from this pointof view by employing a suitable regularization prescription such as ζ-functionregularization. In the next section we shall calculate the partition function fora single boson from this point of view.

The generalization of the genus oneresults (7.13a–d) to partition functions (equivalently fermion determinants) onhigher genus Riemann surfaces, as well as some of the later results to appearhere, may be found in [41],[42].Finally we can use (7.13a–d) to write the c = 12 Virasoro characters deﬁnedin (7.8) and (7.10) asχ0 = 12 AA+ PA!= 12 sϑ3η +sϑ4η!χ1/2 = 12 AA−PA!= 12 sϑ3η −sϑ4η!χ1/16 =1√2 AP± PP!=1√2sϑ2η ,(7.16a)94

or conversely we can writeAA= χ0 + χ1/2PA= χ0 −χ1/2AP=√2 χ1/16PP= 0 . (7.16b)7.5.

Critical Ising model on the torusWe now proceed to employ the formalism developed thus far to describe theIsing model on the torus at its critical point. As explained in Cardy’s lectures,this is a theory with c = c = 12 and a necessarily modular invariant partitionfunction.

(The role of modular invariance in statistical mechanical systems ona torus was ﬁrst emphasized in [43].) Thus we should expect to be able torepresent it in terms of a modular invariant combination of spin structures fortwo fermions ψ(w), ψ(w).

It will turn out to be suﬃcient for (in fact requiredby) modular invariance to consider only those spin structures for which bothfermions have the same boundary conditions on each of the two cycles. Thecalculation of the partition function for the various spin structures can then beread oﬀdirectly from the purely holomorphic case.

For anti-periodic boundaryconditions for both fermions on the two cycles, for example, we use (7.13a) towriteAAAA≡AAAA=sϑ3ηsϑ3η =ϑ3η .There is one minor subtlety in the PP Hamiltonian sector (i.e. with PPboundary conditions along the “spatial” (σ1) direction), since we need to treatthe zero mode algebra of ψ0 and ψ0.

Restricting to a non-chiral theory meansthat we allow no operator insertions of separate (−1)FL or (−1)FR’s, i.e. weexclude boundary conditions for example of the form AP, and allow onlyAAor PP.Then we need to represent only the non-chiral zero modealgebra {(−1)F, ψ0} = {(−1)F, ψ0} = {ψ0, ψ0} = 0.According to (6.20), the representation of the non-chiral zero mode algebrarequires only a two-dimensional ground state representationh =116, h =116±,95with eigenvalues ±1 under (−1)F .

These two states can be identiﬁed with two(non-chiral) primary twist ﬁelds σ(w, w), µ(w, w) such thatσ(0)|0⟩= 116, 116+andµ(0)|0⟩= 116, 116−. (7.17)The exact form of the operator product expansions of ψ and ψ with these twoﬁelds can be determined by considering 4-point correlation functions (as Cσσεwas determined from (5.11)).

The x →0 limit of (5.13) determines that theshort distance operator product expansion of σ and µ take the formσ(z, z) µ(w, w) =1√2 |z −w|1/4e−iπ/4(z −w)1/2 ψ(w)+ eiπ/4(z −w)1/2 ψ(w). (7.18)Either by taking operator products on both sides with µ or by using permu-tation symmetry of operator product coeﬃcients, we determine that the twistoperators satisfy the operator product algebra*ψ(z) σ(w, w) = eiπ/4√2µ(w, w)(z −w)1/2ψ(z) σ(w, w) = e−iπ/4√2µ(w, w)(z −w)1/2ψ(z) µ(w, w) = e−iπ/4√2σ(w, w)(z −w)1/2ψ(z) µ(w, w) = eiπ/4√2σ(w, w)(z −w)1/2 ,(7.19)consistent with (6.20) under the identiﬁcations (7.17).The remaining non-vanishing operator products of the Ising model can beused to complete the ‘fusion rules’ of (5.4) to[ε][ε] = 1[σ][σ] = 1 + [ε][σ][ε] = [σ][ψ][σ] = [µ][ ψ ][σ] = [µ][ψ][ψ] = 1[µ][µ] = 1 + [ε][µ][ε] = [µ][ψ][µ] = [σ][ ψ ][µ] = [σ][ ψ ][ ψ ] = 1[µ][σ] = [ψ] + [ ψ ][ψ][ ψ ] = [ε][ψ][ε] = [ ψ ][ ψ ][ε] = [ψ](7.20)* (7.18) was derived in [44] from the analog of (5.13) by correcting a sign (a mis-print?) in the corresponding result in [1].

(7.19) here corrects the phases and normal-izations (more misprints?) in eq.

(1.13d) of [44]. I thank P. Di Francesco for guidingme through the typos.96

for all the conformal families of the Ising model. We take this opportunity topoint out that the analysis of such operator algebras has a long history in thestatistical mechanical literature (see for example [15],[45]).

As we noted nearthe end of section 3, the minimal models of [1] gave a class of examples thatclosed on only a ﬁnite number of ﬁelds. In [43], it was shown that modularinvariance on the torus for models with c ≥1 requires an inﬁnite number ofVirasoro primary ﬁelds.

Thus the c < 1 discrete series described in section4 exhausts all (unitary) cases for which the operator algebra can close withonly a ﬁnite number of Virasoro primaries. Rational conformal ﬁeld theories,whose operator algebras close on a ﬁnite number of ﬁelds primary under a largeralgebra, however, can exist and be modular invariant at arbitrarily large valuesof c.Given the two vacuum states (7.17), the analog of (7.13c) for the non-chiralcase is thusAAPP= (qq)−1/48tr qL0qL0 = 2(qq)1/24∞Yn=0(1 + qn)(1 + qn)=sϑ2ηsϑ2η =ϑ2η .We see that the factor of1√2 included in the deﬁnition (7.13c) together withthe factor of 12 reduction in ground state dimension for the non-chiral (−1)Fzero mode algebra results in the simple relation AAPP= APAP.From (7.14), we easily verify that the two combinations of spin structures, AAAA+ PPAA+ AAPP!andPPPP,(7.21)for fermions ψ(w), ψ(w) on the torus are modular invariant.

Although it wouldseem perfectly consistent to retain only one of these two modular orbits to con-struct a theory, we shall see that both are actually required for a consistentoperator interpretation. (In the path integral formulation of string theory thisconstraint, expressed from the point of view of factorization and modular invari-ance of amplitudes on a genus two Riemann surface, was emphasized in [46].

)97As a contribution to the partition function, PPPPof course vanishes due tothe fermion zero mode, but this spin structure does contribute to higher pointfunctions. Hence we shall carry it along in what follows as a formal reminderof its non-trivial presence in the theory.We thus take as our partition functionZIsing = 12 AAAA+ PPAA+ AAPP± PPPP!= (qq)−1/48 trAA121 + (−1)F qL0qL0+ (qq)−1/48 trPP121 ± (−1)F qL0qL0= 12ϑ3η +ϑ4η +ϑ2η ±ϑ1η= χ0χ0 + χ1/2χ1/2 + χ1/16χ1/16 .

(7.22)The overall factor of 12 is dictated by the operator interpretation of the sumover spin structures as a projection, as expressed in the second line of (7.22),and insures a unique ground state in each of the AA and PP sectors. We noticethat the partition function (7.22) neatly arranges itself into a diagonal sumover three left-right symmetric Virasoro characters, corresponding to the threeconformal families that comprise the theory.The projection dictated by modular invariance of (7.21) is onto (−1)F = 1states in the AA sector, i.e.

onto the statesψ−n1 . .

. ψ−nℓψ−nℓ+1 .

. .

ψ−n2k|0⟩. (7.23)In the PP sector the sign for the projection is not determined by modularinvariance, and the two choices of signs, although giving the same partitionfunction, lead to retention of two orthogonal sets of states, as discussed after(7.10).

That these two choices lead to equivalent theories is simply the σ ↔µ(order/disorder) duality of the critical Ising model.In string theory projections onto states in each Hamiltonian sector with agiven value of (−1)F go under the generic name of GSO projection[47]. Such98

a projection was imposed to insure spacetime supersymmetry, among otherthings, in superstring theory, and was later recognized as a general consequenceof modular invariance of the theory on a genus one surface. In the spacetimecontext, the sign ambiguity in the P sector is simply related to the arbitrarinessin conventions for positive and negative chirality spinors.

A general discussionin the same notation employed here may be found in [48].The partition function (7.22) corresponds to boundary conditions on theIsing spins σ = ±1 periodic along both cycles of the torus, i.e. toZPP = PPboundary conditions, where we use italic A, P to denote boundary conditionsfor Ising spins (as opposed to the fermions ψ, ψ).

Depending on the choice of(−1)F projection, the operators that survive in the spectrum are either {1, σ, ε}or {1, µ, ε}, in each case providing a closed operator subalgebra of (7.20).We can also consider the non-modular invariant case of Ising spins twistedalong the “time” direction, which we denoteZPA = AP.This case, as discussed in Cardy’s lectures (section 5.2), corresponds to calcu-lating the trace of an operator that takes the Ising spins σ →−σ, but leaves theidentity 1 and energy ε invariant. In free fermion language, this is equivalent toan operation that leaves the AA sector invariant (the (0,0) and ( 12, 12) families),and takes the PP sector (the ( 116, 116) family) to minus itself.

The resultingpartition function is thusZPA = 12 AAAA+ PPAA!−12 AAPP± PPPP!= |χ0|2 + |χ1/2|2 −|χ1/16|2 . (7.24)The modular transformation τ →−1/τ then allows us to calculate thepartition function for the boundary conditionsZAP = PA,99with Ising spins now twisted in the “space” direction.Applying (7.14b) to(7.24), then using (7.16, ) we ﬁndZAP = 12 AAAA−PPAA!+ 12 AAPP∓PPPP!= χ0χ1/2 + χ1/2χ0 + |χ1/16|2 .

(7.25)We see that the negative sign between the ﬁrst two terms in (7.25) changes thechoice of projection in the AA sector. Now we keep states with odd rather thaneven fermion number as in (7.23), i.e.

states with h−h ∈Z+ 12 rather than withh−h ∈Z. This change is easily seen reﬂected in the oﬀ-diagonal combinations of0 and 12 characters in (7.25).

Changing boundary conditions on the Ising spinsthus allows us to focus on the operator content (ψ, ψ, and µ) of the theory thatwould not ordinarily survive the projection. Playing with boundary conditionsis also a common practice in numerical simulations, so results such as these allowa more direct contact between theory and “experiment” in principle.

Furtheranalysis of partition functions with a variety of boundary conditions in c < 1models, showing how the internal symmetries are tied in with their conformalproperties, may be found in [49].While neither ZPA nor ZAP is modular invariant, we note that the com-bination ZPA + ZAP = AAAA= |χ0 + χ1/2|2 is invariant under a subgroup ofthe modular group, namely that generated by τ →−1/τ and τ →τ + 2. Theoperator content surviving the projection for this combination is {1, ψ, ψ, ǫ},again forming a closed operator subalgebra of (7.20).Finally, from (7.25) the modular transformation τ →τ + 1 can be used todetermine the result for boundary conditionsZAA = AA,for anti-periodic Ising spins along both cycles of the torus.

But from (7.14a)we see that this just exchanges the ﬁrst two terms in (7.25),ZAA = −12 AAAA−PPAA!+ 12 AAPP∓PPPP!= −χ0χ1/2 −χ1/2χ0 + |χ1/16|2 ,(7.26)100

giving the Z2 transformation properties of the operators ψ, ψ, and µ in the Asector of the theory.7.6. Recreational mathematics and ϑ-function identitiesIn this subsection we detail some of the properties of Jacobi elliptic func-tions that will later prove useful.

To illustrate the ideas involved, we begin witha proof of the Jacobi triple product identity,∞Yn=1(1 −qn)(1 + qn−1/2w)(1 + qn−1/2w−1) =∞Xn=−∞q12n2wn ,(7.27)for |q| < 1 and w ̸= 0. (For |q| < 1 the expressions above are all absolutelyconvergent so naive manipulations of sums and products are all valid.

)Rather than the standard combinatorial derivation* of (7.27), we shall tryto provide a more “physical” treatment. To this end, we consider the partitionfunction for a free electron-positron system with linearly spaced energy levelsE = ε0(n −12), n ∈Z, and total fermion number N = Ne −Ne.

If we rewritethe energy E and fugacity µ respectively in terms of q = e−ε0/T and w = eµ/T ,then the grand canonical partition function takes the formZ(w, q) =Xfermionoccupationse−E/T + µN/T =∞XN=−∞wN ZN(q)=∞Yn=1(1 + qn−1/2w)(1 + qn−1/2w−1) ,(7.28)where ZN(q) is the canonical partition function for ﬁxed total fermion numberN. The lowest energy state contributing to Z0 has all negative energy levelsﬁlled (and by deﬁnition of the Fermi sea has energy E = 0).

States excited toenergy E = Mε0 are described by a set of integers k1 ≥k2 ≥· · · ≥kℓ> 0,with Pℓi=1 ki = M (these numbers specify the excitations of the uppermost ℓ* following from the recursion relation P(qw, q) = 1+q−1/2w−11+q1/2wP(w, q) =1q1/2wP(w, q),satisﬁed by the left hand side P(w, q) of (7.27) (see e.g. [50]).101particles in the Fermi sea, starting from the top).

The total number of suchstates is just the number of partitions P(M), so thatZ0 =∞XM=0P(M)qM =1Q∞n=1(1 −qn) .The lowest energy state in the sector with fermion number N, on the otherhand, has the ﬁrst N positive levels occupied, contributing a factorq1/2 · · · qN−3/2qN−1/2 = qPNn=1(j −12)= qN 2/2 .Excitations from this state are described exactly as for Z0, so that ZN =qN 2/2Z0. Combining results givesZ(w, q) =∞XN=−∞wN ZN(q) =∞XN=−∞wNqN 2/2Q∞n=1(1 −qn) ,thus establishing (7.27).The basic result (7.27) can be used to derive a number of subsidiary identi-ties.

If we substitute w = ±1, ±q−1/2, (7.27) allows us to express the ϑ-functionsin (7.13a–d) as the inﬁnite summationsϑ3 =∞Xn=−∞qn2/2ϑ4 =∞Xn=−∞(−1)nqn2/2ϑ2 =∞Xn=−∞q12 (n−12 )2ϑ1 = i∞Xn=−∞(−1)nq12 (n−12 )2 (= 0) . (7.29)We can also express the Dedekind η function as an inﬁnite sum.

We sub-stitute q →q3, w →−q−1/2 in (7.27) to ﬁnd∞Yn=1(1 −q3n)(1 −q3n−2)(1 −q3n−1) =∞Xn=−∞q3n2/2(−1)n q−n/2 ,or equivalently∞Yn=1(1 −qn) =∞Xn=−∞(−1)n q12 (3n2−n) . (7.30)102

Multiplying by q1/24 then givesη(q) = q1/24∞Yn=1(1 −qn) =∞Xn=−∞(−1)n q32 (n−1/6)2 . (7.31)The identity (7.30) is known as the Euler pentagonal number theorem.Someone invariably asks why.

Those readers* with a serious interest in recre-ational mathematics will recall that there exists a series of k-gonal numbersgiven by(k −2)n2 −(k −4)n2.They describe the number of points it takes to build up successive embeddedk-sided equilateral ﬁgures (see ﬁg. 11 for the cases of triagonal (k = 3) num-bers, (n2 + n)/2 = 1, 3, 6, .

. .

; square (k = 4) numbers, n2 = 1, 4, 9, . .

. ; andpentagonal (k = 5) numbers, 12(3n2 −n) = 1, 5, 12, .

. .

). Generating functionsfor some of the other k-gonal numbers may be found in [50].••• ••• •••••• •• ••••••••••••••• ••••••••••• • •Fig.

11. First three triagonal, square, and pentagonal numbers.

(One of Euler’s original interests in (7.30) was evidently its combinatorialinterpretation. The left hand side is the generating function for E(n) −U(n),where E(n) is the number of partitions of n into an even number of unequalparts, and U(n) that into an odd number.

Thus (7.30) states that E(n) = U(n)except when n = 12(3k2 ± k), in which case E(n) −U(n) = (−1)k.)* I am grateful to M. Peskin for initiation in these matters.103To treat modular transformation properties of the ϑ’s and η under τ →−1/τ, we introduce the Poisson resummation formula in the form∞Xn=−∞f(nr) = 1r∞Xm=−∞efmr,(7.32)where the Fourier transform ef is deﬁned asef(y) =Z ∞−∞dx e−2πixyf(x) . (7.32) is easily established by substituting ef on the right hand side.

(Thenatural generalization of (7.32) to higher dimensions isXv∈Γf(v) = 1VXw∈Γ∗ef(w) ,where Γ is a lattice, Γ∗its dual (reciprocal), and V the volume of its unit cell. )Using the sum form (7.31) of the η function, we may apply (7.32) to ﬁndηq(−1/τ)= (−iτ)1/2ηq(τ).

(7.33)Similarly, from (7.29) we ﬁnd that under τ →−1/τ,ϑ2 →(−iτ)1/2ϑ4ϑ4 →(−iτ)1/2ϑ2ϑ3 →(−iτ)1/2ϑ3 . (7.34)We see that (7.12b) follows from (7.33) and (7.34).

For completeness, we tabu-late here as well the transformation properties under τ →τ + 1,ϑ3 ↔ϑ4ϑ2 →√i ϑ2η →eiπ12 η ,(7.35)as already used in (7.14a).We also note that the right hand side of (7.27) with w = e2πiz deﬁnes thefunction ϑ3(z, τ), in terms of which generalizations of all the ϑi ≡ϑi(0, τ)’s are104

writtenϑ3(z, τ) =∞Xn=−∞qn2/2e2πinzϑ4(z, τ) = ϑ3(z + 12, τ) =∞Xn=−∞(−1)nqn2/2e2πinzϑ1(z, τ) = −ieizq1/8ϑ4(z + τ2, τ) = i∞Xn=−∞(−1)nq12 (n−12 )2eiπ(2n−1)zϑ2(z, τ) = ϑ1(z + 12, τ) =∞Xn=−∞q12 (n−12 )2eiπ(2n−1)z . (7.36)The parameter z is useful for expressing the functional integral for complexfermions with boundary conditions twisted by an arbitrary phase, as mentionedat the end of subsection 7.1.

For representations of aﬃne algebras in terms offree fermions, z also plays the role of the Cartan angle in the aﬃne characters.In string theory where spacetime gauge symmetries are realized as aﬃne alge-bras on the worldsheet, the z dependence would then provide the dependenceof the partition function on background gauge ﬁelds. Properties of spacetimegauge and gravitational anomalies may then be probed via the modular trans-formation properties of the functions (7.36) (see [48] for more details).

The zdependence of the ϑ-functions also provides the coordinate dependence of cor-relation functions on the torus (for the critical Ising model for example, see[44]).Some other popular modular invariants are also readily constructed interms of free fermions. For eight chiral fermions, ψµ=1,8(z), all with the samespin structure, we ﬁnd12 A8A8−P8A8−A8P8!= 121η4ϑ43 −ϑ44 −ϑ42= 0 ,(7.37)where the signs are determined by invariance under (7.34) and (7.35).Astraightforward way to understand the vanishing of this quantity is to rec-ognize that 12ϑ43 −ϑ44= Pv∈Γ q12 v2, where Γ is the lattice composed of 4-vectors whose components vi ∈Z satisfy P4i=1 vi = 1 mod 2.

We also recognize10512ϑ42 = 12ϑ42+ϑ41= Pv∈Γ′ q12 v2 where Γ′ is composed of vectors with vi ∈Z+ 12and P4i=1 vi = 0 mod 2. But these two lattices are related by Γ′ = MΓ, whereM is the SO(4) transformationM =121212121212−12−1212−12−121212−1212−12,so it follows that Pv∈Γ q12 v2 = Pv∈Γ′ q12 v2.

(Acting on the weight lattice ofSO(8), the transformation M above is the triality rotation that exchanges thevector with one of the two spinors. )In superstring theory, the vanishing of (7.37) is the expression of spacetimesupersymmetry at one-loop order.

The ﬁrst two terms represent the contribu-tion to the spectrum of (GSO projected) spacetime bosons, and the last termthe spacetime fermions. Another way to see that (7.37) has to vanish is to recall[51] that a basis for modular forms of weight 2k is given by Gα2 Gβ3 (α, β ∈Z+,2α + 3β = k), where the Gk(τ) = P{m,n}̸={0,0}(mτ + n)−2k are the Eisensteinseries of weights 4 and 6 for k = 2, 3 respectively.

(A modular form of weight 2ksatisﬁes faτ+bcτ+d= (cτ + d)2kf(τ), so that f(τ)(dτ)k is invariant.) From themodular transformation properties (7.34) and (7.35), we see that ϑ43 −ϑ44 −ϑ42is a modular form of weight 2, of which there are none non-trivial, and hencemust vanish.For 16 chiral fermions, ψµ=1,16(z), we ﬁnd12 A16A16+ P16A16+ A16P16!= 121η8ϑ83 + ϑ84 + ϑ82=Pv∈Γ8 qv2/2η8,where the summation is over lattice vectors v in Γ8, the E8 root lattice.

Thisis a lattice composed of vectors whose components vi are either all integral,vi ∈Z, or half-integral, vi ∈Z + 12, and in either case their sum is even,P8i=1 vi = 0 mod 2 (the last a consequence of the GSO projection on evenfermion number in the A and P sectors).Actually, since 16 chiral fermions have c = 8, c = 0, the above combinationof spin structures has a leading q behavior of q−c/24 ∼q−1/3 so it is strictly106

speaking only modular covariant. (In this case that means that it picks up acube root of unity phase under τ →τ + 1; since S2 = 1, the only possiblenon-trivial phase for S would be −1, but this is excluded here by the otherrelation (ST )3 = 1.) To get a modular invariant, we cube the E8 character toﬁnd123 A16A16+ P16A16+ A16P16!3= 181η24ϑ83 + ϑ84 + ϑ823= j(q) = 1q + 744 + 196884q + .

. .

,where j is the famous modular invariant function (the coeﬃcients in whose q-expansion, excepting the constant term 744, are simply expressed in terms ofthe dimensions of the irreducible representations of the monster group (see [52]for a recent treatment with physicists in mind and for further references)).We can also generalize this construction to 16k chiral fermions, ψµ=1,16k(z),to get12 A16kA16k+ P16kA16k+ A16kP16k!= 121η8k (ϑ8k3 + ϑ8k4 + ϑ8k2 ) =Pv∈Γ8k qv2/2η8k,where the lattice Γ8k is deﬁned analogously to Γ8, i.e. again a lattice composedof vectors whose components vi are either all integral, vi ∈Z, or half-integral,vi ∈Z + 12, such that in either case their sum is even, P8ki=1 vi = 0 mod 2.The Γ8k are examples of even self-dual integer lattices.

(An integer lattice Γis such that vectors v ∈Γ have v2 ∈Z. The dual lattice Γ∗consists of allvectors w such that w · v ∈Z, and a self-dual lattice satisﬁes Γ = Γ∗.

See [51]for more details.) Modular covariant fermionic partition functions of the formconsidered here generically bosonize to theories of chiral bosons compactiﬁedon such lattices.1078.

Free bosons on a torusWe now continue our study of conformal ﬁeld theory on the torus to thenext simplest case, that of free bosons. This case aﬀords a surprising richnessof structure that begins to hint at the complexity of more general conformalﬁeld theories.8.1.

Partition functionIn the previous section, we calculated the partition functions (7.13) for freefermions with assorted boundary conditions on a torus by means of the Hamil-tonian interpretation in which the sum over Hilbert space states is implementedwith appropriate operator insertions. A similar procedure could be employedto calculate free bosonic partition functions.

To illustrate the alternative in-terpretation of partition functions as determinants of operators, however, weshall instead calculate the bosonic partition functions by means of a Lagrangianformulation in this section.Since we are dealing with a free ﬁeld theory with actionS = 12πZ∂X∂X ,(8.1)we can calculate functional integral exactly simply by taking proper accountof the boundary conditions. We assume a bosonic coordinate X ≡X + 2πrcompactiﬁed on a circle of radius r. That means when we calculate the func-tional integral, we need to consider all “instanton” sectors n′nwith boundaryconditionsX0(z + τ, z + τ) = X0(z, z) + 2πrn′X0(z + 1, z + 1) = X0(z, z) + 2πrn .The solutions to the classical equations of motion, ∂∂X0 = 0, with the aboveboundary conditions, areX(n′,n)0(z, z) = 2πr 12iτ2n′(z −z) + n(τz −τz).

(8.2)In each such sector, we also have a contribution from the ﬂuctuations aroundthe classical solution.108

The functional integral is easily evaluated using the normalization conven-tions of [53]. * (In general, functional integrals are deﬁned only up to an inﬁniteconstant so only their ratios are well-deﬁned, and any ambiguities are resolvedvia recourse to canonical quantization.

The prescription here is chosen to give aτ2 dependence consistent with modular invariance, and an overall normalizationconsistent with the Hamiltonian interpretation. A related calculation may befound in [54].) To carry out the DX integration, we separate the constant pieceby writing X(z, z) = eX +X′(z, z), where X′(z, z) is orthogonal to the constanteX, and write DX = d eX DX′.

We normalize the gaussian functional integral toRDδX e−12πR(δX)2 = 1, so thatZDδX′ e−12πR(δX)2= Zdx e−12πRx2 !−1=π12πR 1−1/2=√2τ2π.In (8.1), we have taken the measure to be 2idz∧dz (=4τ2 dσ1∧dσ0 in coordinatesz = σ1 + τσ0), so the integral on the torus is normalized toR1 = 4τ2. Theintegral over the constant piece eX, on the other hand, just gives 2πr.Now from (8.2), we have that ∂X(n′,n)0= 2πr2iτ2 (n′ −τn).

Substituting intothe action (8.1), together with the above normalization conventions, allows usto express the functional integral in the formZe−S = 2πr√2τ2π1det′ 1/2∞Xn,n′=−∞e−SX(n′,n)0= 2r√2τ21det′ 1/2∞Xn,n′=−∞e4τ2 12π2πr2iτ22(n′ −τn)(n′ −τn)= 2r√2τ21det′ 1/2∞Xn,n′=−∞e−2π1τ2 (n′r −τ1nr)2 + τ2 n2r2,(8.3)where≡−∂∂, and det ′ is a regularized determinant. * I am grateful to A. Cohen for his notes on the subject.109To evaluate det′as a formal product of eigenvalues, we work with abasis of eigenfunctionsψnm = e2πi12iτ2n(z −z) + m(τz −τz),single-valued under both z →z + 1, z →z + τ.

The regularized determinant isdeﬁned by omitting the eigenfunction with n = m = 0,det ′≡Y{m,n}̸={0,0}π2τ 22(n −τm)(n −τm) . (8.4)The inﬁnite product may be evaluated using ζ-function regularization (recallthat ζ(s) = P∞n=1 n−s, ζ(−1) = −112, ζ(0) = −12, ζ′(0) = −12 ln 2π).

In thisregularization scheme we have for example∞Yn=1a = aζ(0) = a−1/2and∞Yn=−∞a = a2ζ(0)+1 = 1 ,so that in particular for the product in (8.4), with m = n = 0 excluded, weﬁnd Q′(π2/τ 22 ) = τ22 /π2. Another identity in this scheme that we shall need isQ∞n=1 nα = e−αζ′(0) = (2π)α/2.The remainder of (8.4) is evaluated by means of the product formulaQ∞n=−∞(n + a) = a Q∞n=1(−n2)(1 −a2/n2) = 2i sinπa.

The result isdet ′=Y{m,n}̸={0,0}π2τ 22(n −mτ)(n −mτ)= τ 22π2 Yn̸=0n2Ym̸=0, n∈Z(n −mτ)(n −mτ)= τ 22π2 (2π)2Ym>0,n∈Z(n −mτ)(n + mτ)(n −mτ)(n + mτ)= 4τ 22Ym>0(e−πimτ −eπimτ)2(e−πimτ −eπimτ)2= 4τ 22Ym>0(qq)−m(1 −qm)2(1 −qm)2= 4τ 22 (qq)1/12 Ym>0(1 −qm)2(1 −qm)2 = 4τ 22 η2η2 ,110

so the relevant contribution to (8.3) is2r√2τ21det ′1/2=r2τ2r 1ηη . (8.5)Since under the modular transformation τ →−1/τ, we have τ2 →τ2/|τ|2, weverify modular invariance of (8.5) from the modular transformation property(7.33) of η.

Techniques identical to those used to derive (8.5) could also havebeen used to derive the fermion determinants (7.13). ((8.5) can also be com-pared with the result of section 4.2 of Cardy’s lectures.

For a general actiong4πR∂φ∂φ, with φ ≡φ+2πR, the “physical” quantity r =p g2 R is independentof rescaling of φ, and coincides with the usual radius for g = 2, as desired fromthe normalization of (2.14). We see that the right hand side of (8.5) takes theform g1/2R/(τ 1/22ηη), and for R = 1 agrees with Cardy’s eq.

(4.10)).We have now to consider the eﬀect of summing over the instanton sectors,or equivalently the interpretation of the momentum zero modes pL ≡α0, pR ≡α0. As usual in making the comparison between Lagrangian and Hamiltonianformulations, the summation over the winding n′ in the “time” direction in(8.3) can be exchanged for a sum over a conjugate momentum by performinga Poisson resummation (7.32).Thus we ﬁrst take the Fourier transform off(n′r) = e−(2π/τ2)(n′r−τ1nr)2,˜f(p) =Z ∞−∞dx e2πixpf(x) =rτ22 e2πiτ1nrp −12πτ2p2.Then we substitute (7.32) and (8.5) to express (8.3) asZe−S = 1ηη∞Xn,m=−∞e−2πτ2n2r2 + 2πiτ1nm −12πτ2(m/r)2= 1ηη∞Xn,m=−∞q12( m2r + nr)2q12( m2r −nr)2= 1ηη∞Xn,m=−∞q12( p2 + w)2q12( p2 −w)2.

(8.6)In the last line we have introduced the momentum p = m/r and the windingw = nr. We see that this conjugate momentum is quantized in units of 1/r.

It111is convenient to deﬁne as well pL,R = p/2 ± w = m/2r ± nr, and express theresult for the partition function in the formZcirc(r) =Ze−S = 1ηη∞Xn,m=−∞q12p2Lq12p2R . (8.7)(Generalizations of (8.7) to higher dimensions and additional background ﬁeldsare derived from the Hamiltonian and Lagrangian points of view in [55].

)To complete the identiﬁcation with the Hamiltonian trace over Hilbertspace states, we now recall the alternative interpretation of (8.7) as (qq)−c/24tr qL0qL0.We infer an inﬁnite number of Hilbert space sectors |m, n⟩, labeled by m, n =−∞, ∞, for whichL0|m, n⟩= 12 m2r + nr2|m, n⟩andL0|m, n⟩= 12 m2r −nr2|m, n⟩. (8.8)We see that L0 = P α−mαm + 12p2L, with α0 ≡pL = ( p2 + w), and L0 =P α−mαm + 12p2R, with α0 ≡pR = p2 −w.

We also see that the |m, n⟩state hasenergy and momentum eigenvaluesH = L0 + L0 = 12(p2L + p2R) = 14p2 + w2 = m24r2 + n2r2P = L0 −L0 = 12(p2L −p2R) = pw = mn ∈Z . (8.9)(We note brieﬂy how the eigenvalues of α0 and α0 can also be determineddirectly in the Hamiltonian point of view.

Since α0 + α0 is the zero mode ofthe momentum ∂X conjugate to the coordinate X, with periodicity 2πr, ithas eigenvalues quantized as p = m/r (m ∈Z). Mutual locality, i.e.

integereigenvalues of L0 −L0, of the operators that create momentum/winding statesthen ﬁxes the diﬀerence α0 −α0 = 2w = 2nr. )The factor of (ηη)−1 in (8.5) also has a straightforward Hamiltonian inter-pretation.

The bosonic Fock space generated by α−n consists of all states ofthe form |m, n⟩, α−n|m, n⟩, α2−n|m, n⟩, . .

. .

Calculating as for the fermionic112

case (before (7.13)) and ignoring for the moment the zero mode contribution,we ﬁnd for the trace in the |m, n⟩Hilbert space sectortrqL0 = tr qP∞n=1 α−nαn =∞Yn=1(1 + qn + q2n + . .

.) =∞Yn=111 −qn ,as expected for Bose-Einstein statistics.

Including the α−n’s as well, we have(qq)−c/24 trqL0qL0 = (qq)−1/24∞XN,M=0P(N) P(M) qNqM = 1ηη ,where the product P(N) P(M) just counts the total number of statesα−n1 . .

. α−nm α−m1 .

. .

α−mk|m, n⟩with Pmi=1 ni = N, Pkj=1 mj = M.The result (8.6) is easily veriﬁed to be modular invariant. Under τ →τ +1,each term in (8.6) acquires a phase exp 2πi 12(p2L −p2R), which is equal to unityby the second relation in (8.9).

Under τ →−1/τ, we note that the boundaryconditions in the Lagrangian formulation transform as n′n→(−n)n′, sowe see how summation over n′ and n may result in a modular invariant sum.We see moreover that the roles of “space” and “time” are interchanged byτ →−1/τ, so it is clear that to verify modular invariance we should performa Poisson resummation over both m and n in (8.6). Doing that and using thetransformation property (7.33) of η indeed establishes the modular invarianceof (8.6).

(Modular invariance of (8.6) can be understood in a more general frame-work as follows[56]. Consider (pL, pR) to be a vector in a two-dimensional spacewith Lorentzian signature, so that (pL, pR) · (p′L, p′R) ≡pLp′L −pRp′R.

We maywrite arbitrary lattice vectors as(pL, pR) = m 12r, 12r+ n(r, −r) = mk + nk ,where the basis vectors k, k satisfy kk = 1, k2 = k2 = 0. k and k generate whatis known as an even self-dual Lorentzian integer lattice Γ1,1. (Self-duality here113is deﬁned for Lorentzian signature just as was deﬁned for Euclidean signatureat the end of section 7.) The general statement is that partition functions ofthe formZΓr,s =1ηrηsX(pL,pR)∈Γr,sq12p2Lq12p2Rare modular covariant provided that Γr,s is an r + s dimensional even self-dual Lorentzian lattice of signature (r, s).

The even property, p2L −p2R ∈2Z,guarantees invariance under τ →τ + 1 (up to a possible phase from η−rη−swhen r −s ̸= 0 mod 24), while the self-duality property guarantees invarianceunder τ →−1/τ. Such lattices exist in every dimension d = r −s = 0 mod 8,and for r, s ̸= 0 are unique up to SO(r, s) transformations.

In the Euclidean casediscussed at the end of section 7, on the other hand, there are a ﬁnite number ofsuch lattices for every d = r = 0 mod 8, unique up to SO(d) transformations. )We close here by pointing out that the partition function (8.7) can also beexpressed in terms of c = 1 Virasoro characters.

To see what these characterslook like, we recall from the results of section 4 that there are no null statesfor c > 1 except at h = 0, and none at c = 1 except at h = n2/4 (n ∈Z). Forc > 1, this means that the Virasoro characters take the formχh̸=0(q) = 1η qh−(c−1)/24(8.10a)χ0(q) = 1η q−(c−1)/24(1 −q)(8.10b)(the extra factor of (1 −q) in the latter due to L−1|0⟩= 0).

At c = 1 (8.10a)remains true for h ̸= n2/4 but for h = n2/4, due to the null states the charactersare insteadχn2/4(q) = 1ηqn2/4 −q(n+2)2/4= 1η qn2/41 −qn+1. (8.11)Unlike the Ising partition function (7.22), which was expressible in terms of aﬁnite number of Virasoro characters, the expression for (8.7) would involve aninﬁnite summation.

This is consistent with result of [43] cited after (7.20), thatfor c ≥1 modular invariance requires an inﬁnite number of Virasoro primaries.114

8.2. FermionizationIn earlier sections we have alluded to the fact that two chiral (c =12)fermions are equivalent to a chiral (c = 1) boson.In this subsection weshall illustrate this correspondence explicitly on the torus.

Consider two Diracfermions comprised of ψ1(z), ψ2(z) and ψ1(z), ψ2(z). By Dirac fermion on thetorus [57], we mean to take all these fermions to have the same spin structure.The partition function for such fermions is consequently given by the modularinvariant combination of spin structuresZDirac = 12 A2A2A2A2+ P2P2A2A2+ A2A2P2P2+ P2P2P2P2!= 12ϑ3η2+ϑ4η2+ϑ2η2+ϑ1η2,(8.12)where we have for convenience chosen the projection on (−1)F = +1 states inthe PP sector.The partition functions (7.13) were all derived from the standpoint of theexpressions of the ϑ-functions as inﬁnite products.

In (7.29), however, we haveseen that these functions also admit expressions as inﬁnite sums via the Jacobitriple product identity. We shall now see that this equivalence is the expressionof bosonization of fermions on the torus.

Substituting the sum forms of theϑ-functions in (8.12), we ﬁndZDirac = 1ηη∞Xn,m=−∞q12 n2q12 m2 + q12 (n+ 12 )2q12 (m+ 12 )2 121 + (−1)n+m= 1ηη∞Xn,m′=−∞q12 (n+m′)2q12 (n−m′)2 + q12 (n+ 12 +m′)2q12 (n+ 12 −m′)2= 1ηη∞Xn,m=−∞q12( m2 + n)2q12( m2 −n)2= Zcirc(r = 1) ,(8.13)equal to the bosonic partition function (8.7) at radius r = 1. (In (8.13) we haveused the property that 121+(−1)n+macts as a projection operator, projectingonto terms in the summation with n + m even, automatically implemented in115the next line by the reparametrization of the summation in terms of n andm′.) Recalling that the vertex operators e±ix(z) have conformal weight h = 12,it is not surprising that (8.12) emerges as the bosonic partition function atradius r = 1.

It is precisely at this radius that the vertex operators e±ix(z)are suitably single-valued under x →x + 2π/r = x + 2π. The connection withthe real fermions above is given, as in (6.14), by e±ix(z) =i√2ψ1(z) ± iψ2(z),e±ix(z) =i√2ψ1(z) ± iψ2(z).By comparing (8.12) and (8.13) we can identify the states in the bosonicform of the partition function that correspond to the states in the various sectorsof the fermionic form.

The partition function only includes states that survivethe GSO projection onto (−1)F = +1 (where F = F1 + F2 + F 1 + F 2 is thetotal fermion number). Thus we need to extend the range of n in the last lineof (8.13) to n ∈Z/2 to construct a non-local covering theory that includes aswell the (−1)F = −1 states prior to projection.

Then the states of the A2A2fermionic sector with (−1)F = ±1 are given respectively by {n ∈Z, m ∈2Z}and {n ∈Z+ 12, m ∈2Z+1}; while the states of the P2P2 fermionic sector with(−1)F = ±1 are given respectively by {n ∈Z, m ∈2Z + 1} and {n ∈Z + 12,m ∈2Z}. Thus we have seen how the classical identity (7.27) becomes thestatement of bosonization of fermions on the torus.

(The generalization ofthese results to arbitrary genus Riemann surfaces, including the interpretationof modular invariance at higher genus as enforcing certain projections, may befound in [41][58]. )If we relax the restriction in (8.12) that all fermions ψ1,2, ψ1,2 have thesame spin structure, then we can construct another obvious c = c = 1 modularinvariant combination,Z2Ising = 122 AAAA+ PPAA+ AAPP!

AAAA+ PPAA+ AAPP!= 14ϑ3η +ϑ4η +ϑ2η2. (8.14)Following [57], we refer to the choice of independent boundary conditions forψ1, ψ1 and ψ2, ψ2 as specifying two Majorana fermions (as opposed to a single116

Dirac fermion). The partition function (8.14) is of course the square of the Isingpartition function (7.22).It is natural to ask whether (8.14) as well has a representation in termsof a free boson.

It is ﬁrst of all straightforward to see that (8.14) does notcorrespond to (8.7) for any value of r. (For example, one may note that thespectrum of (8.14) has two ( 116, 116) states. But (8.7) has two such states onlyfor r =√2 and r = 1/2√2, at which points it is easy to see that there are no( 12, 12) states.) The distinction between (8.12) and (8.14) is the decoupling ofthe spin structures of the two Majorana fermions.

Due to the correspondenceψ1,2 ∼(eix ± e−ix), we see that the bosonic operation x →−x, taking ψ1 →ψ1and ψ2 →−ψ2 (and similarly for ψ1,2), distinguishes between ψ1, ψ1 and ψ2, ψ2.The key to constructing a bosonic realization of (8.14), then, is to implementsomehow the symmetry action x →−x on (8.7). This is provided by the notionof an orbifold, to which we now turn.8.3.

Orbifolds in generalOrbifolds arise in a purely geometric context by generalizing the notion ofmanifolds to allow a discrete set of singular points. Consider a manifold Mwith a discrete group action G : M →M.

This action is said to possess a ﬁxedpoint x ∈M if for some g ∈G (g ̸= identity), we have gx = x. The quotientspace M/G constructed by identifying points under the equivalence relationx ∼gx for all g ∈G deﬁnes in general an orbifold.

If the group G acts freely(no ﬁxed points) then we have the special case of orbifold which is an ordinarymanifold. Otherwise the points of the orbifold corresponding to the ﬁxed pointset have discrete identiﬁcations of their tangent spaces, and are not manifoldpoints.

(A slightly more general deﬁnition of orbifold is to require only that theabove condition hold coordinate patch by coordinate patch.) A simple exampleis provided by the circle, M = S1, coordinatized by x ≡x + 2πr, with groupaction G = Z2 : S1 →S1 deﬁned by the generator g : x →−x.

This groupaction has ﬁxed points at x = 0 and x = πr, and we see in ﬁg. 12 that theS1/Z2 orbifold is topologically a line segment.117xxxxxFig.

12. The orbifold S1/Z2.In conformal ﬁeld theory, the notion of orbifold acquires a more generalizedmeaning.

It becomes a heuristic for taking a given modular invariant theoryT , whose Hilbert space admits a discrete symmetry G consistent with the in-teractions or operator algebra of the theory, and constructing a “modded-out”theory T /G that is also modular invariant[59].Orbifold conformal ﬁeld theories occasionally have a geometric interpre-tation as σ-models whose target space is the geometrical orbifold discussed inthe previous paragraph. This we shall conﬁrm momentarily in the case of theS1/Z2 example.

We shall also see examples however where the geometrical in-terpretation is either ambiguous or non-existent. Consequently it is frequentlypreferable to regard orbifold conformal ﬁeld theories from the more abstractstandpoint of modding out a modular invariant theory by a Hilbert space sym-metry.

(Historically, orbifolds were introduced into conformal ﬁeld theory [59](see also [60]) via string theory as a way to approximate conformal ﬁeld the-ory on “Calabi-Yau” manifolds. Even before the “phenomenological” interestin the matter subsided, orbifold conformal ﬁeld theories were noted to possessmany interesting features in their own right, and in particular enlarged theplayground of tractable conformal ﬁeld theories.

)The construction of an orbifold conformal ﬁeld theory T /G begins with aHilbert space projection onto G invariant states. It is convenient to representthis projection in Lagrangian form as1|G|Xg∈Gg1,(8.15)118

where g1represents boundary conditions on any generic ﬁelds x in the theorytwisted by g in the “time” direction of the torus, i.e. x(z+τ) = gx(z).

In Hamil-tonian language such twisted boundary conditions correspond to insertion ofthe operator realizations of group elements g in the trace over states, and hence(8.15) corresponds to the insertion of the projection operator P =1|G|Pg∈G g.But (8.15) is evidently not modular invariant as it stands since under S :τ →−1/τ for example we have g1→1g(this is easily veriﬁed by shiftingappropriately along the two cycles of the torus using the representation S =T−1UT−1 given before (7.12)). Under τ →τ +n we have moreover that 1g→gng, so we easily infer the general resultgh→gahbgchdunderτ →aτ + bcτ + d ,(8.16)for g, h ∈G such that gh = hg.

(We note that there seems an ambiguity in(8.16) due to the possibility of taking a, b, c, d to minus themselves. But for self-conjugate ﬁelds, for which charge conjugation C = 1 and the modular groupis realized as PSL(2,Z), ghand g−1h−1are equal.

In a more general contextone would have to implement S2 = (ST )3 = C.)To have a chance of recovering a modular invariant partition function, wethus need to consider as well twists by h in the “space” direction of the torus,x(z + 1) = hx(z), and deﬁneZT /G ≡Xh∈G1|G|Xg∈Ggh=1|G|Xg,h∈Ggh. (8.17)The boundary conditions in individual terms of (8.17) are ambiguous forx(z + τ + 1) unless gh = hg.

Thus in the case of non-abelian groups G, thesummation in (8.17) should be restricted only to mutually commuting bound-ary conditions gh = hg.From (8.16) we see that modular transformationsof such boundary conditions automatically preserve this property. Moreoverwe see that (8.17) contains closed sums over modular orbits so it is formally119invariant under modular transformations.

(In the following we shall considerfor simplicity only symmetry actions that act symmetrically on holomorphicand anti-holomorphic ﬁelds, so modular invariance of (8.17) is more or less im-mediate. For more general asymmetric actions, additional conditions must beimposed on the eigenvalues of the realizations of the group elements to insurethat no phase ambiguities occur under closed loops of modular transformationsthat restore the original boundary conditions [59][61][62].) We also note thatthe orbifold prescription, changing only boundary conditions of ﬁelds via a sym-metry of the stress-energy tensor, always gives a theory with the same value ofthe central charge c.For G abelian, the operator interpretation of (8.17) is immediate.TheHilbert space decomposes into a set of twisted sectors labeled by h, and in eachtwisted sector there is a projection onto G invariant states.

A similar interpre-tation exists as well for the non-abelian case, although then it is necessary torecognize that twisted sectors should instead be labeled by conjugacy classesCi of G. This is because if we consider ﬁelds hx(z) translated by some h, thenthe g twisted sector, hx(z + 1) = ghx(z), is manifestly equivalent to the h−1ghtwisted sector, x(z + 1) = h−1ghx(z). Now the number of elements g ∈Ni ⊂Gthat commute with a given element h ∈Ci ⊂G depends only on the conjugacyclass Ci of h (the group Ni is known as the stabilizer group, or little group, of Ciand is deﬁned only up to conjugation).

This number is given by |Ni| = |G|/|Ci|,where |Ci| is the order of Ci. In the non-abelian case, we may thus rewrite thesummation in (8.17) as1|G|Xhg=ghgh=Xi1|Ni|Xg∈NigCi,manifesting the interpretation of the summation over g as a properly normalizedprojection onto states invariant under the stabilizer group Ni in each twistedsector labeled by Ci.While we have discussed here only the construction of the orbifold par-tition function (8.17), we point out that the orbifold prescription (at least in120

the abelian case) also allows one to construct all correlation functions in prin-ciple[63]. We also point out that we have been a bit cavalier in presenting thesummation in (8.17).

In general such a summation will decompose into dis-tinct modular orbits, i.e. distinct groups of terms each of which is individuallymodular invariant.

The full summation in (8.17) is nonetheless required for aconsistent operator interpretation of the theory (or equivalently for modularinvariance on higher genus Riemann surfaces). There may remain however dis-tinct choices of relative phases between the diﬀerent orbits in (8.17) (just asin the case of the Ising model (7.22)), corresponding in operator language todiﬀerent choices of projections in twisted sectors.

In [61], the diﬀerent possibleorbifold theories T /G that may result in this manner were shown to be classi-ﬁed by the second cohomology group H2(G, U(1)), which equivalently classiﬁesthe projective representations of the group G. (Torsion-related theories canalso be viewed to result from the existence of an automorphism of the fusionrules of the chiral algebra of a theory. Instead of a diagonal sesquilinear com-bination P χiχi of chiral characters as the partition function, we would haveP χi Pij χj, where P is a permutation of the chiral characters that preservesthe fusion rules.)8.4.

S1/Z2 orbifoldWe now employ the general orbifold formalism introduced above to con-struct a G = Z2 orbifold conformal theory of the free bosonic ﬁeld theory (8.1).We ﬁrst note that the action (8.1) is invariant under g : X →−X, under whichαn →−αn and αn →−αn. (Recall that X(z, z) = 12x(z) + x(z), and theαn’s and αn’s are respectively the modes of i∂x(z) and i∂x(z).) These includethe momentum zero modes pL = α0 and pR = α0 so the action of g on theHilbert space sectors |m, n⟩of (8.8) is given by |m, n⟩→| −m, −n⟩.121The general prescription (8.17) for the T /G orbifold partition functionreduces for G = Z2 toZorb(r) = 12 +++ −++ +−+ −−!= (qq)−1/24tr(+)12(1 + g)qL0qL0+ (qq)−1/24tr(−)12(1 + g)qL0qL0 .

(8.18)In the ﬁrst line of (8.18), we use ± to represent periodic and anti-periodicboundary conditions on the free boson X along the two cycles of the torus. Inthe second line tr(+) denotes the trace in the untwisted Hilbert space sectorH(+)corresponding to X(z + 1, z + 1) = X(z, z), and tr(−) denotes the tracein the twisted sector H(−)corresponding to X(z + 1, z + 1) = −X(z, z).The above symmetry actions induced by g : X →−X imply that theuntwisted Hilbert space H(+) decomposes into g = ±1 eigenspaces H±(+) asH+(+) =nα−n1 · · · α−nℓα−nℓ+1 · · · α−n2k|m, n⟩+ | −m, −n⟩o+nα−n1 · · · α−nℓα−nℓ+1 · · · α−n2k+1|m, n⟩−| −m, −n⟩o,H−(+) =nα−n1 · · · α−nℓα−nℓ+1 · · · α−n2k+1|m, n⟩+ | −m, −n⟩o+nα−n1 · · · α−nℓα−nℓ+1 · · · α−n2k|m, n⟩−| −m, −n⟩o,(8.19)where ni ∈Z+.

We see that in each sector with {m, n} ̸= {0, 0}, exactly halfthe states at each level of L0 and L0 have eigenvalue g = +1. To calculatetr(+)12(1 + g)qL0qL0, we note that g|m, n⟩= | −m, −n⟩, so that the trace withg inserted receives only contributions from the states built with α’s and α’s on|0, 0⟩.

The overall trace over states with eigenvalue g = +1 in the untwisted122

sector is thus given by(qq)−1/24 trH+(+)qL0qL0 = (qq)−1/24 tr(+)12(1 + g)qL0qL0= 121ηη∞Xm,n=−∞q12( m2r + nr)2q12( m2r −nr)2+ 12(qq)−1/24Q∞n=1(1 + qn)(1 + qn)= 12Zcirc(r) +ηϑ2 . (8.20)Next we need to construct the twisted Hilbert space H(−).

The ﬁrst sub-tlety is that there are actually two dimension ( 116, 116) twist operators σ1,2,satisfying∂x(z) σ1,2(w, w) ∼(z −w)−1/2 τ1,2(w, w)∂x(z) σ1,2(w, w) ∼(z −w)−1/2 eτ1,2(w, w)(8.21)as in (6.11). (Here the dimensions of the excited twist operators τ1,2 and eτ1,2 aregiven respectively by 916, 116and 116, 916.

The states identiﬁed with τ1,2(0)|0⟩and eτ1,2(0)|0⟩can also be written α−1/2 116, 1161,2 and α−1/2 116, 1161,2.) Geo-metrically the existence of two twist operators results from the two ﬁxed pointsof the symmetry action g : X →−X, as depicted in ﬁg.

12, and two distinctHilbert spaces are built on top of each of these two ﬁxed point sectors. Equiv-alently, we note two ways of realizing g, either as x →−x or as x →2π −x,and each realization is implemented by a diﬀerent twist operator.

The multi-plicity is also easily understood in terms of the fermionic form of the current,∂x ∼ψ1ψ2. Then the two twist operators may be constructed explicitly interms of the individual twist operators for each of the two fermions.

Finally themultiplicity of vacuum states can also be veriﬁed by performing the modulartransformationτ →−1/τ :−+→+−to construct the trace +−over the spectrum of the unprojected twisted sectorfrom the trace −+over the untwisted sector with the operator insertion of g.123Denoting the two 116, 116twisted sector ground states by 116, 1161,2, weﬁnd that the twisted Hilbert space H(−) decomposes into g = ±1 eigenspacesH±(−) asH+(−) =nα−n1 · · · α−nℓα−nℓ+1 · · · α−n2k 116, 1161,2oH−(−) =nα−n1 · · · α−nℓα−nℓ+1 · · · α−n2k+1 116, 1161,2o,(8.22)where the moding is now ni ∈(Z + 12)+. The overall trace over states witheigenvalue g = +1 in the twisted Z2 sector is thus given by(qq)−1/24 trH+(−)qL0qL0 = (qq)−1/24 tr(−)12(1 + g)qL0qL0= 2 12 (qq)1/48Q∞n=1(1 −qn−1/2)(1 −qn−1/2)+(qq)1/48Q∞n=1(1 + qn−1/2)(1 + qn−1/2)!=ηϑ4 +ηϑ3 .

(8.23)Now if we substitute (8.20) and (8.23) into (8.18), and use the identityϑ2ϑ3ϑ4 = 2η3, we ﬁnd that the orbifold partition function satisﬁesZorb(r) = 12 +++ −++ +−+ −−!= 12Zcirc(r) + |ϑ3ϑ4|ηη+ |ϑ2ϑ3|ηη+ |ϑ2ϑ4|ηη. (8.24)We note that modular invariance of (8.24) can be easily veriﬁed from the trans-formation properties (7.14).We may now at last return to the point left open earlier, namely the bosonicrealization of the Ising2 partition function (8.14).

From (8.12) and (8.24) weevaluate Zorb(r = 1),Zorb(1) = 12|ϑ3|2 + |ϑ4|2 + |ϑ2|22|η|2+ 12|ϑ3ϑ4||η|2+ |ϑ2ϑ3||η|2+ |ϑ2ϑ4||η|2= 14ϑ3η +ϑ4η +ϑ2η2= Z2Ising .124

We thus see that two Majorana fermions bosonize onto an S1/Z2 orbifold atradius r = 1. The Z2Ising theory can also be constructed directly as an orbifoldfrom the ZDirac theory by modding out by the Z2 symmetry ψ2 →−ψ2, ψ2 →−ψ2.It is useful to consider the generic symmetry possessed by the family oftheories (8.24).

The two twist operators σ1,2 of (8.21) and their operator al-gebras are unaﬀected by changes in the radius r.The theory consequentlyadmits a generic symmetry generated by separately taking either σ1 →−σ1 orσ2 →−σ2, or interchanging the two, σ1 ↔σ2. The group so generated is iso-morphic to D4, the eight element symmetry group of the square.

(This groupmay also be represented in terms of Pauli matrices as {±1, ±σx, ±iσy, ±σz},with the order four element iσy, say, corresponding to σ1 →−σ2, σ2 →σ1).D4 is also the generic symmetry group of a lattice model constructed bycoupling together two Ising models, known as the Ashkin-Teller model. If wedenote the two Ising spins by σ and σ′, then the Ashkin-Teller action is givenbySAT = −K2X⟨ij⟩σiσj + σ′iσ′j−K4X⟨ij⟩σiσjσ′iσ′j ,(8.25)where the summation is over nearest neighbor sites ⟨ij⟩on a square lattice.The D4 symmetry group in this case is generated by separately taking eitherσ →−σ or σ′ →−σ′, or interchanging the two, σ ↔σ′, on all sites.

Sincethere are now two parameters, (8.25) has a line of critical points, given by theself-duality condition exp(−2K4) = sinh 2K2. As shown in [64], the criticalpartition function for the Ashkin-Teller model on a torus takes identically theform (8.24), with sin(πr2/4) = 12 coth 2K2.

For K4 = 0, (8.25) simply reducesto two uncoupled copies of the Ising model, with critical point partition function(8.14). That is the point r = 1 on the orbifold line.

Calculations of the criticalcorrelation functions in the Ashkin-Teller model from the bosonic point of viewmay be found in [65].In general the Ashkin-Teller model can be regarded as two Ising modelscoupled via their energy densities ε1 and ε2. On the critical line this inter-action takes the form of a four-fermion interaction ε1ε2 = ψ1ψ1ψ2ψ2.

This125four-fermion interaction deﬁnes what is known as the massless Thirring model.Although seemingly an interacting model of continuum fermions, properly de-scribed it is really just a free theory since in bosonic form we see that theinteraction simply changes the radius of a free boson. (A recent pedagogicaltreatment with some generalizations and references to the earlier literature maybe found in [66].) At radius r =√2 the partition function Zorb(√2) turns outto have a full S4 permutation symmetry and coincides with the critical partitionfunction of the 4-state Potts model on the torus [67][68].8.5.

Orbifold commentsIt may seem that an orbifold theory is somehow less fundamental than theoriginal theory. In the case of abelian orbifolds we shall now see that a theoryand its orbifold stand on equal footing.

Let us ﬁrst consider the case of a G = Z2orbifold. Then the orbifold theory always possesses as well a Z2 symmetry,generated by taking all states in the Z2 twisted sectors (or equivalently theoperators that create them) to minus themselves, i.e.eg :±−→−±−.From the geometrical point of view, for example, it is clear that acting twicewith the twist X →−X takes us back to the untwisted sector.

This is reﬂectedin the interactions (operator products) of twist operators.If we denote the partition function for the orbifold theory by ++′, thenwe can mod out the orbifold theory by its Z2 symmetry by constructing in turn,++′ = 12 +++ −++ +−+ −−!,−+′ = 12 +++ −+−+−−−−!,τ →−1τ⇒+−′ = 12 +++ +−−−+−−−!,τ →τ + 1⇒−−′ = 12 +++ −−−−+−+−!.126

The second line follows from the deﬁnition of the operator insertion of thesymmetry generator eg, and the third and fourth lines follow by performing theindicated modular transformations. The result of orbifolding the orbifold isthus12 ++′ + −+′ + +−′ + −−′!=++,and we see that the original theory ++and the orbifold theory ++′ standon symmetrical footing, each a Z2 orbifold of the other.It is easy to generalize this to a Zn orbifold, and consequently to an ar-bitrary abelian orbifold.

If we let the Zn be generated by an element g ∈Zn,with gn=identity, then the spectrum of the orbifold theory is constructed byprojecting onto Zn invariant states in each of the n twisted sectors labeled bygj (j = 0, . .

. , n −1).

The orbifold theory in this case has an obvious Zn sym-metry, given by assigning the phase ωj to the gj twisted sector, where ωn = 1.The statement that this is a symmetry of the operator algebra of the orbifoldtheory is just the fact that the selection rules allow three point functions for agj1 twist operator and a gj2 twist operator only with a g−j1−j2 twist operator.Straightforward generalization of the argument given above for the G = Z2case shows that modding out a Zn orbifold by this Zn symmetry gives back theoriginal theory. For a non-abelian orbifold, on the other hand, the symmetrygroup is only G/[G, G], where [G, G] is the commutator subgroup (generatedby all elements of the form ghg−1h−1 ∈G), so in general this procedure cannotbe used to undo a non-abelian orbifold (except if the group is solvable).As another class of examples of Z2 orbifolds, this time without an obviousgeometrical interpretation, we consider conformal ﬁeld theories built from anymember of the c < 1 discrete series.

To identify the Z2 symmetry of their op-erator algebras, it is convenient to retain the operators of the (double-counted)conformal grid with p + q = even, as indicated by ± in the checkerboard pat-tern of ﬁg. 13.

We indicate the operators ϕ(+) with both p and q even by +,and operators ϕ(−) with both p and q odd by −.The operators left blankare redundant in the conformal grid. The only non-vanishing operator prod-uct coeﬃcients allowed by the selection rules described in subsection 5.3 are of127the form C+++ and C+−−(i.e.

with an even number of (−)-type operators, inaccord with their “spinorial” nature). The conformal ﬁeld theories built fromthese models therefore possess an automatic Z2 symmetry ϕ(±) →±ϕ(±).↑qp →++−−++−−++Fig.

13. Z2 symmetry of c < 1 fusion rules.We can thus take for example any of the c < 1 theories with partitionfunction given by the diagonal modular invariant combination of characters,i.e.

any member of what is known as the A series, and mod out by this Z2symmetry acting say only on the holomorphic part.That means we throwout the odd p, q operators, non-invariant under the symmetry, and then usea τ →−1/τ transformation to construct the twisted sector.The resultingorbifold theory turns out to have a non-diagonal partition function, representingthe corresponding member of the D series. The D series models equally haveZ2 symmetries, modding out by which takes us back to the corresponding Aseries models.

Further discussion of the A and D series may be found in Zuber’slectures and in section 9.8.6. Marginal operatorsA feature that distinguishes the c = 1 models Zcirc(r) and Zorb(r) consid-ered here from the c < 1 models is the existence of a parameter r that labelsa continuous family of theories.

This is related to the possession by the formermodels of dimension (1,1) operators, known as marginal operators. (More gen-erally, operators of conformal weight (h, h) are said to be relevant if h + h < 2and irrelevant if h+h > 2.) Deformations of a conformal ﬁeld theory, preserving128

the inﬁnite conformal symmetry and central charge c, are generated by ﬁeldsVi of conformal dimension (1,1) [69]. To ﬁrst order, the perturbations theygenerate can be represented in the path integral as an addition to the action,δS = δgiRdzdz Vi(z, z), or equivalently in the correlation function of productsof operators O as δ⟨O ⟩= δgiRdzdzVi(z, z)O.

It is clear that a conformalweight (1,1) operator is required to preserve conformal invariance of the actionat least at the classical level.In the case of the circle theory (8.1), we have the obvious (1,1) operatorV = ∂X ∂X. We see that perturbing by this operator, since it is proportional tothe Lagrangian, just changes the overall normalization of the action, which bya rescaling of X can be absorbed into a change in the radius r. The operator V ,invariant under X →−X, evidently survives the Z2 orbifold projection in theuntwisted sector, and remains to generate changes in the radius of the orbifoldtheory (8.24).

(See [70] for further details concerning the marginal operators inc = 1 theories.) (In the Ashkin-Teller language of (8.25), the marginal operatorat the two Ising decoupling point is given by V = ε1ε2.

This is the Ashkin-Tellerinteraction coupling the two Ising energy operators. )In general whenever there exists a generic symmetry of a continuous familyof modular invariant conformal ﬁeld theories, modding out by the symmetrygives another continuous family of (orbifold) theories.

From the operator pointof view, this may be expressed as the fact the marginal operators generating theoriginal family of theories are invariant under the symmetry. Hence they survivethe projection in the untwisted sector of the orbifold theory and continue togenerate a family of conformal theories.The mere existence of (1,1) operators is not suﬃcient, however, to resultin families of conformal theories.

An additional “integrability condition” mustbe satisﬁed [69] to guarantee that the perturbation generated by the marginaloperator does not act to change its own conformal weight from (1,1). In thecase of a single marginal operator V as above, this reduces in leading order tothe requirement that there be no term of the form CV V V (z −w)−1(z −w)−1 V129in the operator product of V with itself.Otherwise the two-point functionV (z, z)V (w, w)= (z −w)−2(z −w)−2 varies according toδV (z, z)V (w, w)= δgZd2z′ V (z, z) V (w, w) V (z′, z′)= δg 2πCV V V (z −w)−2(z −w)−2 log |z −w|2,showing that the conformal weight of V is shifted to (h, ¯h) = (1−δg πCV V V , 1−δg πCV V V ) under the perturbation generated by V .V would therefore notremain marginal away from the point of departure, and could not be used togenerate a one-parameter family of conformal theories.To higher orders, we need to require as well the vanishing of integrals of(n + 2)-point functions (δg)nV (z, z)V (w, w) QiRd2z′i V (z′i, z′i)to insure thatthe 2-point function remains unperturbed.

If this is the case, so the operatorV generates a one-parameter family of conformal theories, then it is calledeither exactly marginal, truly marginal, critical, persistent, or integrable, etc.In general, it is diﬃcult to verify by examination of (n + 2)-point functionsthat an operator remains marginal to all orders. In some cases, however, it ispossible[71] to show integrability to all orders just by verifying that the 4-pointfunction takes the form of that of the marginal operator ∂X∂X for a free boson.8.7.

The space of c = 1 theoriesIt can be veriﬁed from (8.7) and (8.24) that the circle and orbifold partitionfunctions coincide atZorbr =1√2= Zcircr =√2. (8.26)Although such an analysis of the partition functions shows the two theoriesat the above radii have identical spectra, it is not necessarily the case thatthey are identical theories, i.e.

that their operator algebras are as well identical(although two conformal ﬁeld theories whose partition functions coincide onarbitrary genus Riemann surfaces can probably be shown to be equivalent inthis sense). We shall now proceed to show that the equivalence (8.26) doesindeed hold at the level of the operator algebras of the theories by making130

use of a higher symmetry, in this case an aﬃne SU(2) × SU(2) symmetry,possessed by the circle theory at r = 1/√2. Equivalences such as (8.26) showthat geometrical interpretations of the target spaces of these models, as alludedto earlier, can be ambiguous at times.

The geometrical data of a target spaceprobed by a conformal ﬁeld theory (or a string theory) can be very diﬀerentfrom the more familiar point geometry probed by maps of a point (as opposedto loops) into the space.We ﬁrst note from (8.6) that Zcirc(r) possesses a duality symmetryZcirc(r) = Zcirc(1/2r), in which the roles of winding and momentum are simplyinterchanged. (From (8.24), we recognize this as a symmetry also of the orbifoldtheory Zorb(r).) At the self-dual point r = 1/√2, we read oﬀfrom (8.8) theeigenvalues of L0 and L0 for the |m, n⟩states as 14(m±n)2.

For m = n = ±1 wethus ﬁnd two (1,0) states, and for m = −n = ±1 two (0,1) states. In operatorlanguage these states are created by the operatorsJ±(z) = e±i√2 x(z)andJ±(z) = e±i√2 x(z) ,(8.27a)with conformal weights (1,0) and (0,1).

They become suitably single-valuedunder x →x + 2πr only at the radius r = 1/√2. At arbitrary radius, on theother hand, we always have the (1,0) and (0,1) oscillator states α−1|0⟩andα−1|0⟩, created by the operatorsJ3(z) = i∂x(z)andJ3(z) = i∂x(z) .

(8.27b)The operators J±, J3 in (8.27a, b) are easily veriﬁed to satisfy the operatorproduct algebraJ+(z) J−(w) ∼ei√2(x(z)−x(w))(z −w)2∼1(z −w)2 + i√2z −w ∂x(w) ,J3(z) J±(w) ∼√2z −w J±(w) ,and similarly for J±, J3. If we deﬁne J± =1√2(J1 ±iJ2), then this algebra canbe written equivalently asJi(z) Jj(w) =δij(z −w)2 + i√2 ǫijkz −w Jk(w) .

(8.28)131(8.28) deﬁnes what is known as the algebra of aﬃne Kac-Moody SU(2) at levelk = 1 (level k would be given by substituting δij →kδij in the ﬁrst term onthe right hand side of (8.28)).For the terms in the mode expansionsJi(z) =Xn∈ZJin z−n−1 ,whereJin =Idz2πi zn Ji(z) ,we ﬁnd by the standard method (as employed to determine (3.8)) the commu-tation relations[Jin, Jjm] = i√2 ǫijk Jkn+m + n δij δn+m,0 .We see that the zero modes Ji0 satisfy an ordinary su(2) algebra (in a slightlyirregular normalization of the structure constants corresponding to length-squared of highest root equal to 2), and the remaining modes Jin provide aninﬁnite dimensional generalization (known as an aﬃnization) of the algebra.The generalization of this construction to arbitrary Lie algebras will be dis-cussed in detail in the next section.So we see that the circle theory Zcirc(r) at radius r = 1/√2 has an aﬃneSU(2) × SU(2) symmetry. It possesses at this point nine marginal operators,corresponding to combinations of the SU(2) × SU(2) currents JiJj (i, j =1, 2, 3).

But these are all related by SU(2) × SU(2) symmetry to the singlemarginal operator J3J3 = ∂X∂X, which simply changes the compactiﬁcationradius r. In fact, it is no coincidence that the enhanced symmetry occurs atthe self-dual point since either of the chiral SU(2) symmetries also relates themarginal operator ∂X∂X to minus itself, rendering equivalent the directions ofincreasing and decreasing radius at r = 1/√2. (So one might say that there isonly “half” a marginal operator at this point.

)To return to establishing the equivalence (8.26), we consider two possibleways of constructing a Z2 orbifold of the theory Zcirc(1/√2). Under the sym-metry X →−X (so that x →−x, x →−x) discussed in detail earlier, we seethat the aﬃne SU(2) generators (8.27) transform as J± →J∓, J3 →−J3.The shift X →X + 2π/(2√2) (shifting x and x by the same amount) is also a132

symmetry of the action (8.1), and instead has the eﬀect J± →−J±, J3 →J3.The eﬀect of these two Z2 symmetry actions thus can be expressed asJ1 →J1J2 →−J2J3 →−J3J1 →J1J2 →−J2J3 →−J3andJ1 →−J1J2 →−J2J3 →J3J1 →−J1J2 →−J2J3 →J3 .But by aﬃne SU(2) symmetry, we see that these two symmetry actions areequivalent, one corresponding to rotation by π about the 1-axis, the other torotation by π about the 3-axis.The ﬁnal step in demonstrating (8.26) is to note that modding out thecircle theory at radius r by a Zn shift X →X + 2πr/n in general reproducesthe circle theory, but at a radius decreased to r/n. Geometrically, the ZN groupgenerated by a rotation of the circle by 2π/n is an example of a group actionwith no ﬁxed points, hence the resulting orbifold S1/Zn is a manifold — in thiscase topologically still S1, but at the smaller radius.

From the Hilbert spacepoint of view, the projection in the untwisted sector removes the momentumstates allowed at the larger radius, and the twisted sectors provide the windingsappropriate to the smaller radius.Modding out Zcirc(1/√2) by the Z2 shift X →X+2π/(2√2) thus decreasesthe radius by a factor of 2, giving Zcirc(1/2√2), which by r ↔1/2r symmetryis equivalent to Zcirc(√2). Modding out Zcirc(1/√2) by the reﬂection X →−X, on the other hand, by deﬁnition gives Zorb(1/√2).

Aﬃne SU(2) × SU(2)symmetry thus establishes the equivalence (8.26) as a full equivalence betweenthe two theories at the level of their operator algebras.The picture[70][72][73] of the moduli space of c = 1 conformal theories thatemerges is depicted in ﬁg. 14.

The horizontal axis represents compactiﬁcation ona circle S1 with radius rcircle, and the vertical axis represents compactiﬁcationon the S1/Z2 orbifold with radius rorbifold. As previously mentioned, the formeris also known as the gaussian model, and the latter is equivalent to the criticalAshkin-Teller model (which also encompasses two other of the models describedin Cardy’s lectures, namely the 6-vertex model and the 8-vertex model on itscritical line).

The regions represented by dotted lines are determined by theduality r ↔12r.133TOIrorbifoldn=pc==pppp=p==p&0"DnDtwistedN=susyD,-statePottsmodelZparafermions(Ising)=(freemajorana)twistedN=susy.Doo!0=pSU()p=N=susyfreeDiracpCKTpointpN=susy=pCn=pCnrcircleFig. 14.

Survey of conformal ﬁeld theory at c = 1.In ﬁg. 14, we have indicated some of the special radii r = 1/√2, 1,√2 thatwe have discussed.

The partition function at the common point (8.26) of thetwo lines turns out to correspond to the continuum limit Kosterlitz-Thoulesspoint of the X-Y model on the torus[69]. At this point there are ﬁve marginaloperators, J1J1 and JiJj (i, j = 2, 3), that survive the projection under thegroup action x →−x.

In this language, J3J3 again generates changes in thecircle radius r, and the remaining 4 operators, all equivalent to one anotherdue to the U(1) × U(1) symmetry generated by J3 and J3, instead deform thetheory in the orbifold direction of ﬁg. 14.

This is the only such multicriticalpoint in the ﬁgure where there exist inequivalent directions of deformation[70].Two other special radii for circle compactiﬁcations are r =√3/2 and√3,where four operators of dimension ( 32, 32) appear, corresponding to a GSO pro-jected system with N = 2 supersymmetry[74][75]. (The chiral spin-3/2 vertexoperators take the form exp±i√3 x(z), exp±i√3 x(z).) The corresponding134

points r =√3/2,√3 on the orbifold line realize a twisted N = 2 supersym-metry algebra[35][36] that contains an N = 1 supersymmetry surviving the Z2projection[75][76]. (Actually the partition functions at the points r =√3/2and r =√3 on the circle line diﬀer by a constant, equal to 2 (and for the samepoints on the orbifold line the diﬀerence of the partition functions is 1).

Thisis because these theories are actually Z2 orbifolds of one another[77], and thediﬀerence of their partition functions is tr(−1)F in the Ramond sector, which isa constant due to superconformal invariance. By examination of the partitionfunctions (8.7), this relationship can be used to provide a simple superconfor-mal proof of the Euler pentagonal number theorem (7.30).) r =√6/2 on theorbifold line realizes a modular invariant combination of Z4 parafermions[78].

(Other properties of c = 1 models have also been considered in [79]. )The Z2 orbifolding that took us from the aﬃne SU(2) × SU(2) point tothe multicritical point at r =√2 on the circle line can be generalized.

Indeedwe can mod out by any of the discrete subgroups Γ of the diagonal SU(2). Itis easiest think of this in terms of subgroups of SO(3) acting simultaneously onthe vectors Ji(z), Ji(z).

Then the generator of the symmetry group Cn, thecyclic group of rotations of order n about the 3-axis, corresponds to the actionX →X + 2π/(n√2) (i.e. J± →e±2πi/nJ±, J3 →J3, and similarly for J’s).The additional generator adjoined to give the dihedral group Dn corresponds toX →−X (J3 →−J3, J± →J∓).

Modding out by the Cn’s thus gives pointson the circle line at radius r = n/√2, and modding out by the Dn’s gives thecorresponding points on the orbifold line, as indicated in ﬁg. 14.Something special happens, however, for the tetrahedral, octahedral, andicosahedral groups, T, O, and I.

For these it is easy to see that the only (1,1)operator that is invariant under the full discrete group is V = P3i=1 JiJi, whichis hence the only marginal operator that survives the projection. But recallingthat our aﬃne SU(2) currents satisfy (8.28), we easily verify that CV V V = −2for V = P3i=1 JiJi.This means[72] that the SU(2)/Γ orbifold models forΓ = T, O, I are isolated points in the moduli space for c = 1 conformally invari-ant theories, as depicted in ﬁg.

14. This absence of truly marginal operators isintuitively satisfactory for these cases since we are modding out by symmetries135that exist only at a given ﬁxed radius, the SU(2)×SU(2) radius r = 1/√2, andhence modding out by the symmetries eﬀectively freezes the radius.

Furtherproperties of the SU(2) orbifold models are discussed in [80], and an identiﬁ-cation of critical RSOS-type models that have the same partition functions isincluded in [72].Part of the motivation for studying c = 1 systems is that they represent theﬁrst case beyond the classiﬁcation methods discussed in section 4. For systemswith N = 1 superconformal symmetry (5.16), the corresponding boundary casebetween the (classiﬁed) discrete series and (unclassiﬁed) continuum lies at ˆc = 1.The analog of ﬁg.

14 for this case may be found in [77].9. Aﬃne Kac-Moody algebras and coset constructions9.1.

Aﬃne algebrasIn the previous section, we saw the important role played by aﬃne SU(2) atlevel k = 1 in characterizing the enhanced symmetry at the point r = 1/√2 onthe circle line. We now wish to consider the generalization of this constructionto arbitrary groups and arbitrary level.

We begin by considering a set of (1, 0)conformal ﬁelds Ja(z), called currents (where a labels the diﬀerent currents).Dimensional analysis constrains their operator products to take the formJa(z)Jb(w) =ekab(z −w)2 + if abcz −w Jc(w) + . .

. ,(9.1)where the f abc’s are necessarily antisymmetric in a and b.

Furthermore, asso-ciativity of the operator products can be used to show that the f abc’s satisfyas well a Jacobi identity. That means that they constitute the structure con-stants of some Lie algebra G, which we shall assume in what follows to be thatassociated to a compact Lie group G (i.e.

to have a positive deﬁnite Cartanmetric). For each simple component of the algebra we can choose a basis inwhich the central extension ekab = ekaδab.

The operator product (9.1) is theoperator product for what is known as an aﬃne, or aﬃne Kac-Moody, alge-bra (for a recent review, see [3]), or a 2d current algebra. Aﬃne algebras play136

an important role in closed string theory, where they provide the worldsheetrealization of spacetime gauge symmetries. They also provide many new non-trivial examples of exactly solvable quantum ﬁeld theories in two dimensions,and may ultimately play a role in the classiﬁcation program of two dimensionalconformal ﬁeld theories at arbitrary c.In terms of the mode expansion Ja(z) = Pn∈Z Jan z−n−1, we ﬁnd from(9.1) the commutatorsJam, Jbn= if abc Jcm+n + ek m δab δm+n,0 ,(9.2)where we have restricted for simplicity to the case that the f abc are the struc-ture constants associated to a simple Lie group G. (9.2) by deﬁnition deﬁnesthe untwisted aﬃne algebra bG associated with a compact ﬁnite-dimensional liealgebra G, where m, n ∈Z; and a, b, c run over the values 1 to |G| ≡dim G.We see that the subalgebra of zero modes Ja0 constitutes an ordinary Lie al-gebra, known as the horizontal Lie subalgebra, in which the c-number centralextension ek does not appear.The full inﬁnite set of Jan’s provides what isknown as an “aﬃnization” of the ﬁnite dimensional subalgebra of Ja0 ’s.

As in(7.1), we can pull back J(z) to the cylinder, so that we have the Fourier se-ries Jacyl(w) = Pn Jan e−nw. With w real, we recognize the modes Jan as theinﬁnitesimal generators of the group of gauge transformations g(σ) : S1 →Gon the circle.The representation theory of aﬃne algebras shares many features with thatof the Virasoro algebra.

For example, regularity of J(z)|0⟩at z = 0 requiresthatJan|0⟩= 0forn ≥0 .There also exists a notion of primary ﬁeld ϕℓ(r) (actually a multiplet of ﬁelds)with respect to the aﬃne algebra, for which the operator product has the leadingsingularityJa(z) ϕ(r)(w) ∼ta(r)z −w ϕ(r)(w) + . .

. .

(9.3)This should be recognized as the statement that ϕ(r) transforms as some repre-sentation (r) of G, where the right hand side is shorthand for (ta(r))ℓkϕk(r), and137ta(r) are representation matrices for G in the representation (r). These primaryﬁelds create states, called highest weight states,(r)≡ϕ(r)(0)|0⟩(9.4)(again a multiplet of states), that provide a representation of the zero modealgebraJa0(r)= ta(r)(r),withJan(r)= 0(n > 0) .

(9.5)The Ward identities for aﬃne symmetry take the formJa(z) ϕ(r1)(w1, w1) . .

. ϕ(rn)(wn, wn)=nXj=1ta(rj)z −wjϕ(r1)(w1, w1) .

. .

ϕ(rn)(wn, wn). (9.6)These are derived as was (2.22) by computing the contour integralRdz2πiαa(z)Ja(z)inserted in a correlation function of ϕ(rj)’s, where the contour encloses all ofthe points wj (as in ﬁg.

3) and the αa(z)’s parametrize an inﬁnitesimal localG-transformation. Then by deforming the contour to a sum of small contoursaround each of the wj’s we ﬁnd from (9.3)Zdz2πi αa(z)Ja(z) ϕ(r1)(w1, w1) · · · ϕ(rn)(wn, wn)=nXj=1ϕ(r1)(w1, w1) · · · δαϕ(rj)(wj, wj) · · · ϕ(rn)(wn, wn),where δαϕ(rj) = αata(rj)ϕ(rj) is by deﬁnition the change in ϕ(rj) under theinﬁnitesimal G transformation parametrized by α.

We shall see a bit later how(9.6) may be used to derive ﬁrst-order diﬀerential equations for Green functionsinvolving primary ﬁelds ϕ(rj).9.2. Enveloping Virasoro algebraThe algebraic structure (9.1), characterizing an aﬃne or current algebra,turns out to incorporate as well a natural deﬁnition of a stress-energy tensorT (z).

Equivalently, we may construct generators Ln of a Virasoro algebra in138

terms of the modes Jan, thereby making contact with the Virasoro representationtheory detailed earlier.Recall that for a single boson, the natural (2, 0) object was T (z) =−12: ∂x(z)∂x(z): = 12: J3(z)J3(z):, where J3 = i∂x. (In the language of aﬃnealgebras, this is the case G = U(1), with central charge c = 1.) The naturalgroup invariant generalization isT (z) = 1β|G|Xa=1: Ja(z)Ja(z): = limz→w|G|Xa=1Ja(z)Ja(w) −ek|G|(z −w)2 .

(9.7)The constant β above is ﬁxed either by requiring that T (z) satisfy the canonicaloperator product (3.1), or by requiring that the Ja(z)’s indeed transform asdimension (1, 0) primary ﬁelds.Implementing the latter approach, we write the singular terms in the op-erator product expansionT (z)Ja(w) =Ja(w)(z −w)2 + ∂Ja(w)z −w,(9.8a)implying the commutations relations[Lm, Jan] = −nJam+n(9.8b)for the modes of T and J. From (9.7), we haveLn = 1β∞Xm=−∞: Jam+n Ja−m: ,(9.9)so that applying L−1 to a highest weight state and using (9.5) givesL−1(r)= 2β Ja−1 ta(r)(r).We next apply Jb1 to both sides and use (9.2) and (9.8b) to gettb(r)(r)= 2β (if bac Jc0 + ekδab)ta(r)(r)= 2β (if bac 12if dca td(r) + ektb(r))(r)= 2β12CA + ektb(r)(r),139where the quadratic casimir CA of the adjoint representation is deﬁned byf acdf bcd = CA δab.

We conclude that consistency of (9.7) with (9.8) requiresthatβ = 2ek + CA . (9.10)At this point it is now straightforward to check that the stress-energy tensorT (z) =1/2ek + CA/2|G|Xa=1: Ja(z)Ja(z) :(9.11)satisﬁes as well the canonical operator product expansion (3.1), with leadingsingularityT (z)T (w) ∼cG/2(z −w)4 + .

. .given by the central chargecG =ek |G|ek + CA/2.

(9.12)The stress-energy tensor (9.11), quadratic in the currents, is known as the Sug-awara form of the stress-energy tensor. Historically, the normalization (9.10)was the culmination of eﬀort by numerous parties (see [3] for extensive refer-ences).The number CA/2 depends in general on the normalization chosen for thestructure constants f abc.

Since its value plays an important role in what follows,we digress brieﬂy to introduce some of the necessary group theoretic notation.If we writetr ta(r)tb(r) = ℓrδab(9.13)for an arbitrary representation (r) of G of dimension dr, then summing overa, b = 1, . .

. , |G| givesCrdr = ℓr|G| ,(9.14)where Cr is the quadratic Casimir of the representation.

Summing only overthe Cartan subalgebra of G (a, b = 1, . .

. , rG), on the other hand, givesdrXj=1µ2(j) = ℓrrG ,(9.15)140

where rG is the rank of the group G and the µ are the weights of the represen-tation (r).For the adjoint representation, we have dA = |G| and CA = ℓ(A) =r−1GP|G|a=1 α2(a), where α are the roots.If we let ψ denote the highest root,then the normalization independent quantity ehG ≡CA/ψ2, known as the dualCoxeter number, satisﬁesehG ≡CAψ2 = 1rG nL +SL2nS!. (9.16)In (9.16), nS,L are the number of short and long roots of the algebra (the highestroot ψ is always a long root), and (S/L)2 is the ratio of their squared lengths(roots of simple Lie algebras come at most in two lengths).

Those algebrasassociated to Dynkin diagrams with only single lines, i.e. SU(n), SO(2n), E6,7,8,are called “simply-laced”, and have roots all of the same length.

(In moremathematical circles these are known as the (A, D, E) series of algebras. Ingeneral, the Coxeter number itself is the order of the Coxeter element of theWeyl group, by deﬁnition the product of the simple Weyl reﬂections.TheCoxeter number is also equal to the number of (non-zero) roots divided by therank of the algebra, and coincides with the dual Coxeter number only for thesimply-laced algebras.) The remaining algebras have roots of two lengths, theirratio (L/S) either√2 (for SO(2n + 1), Sp(2n), F4) or√3 (for G2).Equation (9.16) allows us to tabulate the dual Coxeter numbers for all thecompact simple Lie algebras:SU(n) (n ≥2) : ehSU(n) = n,ℓ(n) = 12ψ2SO(n) (n ≥4) : ehSO(n) = n −2,ℓ(n) = ψ2E6 : ehE6 = 12,ℓ(27) = 3ψ2E7 : ehE7 = 18,ℓ(56) = 6ψ2E8 : ehE8 = 30,ℓ(248) = 30ψ2Sp(2n) (n ≥1) : ehSp(2n) = n + 1,ℓ(2n) = 12ψ2G2 : ehG2 = 4,ℓ(7) = ψ2F4 : ehF4 = 9,ℓ(26) = 3ψ2 .

(9.17)141We see that the dual Coxeter number is always an integer. In (9.17) we havealso tabulated the index ℓr, as deﬁned in (9.13), for the lowest dimensionalrepresentations as a function of ψ2.9.3.

Highest weight representationsIn what follows, we shall be interested in so-called irreducible unitary high-est weight representations of the algebra (9.2). This means that the highestweight states transform as an irreducible representation of the ordinary Lie al-gebra of zero modes Ja0 (the horizontal subalgebra), as in (9.5).

Since these arealso the states in a given irreducible representation of the aﬃne algebra withthe smallest eigenvalue of L0, we shall frequently refer to the multiplet of states(9.4) as the vacuum states, and (r) as the vacuum representation. The states atany higher level, i.e.

higher L0 eigenvalue, will also transform as some represen-tation of the horizontal subalgebra, although only the lowest level necessarilytransforms irreducibly.Unitarity is implemented as the condition of hermiticity on the generators,Ja†(z) = Ja(z). By the same argument leading to (3.12) in the case of theVirasoro algebra, we see that this implies Jan† = Ja−n.

In a Cartan basis theJa(z)’s are written Hi(z) and E±α(z), where i = 1, . .

. , rG labels the mutuallycommuting generators, and the positive roots α label the raising and loweringoperators.

In this basis the truly highest weight state |λ⟩≡(r), λof thevacuum representation satisﬁesHin|λ⟩= E±αn |λ⟩= 0 ,n > 0 ,Hi0|λ⟩= λi|λ⟩,andEα0 |λ⟩= 0 , α > 0 .New states are created by acting on the state |λ⟩with the E−α0’s or any of theJa−n’s for n > 0.Now we wish to consider the quantization condition on the central extensionek in (9.2). It is evident that ek depends on the normalization of the structureconstants.

We shall show that the normalization independent quantity k ≡2ek/ψ2, known as the level of the aﬃne algebra, is quantized as an integer in ahighest weight representation. (Equivalently, in a normalization in which the142

highest root ψ satisﬁes ψ2 = 2, we have ek = k ∈Z. The normalization conditionψ2 = 2 on the structure constants is easily translated into a condition on theindex ℓr for the lowest dimensional representations listed in (9.17).) In termsof the integer quantities k and ehG, we may rewrite the formula (9.12) for thecentral charge ascG =k |G|k + ehG.

(9.18)As an example, we see from (9.17) that ehSU(2) = 2, so for the lowest levelk = 1 we ﬁnd from (9.18) that cSU(2) = 3/(1 + 2) = 1. Thus we infer that therealization of aﬃne SU(2) provided at radius r = 1/√2 on the (c = 1) circleline is at level k = 1.To establish the quantization condition on k, we ﬁrst consider the caseG = SU(2).

Note that the normalization of structure constants, f ijk =√2ǫijk,in (8.28) corresponds to the aforementioned ψ2 = 2. Because of the√2 in thecommutation rules, we need to takeI± =1√2(J10 ± iJ20)andI3 =1√2J30(9.19a)to give a conventionally normalized su(2) algebra [I+, I−] = 2I3, [I3, I±] =±I±, in which 2I3 has integer eigenvalues in any ﬁnite dimensional representa-tion.

But from (9.2) we ﬁnd thateI+ =1√2(J1+1−iJ2+1) , eI−=1√2(J1−1+iJ2−1) , and eI3 = 12k−1√2J30 (9.19b)as well satisfy [eI+, eI−] = 2eI3, [eI3, eI±] = ±eI±, so 2eI3 = k −2I3 also has integereigenvalues. It follows that k ∈Z for unitary highest weight representations.This argument is straightforwardly generalized by using the canonical su(2)subalgebraI± = E±ψ0,I3 = ψ · H0/ψ2(9.20a)generated by the highest root ψ of any Lie algebra.

From (9.2),eI± = E∓ψ±1 ,eI3 = (ek −ψ · H0)/ψ2(9.20b)143also form an su(2) subalgebra, implying that the level k = 2ek/ψ2 = 2eI3 + 2I3is quantized for unitary highest weight representations of aﬃne algebras basedon arbitrary simple Lie algebras.We pause here to remark that the quantization condition on k also fol-lows [81] from the quantization of the coeﬃcient of the topological termΓ =124πRtr(g−1dg)3 in the Wess-Zumino-Witten lagrangian,S =14λ2Zd2ξ tr(∂µg)(∂µg−1)+kΓ = k 116πZtr(∂µg)(∂µg−1) + Γ, (9.21)for a two dimensional σ-model with target space the group manifold of G. In(9.21) we have substituted the value of the coupling λ for which the modelbecomes conformally invariant. The currents J = Jata ∼∂gg−1, J = Jata ∼g−1∂g, derived from the above action, satisfy the equations of motion ∂J =∂J = 0.

This factorization of the theory was shown in [81] to imply an aﬃneG×G symmetry, and theories of the form (9.21) were analyzed extensively fromthis point of view in [82][83]. More details and applications of these theoriesmay be found in Aﬄeck’s lectures.Before turning to other features of the representation theory of (9.2), wecontinue brieﬂy the discussion of the conformal Ward identities (9.6).

First werecall from (9.11) thatL−1 =1ek + CA/2(Ja−1Ja0 + Ja−2Ja1 + . .

. )(where the factor of 1/2 in the numerator of (9.11) is compensated by theappearance of each term exactly twice in the normal ordered sum (9.9)).

Actingon a primary ﬁeld, we thus ﬁnd the null ﬁeld L−1 −Pa Ja−1ta(r)ek + CA/2!ϕ(r) = 0 . (9.22)(9.22) implies that correlation functions involving n primary ﬁelds satisfy nﬁrst-order diﬀerential equations.

To derive them, we multiply (9.6) by ta(rk),take z →wk and use the operator product expansion (9.1), giving ﬁnally[82] ek + CA/2 ∂∂wk+Xj̸=kata(rj)ta(rk)wj −wkϕr1(w1) . .

. ϕrn(wn)= 0 .

(9.23)144

The ﬁrst-order equations (9.23) for each of the wk, together with their anti-holomorphic analogs, can be solved subject to the constraints of crossing sym-metry, monodromy conditions, and proper asymptotic behavior. The simplestsolution involves a symmetric holomorphic/anti-holomorphic pairing, and cor-responds to the correlation functions of the σ-model (9.21).Returning now to (9.11), we observe that the vacuum state (9.4) in generalhas L0 eigenvalueL0(r)=1/2ek + CA/2Xa,m: JamJa−m:(r)=1/2ek + CA/2Xata(r)ta(r)(r)=Cr/2ek + CA/2(r),(9.24a)where Cr is the quadratic Casimir of the representation (r).

The conformalweight of the primary multiplet ϕ(r)(z) is thushr =Cr/2ek + CA/2= Cr/ψ2k + ehG. (9.24b)For the case G = SU(2) with ground state transforming as the spin-j represen-tation of the horizontal su(2), (9.24) givesL0(j)= j(j + 1)k + 2(j)(9.25)(where the quadratic Casimir satisﬁes C(j) = 2j(j + 1) in a normalization ofsu(2) with ψ2 = 2).

For aﬃne SU(2) at level k = 1 we ﬁnd the values h = 0, 14for j = 0, 12.We can easily see how these conformal weights enter into the partitionfunction at the SU(2) × SU(2) point r = 1/√2 of the circle theory consideredin the previous section. By steps similar to those in (8.13), we can write thepartition function (8.7) in the formZcirc 1√2= χ(0),1χ(0),1 + χ(1/2),1χ(1/2),1 ,(9.26)145whereχ(0),1(q) = 1η∞Xn=−∞qn2 ,χ(1/2),1(q) = 1η∞Xn=−∞q(n+ 12 )2 .

(9.27)We see that the values h = 0, 14 emerge as the conformal weights of the leadingterms of the quantities χ(0),1 and χ(1/2),1. (9.26) corresponds to a decompositionof the partition function in terms of characters of an extended chiral algebra,here aﬃne SU(2) × SU(2).A bit later we will discuss aﬃne characters atarbitrary level.There exists a simple constraint on the possible vacuum representations (r)allowed in a unitary highest weight realization of (9.2) at a given level k. To seethis most easily, we return again to G = SU(2).

We take our “vacuum”(r)inthe spin-j representation of SU(2). The 2j + 1 states of this representation arelabeled as usual by their I3 eigenvalue, I3(j), m= m(j), m, where I3 is asdeﬁned in (9.19a).

Using the other su(2) generators (9.19b), we derive the moststringent condition by considering the state |j⟩≡(j), jwith highest isospinm = j,0 ≤⟨j|eI+eI−|j⟩= ⟨j| eI+, eI−|j⟩= ⟨j|k −2I3|j⟩= k −2j . (9.28)It follows that only ground state representations with2j ≤k(9.29)are allowed.

For a given k, these are the k+1 values j = 0, 12, 1, . .

. , k2.

Thus it isno coincidence that the SU(2) level k = 1 partition function (9.26) is composedof only j = 0, 12 characters.The generalization of (9.29) to arbitrary groups is more or less immedi-ate. Instead of |j⟩we consider |λ⟩, where λ is highest weight of the vacuumrepresentation.

Then from (9.28) using instead the eIi’s of (9.20b), we ﬁnd2ψ · λ/ψ2 ≤k . (9.30)(For SU(n) this condition on allowed vacuum representations turns out in gen-eral to coincide with the condition that the width of their Young tableau be146

less than the level k. For SU(2), for which the spin-j representation is thesymmetric combination of 2j spin- 12 representations, this is already manifest in(9.29). )The assemblage of states created by acting on the highest weight states(r)with the Ja−n’s again constitutes a Verma module.

As was the case for thec < 1 representations of the Virasoro algebra, this module will in general containnull states which must be removed to provide an irreducible representation ofthe aﬃne algebra. In the case at hand, it can be shown that all the null states aredescendants of a single primitive null state.

This state is easily constructed fora general aﬃne algebra by using the generators (9.20b) of the (non-horizontal)su(2) subalgebra. Note that the eigenvalue of 2eI3 acting on the highest weightstate(r), λof the vacuum representation is given by M = k −2ψ · λ/ψ2.For the aﬃne representations of interest, the set of states generated by actingwith successive powers of eI−on(r), λforms a ﬁnite dimensional irreduciblerepresentation of the su(2) subalgebra (9.20b).

Thus M is an integer andeI−M+1(r), λ= 0 .This is the primitive null state mentioned above, whose associated null ﬁeldeI−M+1φ(r),λ can be used to generate all non-trivial selection rules[82][83] inthe theory. In the case of a level k representation of aﬃne SU(2), the above nullstate becomesJ+−1k+1|0⟩= 0 for the basic representation, or more generallyJ+−1k−2j+1(j), j= 0 for the spin-j representation.9.4.

Some free ﬁeld representationsIn the case of the Virasoro algebra, we found a variety of useful represen-tations aﬀorded by free bosons and fermions. Free systems can also be usedto realize particular representations of aﬃne algebras.

For example, we take Nfree fermions ψi with operator product algebraψi(z)ψj(w) = −δijz −w .147We consider these fermions to transform in the vector representation of SO(N),with representation matrices ta. Then for N ≥4, the currentsJa(z) = ψ(z)taψ(z)(9.31)are easily veriﬁed to satisfy (9.1) for SO(N) at level k = 1.

We also verify from(9.17) and (9.18) thatcSO(N),k=1 = 1 12N(N −1)1 + (N −2) = 12N ,(9.32)consistent with the central charge for N free fermions. (For N = 3, we wouldﬁnd instead a level k = 2 representation of SU(2) with c = 32).

The free fermionrepresentation (9.31) provides the original context in which aﬃne algebras aroseas two dimensional current algebras.We could equivalently use N complex fermions taken to transform in thevector representation of SU(N), and construct currents Ja(z) = ψ∗(z)taψ(z)analogous to (9.31). These realize aﬃne SU(N)×U(1), with the SU(N) at levelk = 1.

(The notion of level for an abelian U(1) current algebra is more subtlethan we need to discuss here — for our purposes it will suﬃce to recall that italways has c = 1, and the current has the free bosonic realization J = i∂x. )The central charge comes out ascU(1) + cSU(N),k=1 = 1 + 1(N 2 −1)1 + N= N ,consistent with the result for N free complex fermions.Another example is to take rG free bosons, where rG is the rank of somesimply-laced Lie algebra (i.e., as mentioned earlier, SU(n), SO(2n), or E6,7,8).Generalizing the aﬃne SU(2) construction (8.27), we let Hi(z) = i∂xi(z) repre-sent the Cartan subalgebra and J±α(z) = cα : e±iα·x(z): represent the remainingcurrents, where α are the positive roots all normalized to α2 = ψ2 = 2. cα is acocycle (Klein factor), in general necessary to give correct signs in the commu-tation relations (for more details see [3]).

This realization of simply-laced aﬃne148

algebras is known as the ‘vertex operator’ construction[84] (and was anticipatedfor the case SU(n) in [85]). From (9.16) we infer the general relationehG = |G|rG−1(9.33)for simply-laced groups, and from (9.18) the central charge cG = rG thus comesout appropriate to rG free bosons.There is a generalization of this construction that works for any algebraat any level, but no longer involves only free ﬁelds.

We begin again with rGfree bosons, but now take Hi(z) = i√k ∂xi(z) to represent the Cartan currents(with the factor of√k inserted to get the level correct). Now the exponential: e±iα·x(z)/√k: has the correct operator product with the Cartan currents, butno longer has dimension h = 1 in general.

For the full current we write insteadJ±α(z) = :e±iα·x(z)/√k: χα(z) ,(9.34)where χα is an operator of dimension h = 1 −α2/2k whose operator prod-ucts[86] mirror those of the exponentials so as to give overall the correct op-erator products (9.1). The χα’s are known as ‘parafermions’ and depend onG and its level k. Since the aﬃne algebra is constructed from rG free bosonsand the parafermions, the central charge of the parafermion system is given bycG(k) −rG.A ﬁnal free example is take |G| free fermions to transform in the adjointrepresentation of some group G.Then the currents (in a normalization ofstructure constants with highest root ψ2 = 2)Ja(z) = i2f abcψb(z)ψc(z)(9.35)give a realization of aﬃne G at level k = ehG.

The central charge comes outto be cG = ehG|G|/(ehG + ehG) =12|G|.This case of dim G free Majoranafermions in fact realizes[87][88][19] what is known as a super-aﬃne G algebrawith an enveloping super Virasoro algebra. In general, a super-aﬃne algebrahas, in addition to the structure (9.1) and (9.8), a spin-3/2 super stress tensor149TF satisfying (5.16) and superﬁeld aﬃne generators Ja = J a + θJa, whosecomponents satisfyTF (z)Ja(w) =1/2(z −w)2 J a(w) +1/2z −w∂J a(w)TF (z)J a(w) =1/2z −wJa(w)Ja(z)J b(w) = if abcz −w J c(w)J a(z)J b(w) = kδabz −w .In the free fermionic representation, these operator products are satisﬁed (ataﬃne level k = ehG) by the super stress tensor TF = −112√CA/2f abcψaψbψc,and superpartners J a = i√kψa of the aﬃne currents (9.35).A modular invariant super-aﬃne theory on the torus can be constructedby taking left and right fermions ψa and ψa and summing over the same spinstructure for all the fermions (GSO projecting on (−1)FL+FR = +1 states).

Atc = 3/2, for example, three free fermions ψi taken to transform as the adjointof SU(2) (vector of SO(3)) can be used to represent an N = 1 superconformalalgebra with a super-aﬃne SU(2) symmetry at level k = 2. The supersym-metry generator is given by TF = −112ǫijkψiψjψk = −12ψ1ψ2ψ3, and similarlyfor T F .

(For an early discussion of supersymmetric systems realized by threefermions, see [89].) The sum over fully coupled spin structures gives a theorythat manifests the full super-aﬃne SU(2)2 symmetry.

It has partition function12 A3A3A3A3+ P3P3A3A3+ A3A3P3P3+ P3P3P3P3!= 12 ϑ3η3+ϑ4η3+ϑ2η3!= χ(0),2χ(0),2 + χ(1/2),2χ(1/2),2 + χ(1),2χ(1),2 ,(9.36)which we have also expressed in terms of the level 2 aﬃne SU(2) charactersχ(j=0,1/2,1),k=2. From (9.25), we see that the associated primary ﬁelds have150

conformal weights h = j(j + 1)/(2 + 2) = 0, 316, 12. The characters themselvesmay be calculated just as the c = 12 characters of (7.16a), with the resultχ(0),2 = 12 A3A3+ P3A3!= 12 ϑ3η3/2+ϑ4η3/2 !χ(1),2 = 12 A3A3−P3A3!= 12 ϑ3η3/2−ϑ4η3/2 !χ(1/2),2 =1√2 A3P3± P3P3!=1√2ϑ2η3/2,(9.37)We also point out that we can bosonize two of the fermions of this construction,say ψ1 and ψ2, so that J3 = i∂x.

Then the remaining fermion can be regardedas an SU(2) level 2 parafermion, providing the simplest non-trivial example ofthe general parafermionic construction (9.34).For the free fermion constructions (9.31) and (9.35) of aﬃne currents, wenoted that the central charge came out equal to a contribution of c = 12 fromeach real fermion.This was not necessarily guaranteed, since we were con-sidering theories deﬁned not by a free stress-energy tensor, T = 12Pi ψi∂ψi,but rather by the stress-energy tensor T of (9.7), which is quadratic in the J’sand thus looks quadrilinear in the fermions. The conditions under which theseemingly interacting stress tensor of (9.7) turns out to be equivalent to a freefermion stress tensor were determined in [87].

If we take fermions in (9.31) totransform as some representation (not necessarily irreducible) of G, then theresult is that the Sugawara stress tensor is equivalent to that for free fermionsif and only if there exists a group G′ ⊃G such that G′/G is a symmetric spacewhose tangent space generators transform under G in the same way as thefermions. (This was shown in [87] by a careful evaluation of the normal order-ing prescription in the deﬁnition (9.7), ﬁnding that it reduces to a free fermionform if and only if a quadratic condition on the representation matrices ta of(9.31) is satisﬁed.

The condition turns out to be equivalent to the Bianchi iden-tity for the Riemann tensor of G′/G when the ta’s are in the representation ofthe tangent space generators.) The three free fermion examples considered ear-lier here correspond to the symmetric spaces SN = SO(N + 1)/SO(N), where151the tangent space transforms as the N of SO(N); CP N = SU(N + 1)/U(N),where the tangent space transforms as the N of U(N); and G × G/G, wherethe tangent space transforms as the adjoint of G. Later we will encounter someother interesting examples of symmetric spaces.9.5.

Coset constructionThe question that naturally suggests itself at this point is whether theenveloping Virasoro algebras associated to aﬃne algebras are also related toany of the other representations of the Virasoro algebra discussed here.Inparticular we wish to focus on the c < 1 discrete series of unitary Virasororepresentations. First of all for SU(2) we see from (9.18) thatcSU(2) =3kk + 2(9.38)satisﬁes 1 ≤cSU(2) ≤3 as k ranges from 1 to ∞, so there is no possibility toget c < 1.

From the expression (9.16), we can easily show furthermore for anygroup thatrank G ≤cG ≤dim G ,so c < 1 is never obtainable directly via the Sugawara stress-tensor (9.11) ofan aﬃne algebra. (The lower bound in the above, cG = rank G, is saturatedidentically by simply-laced groups G at level k = 1, i.e.

identically the caseallowing the vertex operator construction of an aﬃne algebra in terms of rGfree bosons. )To increase in an interesting way the range of central charge accessible byaﬃne algebra constructions, we need somehow to break up the stress-tensor(9.11) into pieces each with smaller central charge.

This is easily implementedby means of a subgroup H ⊂G. We denote the G currents by JaG, and theH currents by JiH, where i runs only over the adjoint representation of H, i.e.from 1 to |H| ≡dim H. We can now construct two stress-energy tensors (forthe remainder we shall take all structure constants to be normalized to ψ2 = 2)TG(z) =1/2kG + ehG|G|Xa=1: JaG(z)JaG(z): ,(9.39a)152

and alsoTH(z) =1/2kH + ehH|H|Xi=1: JiH(z)JiH(z): . (9.39b)Now from (9.8) we have thatTG(z) JiH(w) ∼JiH(w)(z −w)2 + ∂JiH(w)z −w,but as well thatTH(z) JiH(w) ∼JiH(w)(z −w)2 + ∂JiH(w)z −w.We see that the operator product of (TG−TH) with JiH is non-singular.

Since THabove is constructed entirely from H-currents JiH, it also follows that TG/H ≡TG −TH has a non-singular operator product with all of TH. This means thatTG = (TG −TH) + TH ≡TG/H + TH(9.40)gives an orthogonal decomposition of the Virasoro algebra generated by TG intotwo mutually commuting Virasoro subalgebras, [TG/H, TH] = 0.To compute the central charge of the Virasoro subalgebra generated byTG/H, we note that the most singular part of the operator expansion of twoTG’s decomposes asTGTG ∼12cG(z −w)4 ∼TG/HTG/H + THTH ∼12cG/H + 12cH(z −w)4.The result is[19][90]cG/H = cG −cH =kG|G|kG + ehG−kH|H|kH + ehH,and we see that a central charge less than the rank of G may be obtained.

(Earlyexamples of related algebraic structures may be found in [91].) Further insightinto G/H models is provided by their realization as Wess-Zumino-Witten mod-els (9.21) with the H currents coupled to a gauge ﬁeld[73][92]; their correlationfunctions are moreover computable in terms of those of WZW models.153If it turns out that cG/H = 0, then the argument of subsection 3.5 showsthat TG/H must act trivially on any highest weight representation.

From (9.40)there follows[3] the quantum equivalence TG = TH between two superﬁciallyvery diﬀerent stress-energy tensors. Classiﬁcations of embeddings which gen-erate cG/H = 0, known as ‘conformal embeddings’, are considered in [93].

Aparticularly simple example is provided by a group divided by its Cartan sub-group, GU(1)rG. If G is simply-laced, then we saw from (9.33) that its aﬃnealgebra realized at level 1 has cG = rG.

This means that TG in this case is equiv-alent to TU(1)rG, i.e. to the stress-energy tensor for rG free bosons, motivatingthe vertex operator construction.

For G not simply-laced or at level k ≥1,TG/U(1)rG is the (non-trivial) stress-energy tensor of level-k G parafermions.Now we turn to the speciﬁc case of coset spaces of the form G × G/G,where the group G in the denominator is the diagonal subgroup. If we call thegenerators of the two groups in the numerator Ja(1) and Ja(2), the generators ofthe denominator are Ja = Ja(1) + Ja(2).

The most singular part of their operatorproduct expansion isJa(z)Jb(w) ∼Ja(1)(z)Jb(1)(w) + Ja(2)(z)Jb(2)(w) ∼(k1 + k2)δab(z −w)2+ . .

. ,so that the level of the G in the denominator is determined by the diagonalembedding to be k = k1 + k2.A simple example of this type is provided byG/H = SU(2)k × SU(2)1SU(2)k+1 ,in which casecG/H =3kk + 2 + 1 −3(k + 1)(k + 1) + 2 = 1 −6(k + 2)(k + 3) .

(9.41)We recognize these as precisely the values of the c < 1 discrete series (4.6a)where m = k + 2 = 3, 4, 5, . .

. .

Using the known unitarity [94] of the rep-resentations of aﬃne SU(2), this construction allowed the authors of [19] todeduce the existence of unitary representations for all the discrete values of c154

and h allowed for c < 1 by the analysis of the Kac determinant formula (4.5). (Unitary coset constructions for which cG/H < 1 must of course always coincidewith some member of the unitary discrete series (4.6a).

)Another example is to take G/H = SU(2)k × SU(2)2SU(2)k+2, givinginsteadcG/H =3kk + 2 + 32 −3(k + 2)(k + 2) + 2 = 321 −8(k + 2)(k + 4). (9.42)These values of the central charge coincide with those of the N = 1 super-conformal discrete series (5.19), with m = k + 2 = 3, 4, 5, .

. ..Again thisshows[19] that unitary representations of the superconformal algebra (5.16)indeed exist at all these values of c. More generally, the coset constructionG/H = SU(2)k × SU(2)ℓSU(2)k+ℓgives other discrete series associated tomore extended chiral algebras[95].

Algebras of this form have been consideredfor a bewildering variety of groups and levels. Their unitary representationtheory is discussed in [96].To understand better the states that arise in the G/H theory, we needto consider how the representations of G decompose under (9.40).

We denotethe representation space of aﬃne G at level kG bycG, λG, where cG is thecentral charge appropriate to kG, and λG is the highest weight of the vacuumrepresentation. (For a coset space of the form G × G/G, for example, we wouldwrite kG →(k(1), k(2)), and λG →(λ(1), λ(2)), where 1,2 denote the two groupsin the numerator.) Under the orthogonal decomposition of the Virasoro algebraTG = TG/H + TH, this space must decompose as some direct sum of irreduciblerepresentations,cG, λG= ⊕jcG/H, hjG/H⊗cH, λjH,(9.43)wherecG/H, hiG/Hdenotes an irreducible representation of TG/H with lowestL0 eigenvalue hiG/H.For the case G/H = SU(2)k × SU(2)1SU(2)k+1 mentioned above, (9.43)takes the explicit form[19](j)k × (ǫ)1 = ⊕qh(c)p,q⊗12q −1k+1 ,155where c is given by (9.41), p = 2j + 1 (1 ≤p ≤k + 1), and the sum is over1 ≤q ≤k + 2 with p −q even (odd) for ǫ = 0 ( 12).

We are thus able to obtainvia the coset construction all representations (4.6b) of the Virasoro algebra atthe values of c in (4.6a) (with m = k + 2). For the ﬁrst non-trivial case k = 1,for example, the coset construction SU(2)1 × SU(2)1SU(2)2 has c = 12.

Theproducts of SU(2)1 representations decompose as(0)1 × (0)1 = (h(1/2)1,1) (0)2 ⊕(h(1/2)2,1) (1)2(0)1 × ( 12)1 = (h(1/2)1,2) ( 12)2( 12)1 × ( 12)1 = (h(1/2)2,1) (0)2 ⊕(h(1/2)1,1) (1)2 .The three allowed Virasoro representations, with conformal weights h(1/2)p,q=0, 116, 12, all appear in the decompositions consistent with the aﬃne SU(2) con-formal weights h(0),k = 0, h(1/2),1 = 14, h(1),2 = 12, h(1/2),2 =316, and the integerspacing of the levels.As a ﬁnal example, we consider G/H = SO(N)1 ×SO(N)1SO(N)2, withcentral chargecG/H = N2 + N2 −2 12N(N −1)2 + (N −2) = N −(N −1) = 1 .This case turns out to be related to speciﬁc points r =√N/2 on the c = 1circle and orbifold lines discussed in section 8. The holomorphic weights thatenter into the circle line partition function (8.6) at this radius areh(m, n) = 12 m2(√N/2)+ n√N2!2=18N (2m + nN)2 .

(9.44)To give a ﬂavor for how to analyze these constructions more generally, we com-pare some of the weights inferred from (9.43) with these h values. * For SO(N),the representations allowed at level 1 are the adjoint, vector, and spinor(s).The representations allowed at level 2 include all of these together with otherrepresentations present in the decompositions of their direct products.

We will* I thank L. Dixon for his notes on the subject.156

concentrate here only on the rank r antisymmetric tensor representations, de-noted [r], which appear in the product of two spinors. From (9.14) and (9.15),we ﬁnd Cv/ψ2 = 12(N −1), Cs/ψ2 = N(N −1)/16, and C[r]/ψ2 = 12r(N −r).

(9.24b) giveshv,1 = 12hv,2 = N −12Nhs,1 = N16hs,2 = N −116h[r],2 = r(N −r)2N,and of course h(0),k = 0. The values of hjG/H obtainable from (9.43) may bedetermined by picking speciﬁc representations λG and λH at the appropriatelevels and taking the diﬀerence of their conformal weights.

In the case underconsideration, λG is speciﬁed by two SO(N)1 representations, and λH by anySO(N)2 representation allowed in their product. Using v × 1 = v, for example,gives the coset conformal weight hv,1 −hv,2 = 1/(2N) = h(±1, 0).

From s×s ⊃[r]+. .

. , we calculate 2hs,1−h[r],2 = (2r−N)2/(8N) = h(r, −1), giving a varietyof the weights of (9.44).

s × 1 = s, on the other hand, gives 2hs,1 −hs,2 =116,the dimension of the twist ﬁeld in the S1/Z2 orbifold model. In fact, takingappropriate modular invariant combinations of SO(N)1 × SO(N)1SO(N)2characters, we can realize either the circle or orbifold partition functions atr =√N/2.

These partition functions are thereby organized into characters ofthe extended algebras that exist at these points.9.6. Modular invariant combinations of charactersWe now turn to discuss the decomposition of aﬃne algebra representationswith respect to the coset space decomposition (9.40) of the stress-energy tensor.To this end, we begin by introducing more formally the notion of a characterof a representation of an aﬃne algebra, analogous to that considered earlier forthe Virasoro algebra.

In the case of aﬃne SU(2) for example, if we considerthe level k representation built on the spin-j vacuum state(j), then the traceχk(j)(θ, τ) ≡q−cSU(2)/24tr(j),k qL0 eiθJ30(9.45)157characterizes the number of states at any given level (as explained before (7.8)).The group structure also allows us to probe additional information, namely theJ30 eigenvalues, by means of the parameter θ.In (9.27), we have given theexplicit forms for the k = 1 characters χk=1(j=0,1/2)(0, τ) and in (9.37) for thek = 2 characters χk=2(j=0,1/2,1)(0, τ).The generalization to arbitrary group G, at level k and vacuum represen-tation with highest weight λ, is given byχk(λ)(θi, τ) = q−cG/24 tr(λ),k qL0 eiθiHi0 . (9.46)(9.46) should be recognized as the natural generalization of ordinary characterformulae except with the Cartan subalgebra, i.e.

the maximal set of commutinggenerators Hi0, extended to include L0 as well. For cases realizable in terms offree bosons or fermions, the characters take simple forms as in (9.27) and (9.37).In other cases, they can be built up from bosonic and parafermionic characters(see e.g.

[86]). In general there exists a closed expression for these characters(see e.g.

[97][83]), known as the Weyl-Kac formula, which generalizes the Weylformula for the characters of ordinary Lie groups.It follows immediately from the decomposition (9.43) that the character ofan aﬃne G representation with highest weight λa satisﬁesχkGλaG(θi, τ) =XjχcG/HhG/H(λaG,λjH)(τ) χkHλjH (θi, τ) ≡χG/H · χHλH(9.47)(where the θi’s are understood restricted to the Cartan subalgebra of H). In(9.47) the L0 eigenvalues hG/H characterizing the TG/H Virasoro representa-tions depend implicitly on the highest weights λaG and λjH characterizing theassociated G and H aﬃne representations.

On the right hand side of (9.47)we have introduced a matrix notation (see for example [98]) in which the Gand H characters, χkGλaG and χkHλjH, are considered vectors labelled by a and jrespectively, and χG/H is considered a matrix in a, j space.Under modular transformationsγ : τ →aτ + bcτ + d ,158

the characters allowed at any given ﬁxed level kG of an aﬃne algebra transformas a unitary representationχkG(τ ′) = M kG(γ) χkG(τ),(9.48)with (M kG)ab a unitary matrix (see e.g. [97][83]).

But from (9.47) we also haveχkG(τ ′) = χG/H(τ ′) M kH(γ) χkH(τ) .Linear independence of the G and H characters then allows us to solve for themodular transformation properties of the TG/H characters, asχG/H(τ ′) = M kG(γ) χG/H(τ) M kH(γ)−1 . (9.49)For example for SU(2) level k characters, the modular transformation ma-trices for γ = S : τ →−1/τ areS(k)jj′ =2k + 21/2sin π(2j + 1)(2j′ + 1)k + 2,(9.50a)with j, j′ = 0, .

. .

, k2 (and we use the notation S ≡M(γ : τ →−1/τ)). Inparticular for k = 1, this givesS(1) =1√2111−1.

(9.50b)Using these results, we can derive the modular transformation propertiesof the characters χp,r(q) for the c < 1 discrete series. These characters werederived in [99] by careful analysis of null states, but we will never need theirexplicit form here.

(The characters can also be derived as solutions of diﬀerentialequations induced by inserting null vectors, a method that generalizes as wellto higher genus[100].) The matrix S for the transformation χp,rq(−1/τ)=Pp′,r′ Sp′r′pr χp′,r′(q) is determined by substituting (9.50a, b) in (9.49).

The resultisSp′r′pr=8m(m + 1)1/2(−1)(p+r)(p′+r′) sin πpp′msin πrr′m + 1 ,(9.51)where m = k + 2 (see eq. (4.27) of Cardy’s lectures, also [43][54][101]).159(9.49) allows us to use known modular invariant combinations of G and Hcharacters to construct modular invariant combinations of TG/H characters.

Forexample the fact that M kG is unitary (i.e. that χG†χG is modular invariant),and similarly for M kH, implies that tr χ†G/HχG/H is modular invariant.

Moregenerally given any two modular invariants for G and H characters at levels kGand kH,χkG † IkGG χkG = χkGλ† IkGλλ′ χkGλ′andχkH † IkHH χkH = χkHλ† IkHλλ′ χkHλ′ ,we see that the combinationtr IkHH† χ†G/H(τ) IkGG χG/H(τ)(9.52)is a modular invariant combination of G/H characters.9.7. The A-D-E classiﬁcation of SU(2) invariantsIt follows from (9.41) and (9.52) that modular invariants for SU(2) at levels1, k, and k+1 can be used to construct modular invariants for the (m = k+2)thmember of the c < 1 discrete series.

Arguments of [102] also combine to showthat all such modular invariants can be so constructed. Thus the challenge ofconstructing all possible modular invariant combinations of the characters of aparticular member of the c < 1 discrete series, originally posed in [43], is reducedto the classiﬁcation of modular invariant combinations of SU(2) characters forarbitrary level k. For physical applications, we are speciﬁcally interested inmodular invariant combinations that take the form of partition functions all ofwhose states have positive integer multiplicities.The problem of ﬁnding all such aﬃne SU(2) invariants was solved in [103]and is discussed further in Zuber’s lectures.

The result is that the SU(2) mod-ular invariants are classiﬁed by the same ADE series that classiﬁes the simply-laced Lie algebras. The invariant associated to a given G = A, D, E occurs foraﬃne SU(2) at level k = ehG −2.

The invariant associated to Aℓ−1 = SU(ℓ), forexample, is just the diagonal SU(2) invariant at level k = ℓ−2. The modular160

invariant combinations of c < 1 characters for the (m = k + 2)th member of theunitary discrete series are given by pairs(G, G′)(9.53)with Coxeter numbers m and m + 1.Using the coset construction (9.42),modular invariant combinations of the characters of the N = 1 superconformaldiscrete series (5.19) have been similarly classiﬁed[104].Although it is not immediately obvious why there should be a relationbetween aﬃne SU(2) invariants and the ADE classiﬁcation of simply-lacedLie algebras, some insight is given by an argument of [105]. First we recallthat an embedding H ⊂G induces a realization of aﬃne H at some integermultiple of the level of aﬃne G. One way of seeing this is to recall that thelevel satisﬁes k = 2ek/ψ2, so the level of H will be related to the level of G bythe ratio of highest roots ψ2G/ψ2H induced by the embedding.

This integer isknown as the index of embedding. It can also be calculated by working in a ﬁxednormalization, and comparing the ℓof (9.13) for a given representation of G withthat for its decomposition into H representations.

For example consider theembedding G ⊂SO(dG), dG = dim G, deﬁned such that the vector of SO(dG)decomposes to the adjoint representation of G.From (9.17), ℓ(dG)/ψ2 = 1for the vector representation of SO(dG), whereas ℓA/ψ2 = ehG for the adjointrepresentation of G. The index of the embedding is the ratio ℓA/ℓdG = ehG, andthe embedding G ⊂SO(dG) thus induces a level kehG representation of aﬃneG from a level k representation of aﬃne SO(dG).For any subgroup H ⊂G of index 1, H ⊂SO(dG) is also index ehG. Thismeans thatXriℓri/ψ2 = 1ψ2Xri,jµ2(j),rirH= ehG ,where the sum is over the weights of all representations ri of H in the decompo-sition of the vector of SO(dG).

Now consider the coset space G/H, of dimensiondG/H = dim G−dim H. With the canonical H-invariant metric and torsion-freeconnection, this space has holonomy group H so there is a natural embedding161H ⊂SO(dG/H) in the tangent space group. The H representations in the de-composition of the vector of SO(dG/H) are the same as for the vector of SO(dG),except for the removal of one occurrence of the adjoint representation of H. Itis easy to calculate the index of the embedding H ⊂SO(dG/H) in the casethat H is simply-laced, for which from (9.33) we have P(adj H) µ2/rH = ehHψ2.Removing a single adjoint representation of H from the equation above, we ﬁndXri′ℓri/ψ2 = 1ψ2Xri,jµ2(j),rirH−Xjµ2(j),adjrH= ehG −ehH ,(9.54)and the index of H ⊂SO(dG/H) is ehG −ehH.Now recall that every simply-laced algebra G has a distinguished SU(2)subalgebra (9.20a), generated by its highest root ψ.

(We sloppily use G to referboth to the Lie group and to its algebra.) If we take H = SU(2) × K, whereK is the maximal commuting subalgebra, then G/H is a symmetric space.Consider a level 1 representation of aﬃne SO(dG/H) given by free fermions inthe vector representation as in (9.31).

This vector representation transformsunder H ⊂SO(dG/H) exactly as do the tangent space generators of G/H underH ⊂G. This is the symmetric space condition[87] cited at the end of subsection9.4, for which cH = cSO(dG/H), and for which the Virasoro algebras based onthe two aﬃne algebras coincide.

(There are actually two steps here: ﬁrst TH isequivalent to the stress-energy tensor for dG/H free fermions, second that thelatter is equivalent to TSO(dG/H)1. )As an example, we consider the case G = E8, for which H = SU(2) × E7and dG/H = 248 −3 −133 = 112.From (9.32), the level 1 representationof SO(112) has cSO(112),1 = 56.

From (9.54), we ﬁnd that the indices of theembeddings of SU(2) and E7 in SO(112) are ehE8 −ehSU(2) = 30 −2 = 28 andehE8 −ehE7 = 30 −18 = 12. It follows from (9.18) thatcSU(2),28 + cE7,12 = 28 · 328 + 2 + 12 · 13318 + 12 = 56 .The diagonal modular invariant for SO(dG/H)1 characters thus decomposesinto a modular invariant combination of SU(2)ehG−ehSU(2) × KehG−ehK characters.162

This combination always contains a piece proportional to the diagonal invariantfor the KehG−ehK characters, whose coeﬃcient is necessarily an SU(2) invariantat level ehG −ehSU(2) = ehG −2. It turns out[105] that this induced invariantis identically the one labeled by the simply-laced algebra G = A, D, E in theclassiﬁcation of [103].

It thus becomes natural that there should be an SU(2)invariant at level k = ehG−2 associated to each of the G = A, D, E algebras: eachhas a canonical SU(2) generated by its highest root and the above constructionassociates to it a particular aﬃne invariant at the required level.It is notyet obvious from this point of view, however, why all the invariants shouldbe generated this way (unless the construction could somehow always be runbackwards to start from an invariant to reconstruct an appropriate symmetricspace). A similar construction has been investigated further in [73][106] to giverealizations of the c < 1 unitary series directly in terms of free fermions.We mentioned before (9.53) that the A series corresponds to the diagonalinvariants.

The ﬁrst non-diagonal case is the D4 = SO(8) invariant that occursat SU(2) level ehSO(8) −2 = 4. It is given by|χ(0),4 + χ(2),4|2 + 2|χ(1),4|2 ,(9.55)and involves only integer spin (SO(3)) representations.

According to the dis-cussion surrounding (9.53), there are thus two possible modular invariants forthe (m = 5)th member of the c < 1 discrete series: (A5, A4) and (D4, A4). From(4.6a, b), m = 5 gives c = 4/5 and characters that we label χa, a = 0, 2/5, 1/40,7/5, 21/40, 1/15, 3, 13/8, 2/3, 1/8.

The (A5, A4) invariant is just the diagonalsum Pa χaχa, and gives the critical partition function on the torus for the ﬁfthmember of the RSOS series of [20] (described in subsection 4.4). From (9.52)and (9.55), we calculate the (D4, A4) invariant|χ0 + χ3|2 + |χ2/5 + χ7/5|2 + 2|χ1/15|2 + 2|χ2/3|2 ,(9.56)identiﬁed in [43] as the critical partition function for the 3-state Potts modelon the torus.In general the RSOS models of [20] at criticality on the torus are describedby the diagonal invariants (Am, Am−1).

The restriction on the heights in these163models can be regarded as coded in the Dynkin diagram of Am, with the nodesspecifying the height values and linked nodes representing pairs of heights al-lowed at nearest neighbor lattice points. Generalized versions[22] of these mod-els, deﬁned in terms of height variables that live on the Dynkin diagrams of anyof the ADE algebras, turn out to have critical points whose partition functionsrealize the remaining invariants.In the extended chiral algebra game, we encounter a variety of coincidences.For example, one can easily check from (9.18) that the central charge c = 2(k −1)/(k + 2) for SU(2)kU(1) coincides with that for SU(k)1 × SU(k)1SU(k)2.One can also check that (E8)1×(E8)1(E8)2 and SO(n)1SO(n−1)1 each havecG/H = 1/2, giving alternative realizations of the critical Ising model.

Anothercoincidence that we omitted to mention is that the N = 2 superconformaldiscrete series (9.38) and the SU(2) level k series (5.20) coincide (with m =k + 2). This is more or less explained by the construction of [107], in which theN = 2 superconformal algebra is realized in terms of SU(2) level k parafermionsand a single free boson (at a radius diﬀerent from what would be used toconstruct level k SU(2) currents).In the present context, we note that the partition function (9.56), whichlooks oﬀ-diagonal in terms of Virasoro characters, is actually diagonal in termsof a larger algebra, the spin-3 W algebra of [108].

This algebra can also berealized as the coset algebra SU(3)1 × SU(3)1SU(3)2 (from (9.18), we ﬁndcentral charge c = 2 + 2 −16/5 = 4/5), the diagonal combination of whosecharacters turns out to coincide with (9.56). (By the comments of the pre-ceding paragraph, there is also a relation to SU(2)3U(1), i.e.

to SU(2) level 3parafermions.) The spin-3 W algebra is generated by the stress-energy tensor Ttogether with the operator φ4,1, with h4,1 = 3 (see ﬁg.

7). These two operatorstransform in a single representation of the chiral algebra, so that the identitycharacter with respect to this larger algebra is χ′0 = χ0 + χ3.

The ﬁelds withh3,1 = 7/5 and h3,5 = 2/5 also transform as a single representation. This is aspecial case of a general phenomenon[12][109] (see also [110]): modular invari-ant partition functions of rational conformal ﬁeld theories (mentioned brieﬂy insubsection (5.3)), when expressed in terms of characters χi of the largest chiral164

algebra present, are either diagonal, P χiχi, or of the form P χi Pij χj, whereP is a permutation of the chiral characters that preserves the fusion rules.9.8. Modular transformations and fusion rulesWe close our treatment of coset theories with a discussion of some otherinformation that can be extracted from the modular transformation propertiesof the characters.

To place the discussion in a more general context, we ﬁrstpoint out that the modular transformation matrix M(γ) of (9.48) generalizes toother rational conformal ﬁeld theories. Recall that for these theories there are bydeﬁnition a ﬁnite number of ﬁelds primary with respect to a possibly extendedchiral algebra.

All coset models are examples of rational conformal ﬁeld theories(and, in fact, all rational conformal ﬁeld theories known at this writing areexpressible either as coset models or orbifolds thereof). The characters χi(q)are given by tracing over the Hilbert space states in the (extended) family ofprimary ﬁeld i, and are acted on unitarily by the matrix M(γ).

For conveniencewe continue to denote the matrix M(S), representing the action of S : τ →−1/τon the characters, by Sij.There is an extremely useful relation (conjectured in [28], proven in [30] (seealso [109]), and discussed further in Dijkgraaf’s seminar) between this matrixand the fusion algebra (5.15). The statement is that S diagonalizes the fusionrules, i.e.

Nijk = Pn Sjn λ(n)iS†nk (where the λ(n)i’s are the eigenvalues of thematrix Ni). This relation can be used to solve for the (integer) Nijk’s in termsof the matrix S. If we use i = 0 to specify the character for the identity family,then we have N0jk = δkj .

It follows that the eigenvalues satisfy λ(n)i= Sin/S0n,so thatNijk =XnSjn Sin S†nkS0n. (9.57)We stress that it is not at all obvious a priori that there should be arelation such as (9.57) between the fusion rules and the modular transformationproperties of the characters of the algebra.

Applied to (9.50a), for example, wederive the fusion rules for aﬃne SU(2),φj1 × φj2 =min(j1+j2, k−(j1+j2) )Xj3=|j1−j2|φj3 ,165in agreement with the result derived alternatively by considering the diﬀerentialequations induced by null states as in [83].We sketched a similar diﬀerential equation method before stating the fusionrules (5.14) for the c < 1 theories. We are now in a position to see how thefusion rules for these theories can instead be inferred directly from the cosetconstruction: the result (5.14) is easily derived directly from (9.51) by using(9.57).

Since the matrix S is eﬀectively factorized into the product of S matricesfor SU(2) at levels k = m −2 and k + 1 = m −1, we see that the fusionrules similarly factorize. This derivation thus explains our earlier observationsconcerning the resemblance of (5.14) to two sets of SU(2) branching rules.10.

Advanced applicationsLecture 10, in which further extensions and likely directions for futureprogress would have been discussed, was cancelled due to weather.166

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