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The Supersymmetric σ-Model and the Geometry

arXiv:hep-th/9109047v1 25 Sep 1991September 1991UCB-PTH 91/37LBL–31093LPTHE 91–43The Supersymmetric σ-Model and the Geometryof the Weyl-Kˇac Character Formula∗Orlando AlvarezDepartment of PhysicsUniversity of California at BerkeleyandTheoretical Physics GroupLawrence Berkeley LaboratoryBerkeley, CA 94720, USAI.M. SingerDepartment of MathematicsMassachusetts Institute of TechnologyCambridge, MA 02139, USAPaul WindeyLPTHE†Universit´e Pierre et Marie CurieTour 16, 1er ´etage4 Place JussieuF–75252 Paris CEDEX 05, FRANCEAbstractField theoretic and geometric ideas are used to construct a chiral supersym-metric field theory whose ground state is a specified irreducible representation of acentrally extended loop group.

The character index of the associated supercharge(an appropriate Dirac operator on LG/T) is the Weyl-Kˇac character formula whichwe compute explicitly by the steepest descent approximation.∗This work was supported in part by the Director, Office of Energy Research, Office of High Energyand Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under contractDE-AC03-76SF00098, in part by the Division of Applied Mathematics of the U.S. Department of Energyunder contract DE-FG02-88ER25066, in part by the National Science Foundation under grant PHY-90-21139, and in part by the Centre National de la Recherche Scientifique.†Laboratoire associ´e No. 280 au CNRS.

1IntroductionMany results in Lie group theory have both an algebraic and a geometrical origin. TheWeyl character formula [1] is no exception and among the many proofs of it some arepurely algebraic in nature, others purely geometrical.This complementarity extendsoften to the case of infinite dimensional loop groups with central extension (for example,[2, 3, 4]).

In view of the current importance of loop groups and of the related affine Liealgebras in theoretical physics it is useful to have, as much as possible, independent proofsof important results based on physical methods. It is well known from past experiencein quantum mechanics that the operator formalism is best suited to obtain algebraicinsight while the path integral is better apt to reveal the geometrical foundations of aparticular theory.

We adopt this viewpoint to derive the central result of this paperwhich is the Weyl-Kˇac character formula. The motivation is to get a better geometricalunderstanding of some related aspects of loop groups and conformal field theory.

Todevelop the necessary machinery in field theory it is always useful to have as much aspossible a quantum mechanical analogy. The corresponding derivation of the classicalWeyl character formula was previously published separately [5] and will be referred to as Ithroughout the rest of this paper.

The reader unfamiliar with the material of Section 2is encouraged to consult it for a pedagogical introduction to the methods used here.We obtain the Weyl-Kˇac character formula by computing the index of a certain Diracoperator on an infinite dimensional manifold. The index obtained is different from previ-ous elliptic genus computations [6, 7, 8, 9, 10] which were associated with Dirac operatorson LM, the loop space of a finite dimensional manifold M. The elliptic genus from thealgebraic topology viewpoint is discussed in [11].

Here we discuss the Dirac operatoron LG/T, a homogeneous space naturally associated to a connected, simply connected,simple, compact Lie group G with maximal torus T. The space LG/T is not the loopspace of any manifold. However, there is still an S1 action on it which plays an importantrole since it is responsible for the affine grading of the character.The paper is organized as follows.

In Section 2 we review the Borel-Weil construc-tion of the representations of a loop group LG. The representations are obtained asholomorphic sections of line bundles over the coset space LG/T.

We also explain thecrucial role of gLG, the central extension of LG. In particular we discuss how the centralextension may be seen as a U(1) bundle over LG [3], an interpretation which is importantfor our purpose.

We also discuss the generic features of the construction of Atiyah andBott [12, 13] which links group characters with fixed point formulas and character indices.This analysis permits us to identify the correct physical theory whose supersymmetricground states will realize the representations of the loop group. We also introduce the1

partition function which will give the character formula for loop groups.The explicit realization of these ideas in the framework of a very special field theorywith chiral supersymmetry as well as the construction of its lagrangian is developed inSection 3. There we introduce the coupling of “matter”, i.e.

the T gauge couplings whichcorrespond to the Borel-Weil bundles mentioned above. This entire construction requiresa delicate extension of the concept of horizontal supersymmetry already introduced in I.Finally the explicit computation of the Weyl-Kˇac character formula is detailed inSection 4.

Most of our notational conventions are defined in Appendix A and we will usethem freely without further notice.Bernard [14] has derived the Weyl-Kˇac character formula by using a mixture of con-formal field theory and mathematical results.He uses mixed Virasoro — affine Liealgebra Ward identities in the WZW model [15], properties of the Macdonald identitiesderived without using affine algebras, and a variety of results related to the heat kernelon G. A field theoretic purely algebraic construction of the Weyl-Kˇac character formulawas presented by Bouwknegt, McCarthy and Pilch [16]. These authors apply the Euler-Poincar´e-Lefshetz principle to certain free field Fock spaces which are built with the aidof BRST operators associated with “screening currents”.

Warner [17] has employed su-persymmetric index technology to give a proof of the Weyl-Kˇac character formula alongmore algebraic lines.2Borel-Weil Theory and Further PreliminariesWe now briefly set the stage for our problem (a complete treatment in the spirit ofthis paper was given in I for the ordinary Lie group case). Firstly we will build therepresentations of the loop group following the method of Borel-Weil:• to each irreducible representation we associate its infinitesimal character in themaximal torus of the group (essentially the highest weight of the representation);• this character uniquely defines a line bundle over the complex manifold formed bythe coset space of the group over its maximal torus;• the holomorphic sections of the line bundle provide an explicit construction of therepresentation.The group we have to consider is gLG, the central extension by U(1) of the loop groupLG (which locally looks like LG × U(1)).

The multiplication of two elements (in local2

coordinates) (g(x), u) and (g′(x), v) of gLG is given by(g(x), u)(g′(x), v) = (g(x)g′(x), uvΦ(g, g′)) ,(2.1)where Φ denotes the cocycle associated with the U(1) central extension of the loop group.In this case the steps outlined above construct the line bundle L(λ,k) over gLG/(T ×U(1))associated with a character (λ, k) of T × U(1). Three important remarks are in orderhere.

Firstly, the space gLG/(T × U(1)) is isomorphic to LG/T. Secondly one can viewthe central extension gLG as a special U(1) bundle over LG1.

Thirdly, the special linebundle L over LG/T arising from the basic central extension of LG (as discussed inthe Introduction) is isomorphic to L(0,1). The transposition of these ideas to a physicalcontext is a priori straightforward: we construct a quantum mechanical system whoseconfiguration space is the coset manifold LG/T and couple it to an external gauge fieldcorresponding to the group T × U(1) via an ordinary minimal coupling Aµ ˙xµ.Thewave functions of this system will be sections of L(λ,k).

The subtlety lies in the properidentification of the U(1) part of this coupling.To elucidate this point it is best toview a quantum mechanical system over LG as a non-linear two dimensional σ-modelwith group G. It can then be shown that the central U(1) coupling corresponds to theaddition of the Wess-Zumino term (see below.) We still have to quotient by T × U(1).This amounts to choosing an appropriate connection as we will see in Section 4.The second main ingredient we need is, mutatis mutandis, the Atiyah-Bott construc-tion [12, 13].

Since the irreducible representation coincides with the holomorphic sectionsof L(λ,k), it is clear that they belong to the kernel of ¯∂⊗IL(λ,k), the ¯∂operator over LG/Ttwisted by the line bundle L(λ,k). The computation of the index of this operator shouldin principle give us the dimension of the representation2 while to find the character of theirreducible representation we have to compute the character index.

We can equivalentlywork with the Dirac operator ∂/ provided we compensate by an extra twist (see I) to makeup for the difference between the Dirac operator and ¯∂.The analogous statements in our physical setting are familiar: we first construct thesupersymmetric extension of the model (the generator of supersymmetry is identifiedwith the Dirac operator3 on the configuration space) and then compute Tr(−1)Fg toobtain the character index formula [19, 20]. Here (−1)F is the fermion parity operator1For more details on these questions we refer the reader to the excellent exposition given in the bookof Pressley and Segal [3].

We remind the reader that the Lie group G is connected, simply connected,simple and compact.2Actually one should prove a vanishing theorem since the irreducible representation is given by thecohomology group H(0,0)(LG/T, L(λ,k)) and the rest of the cohomology groups are required to vanish.3The Dirac operator we consider is the naive Dirac operator plus Clifford multiplication by the naturalS1 vector field [18] plus appropriate gauge couplings.3

and g ∈T. Again this construction carries over to the loop group case; one simplyworks with the Dirac-Ramond operator ¯G0 with appropriate gauge couplings.

The Dirac-Ramond operator is the generator of supersymmetry in our σ-model. Notice that theabove discussion implies that one should have only one supersymmetry generator: ourconstruction will be chiral in an intrinsic way.

The general principles involved in thecomputation of the index of the Dirac-Ramond operator are by now standard from thework on elliptic genera [6, 7, 8, 9, 10]. The main task facing us is thus the constructionof the lagrangian germane to the situation we have just analyzed.

Let us warn the readerthat the description just given is very sketchy, in particular we will see below and in thenext section that the naive σ-model lagrangian is completely unacceptable before we addthe Wess-Zumino term. The introduction of the central extension is forced by reasonsof symmetry.

All these details as well as the boundary conditions will be discussed atlength below and in the next section.Next we note that there is a left action by the maximal torus of the group (hereT × U(1)) on the coset space. The fixed points of this action are the affine Weyl groupWaffdefined in Section 4.

In the loop group case this fixed point set is infinite; still weexpect the index computation to reduce to a neighborhood of the fixed point set [3]. Thefirst step in implementing these ideas should be the construction of a lagrangian whichadmits the loop group LG as a group of symmetries.

As previously mentioned, let us seewhy the obvious attempt at constructing such a lagrangian fails. The simplest choice isthe standard (1 + 1)-dimensional nonlinear σ-model defined by a map g : Σ →G wherethe world sheet Σ will be taken to be a torus.

The dynamics of the model defined by theclassical actionZΣ d2z Tr(g−1∂ag)(g−1∂ag)(2.2)may be interpreted as the motion of a “particle” on the loop group LG. Unfortunately,(2.2) does not have a large enough symmetry group for our purposes.The classicalaction is not invariant under the action by the loop group LG.

In fact, the symmetrygroup of (2.2) is G × G where the group action is defined by g(x, y) 7→hLg(x, y)h−1R for(hL, hR) ∈G × G. The hamiltonian defined by (2.2) does not have the LG symmetrywe require. However, it is well known [15] that the addition of the Wess-Zumino termto the lagrangian (2.2) extends the symmetry group to LG × LG (in Minkowski space).The Wess-Zumino-Witten (WZW) action [15] readsIWZW=k6πiK(hψ, hψ)−6iZΣ d2z Trn(g−1∂zg)(g−1∂¯zg) −γ∂zγ + (g−1∂zg)γγo+ZB Tr(g−1dg)3 + 6iZΣ d2z Tr(g−1∂zg)γγ,(2.3)4

where B is a three manifold such that ∂B = g(Σ), Tr stands for Trad, and k is a positiveinteger; see Appendix A for the notation. This model is conformally invariant and alsoformally admits LGC × LGC as a symmetry group, where GC is the complexificationof the Lie group G.To be more precise one has the following formal symmetry ofthe action: g(z, ¯z) 7→hL(z)g(z, ¯z)hR(¯z)−1 where we think of the left action as beinggenerated by locally holomorphic maps into GC and the right action being generatedby locally antiholomorphic maps into GC.

For many practical purposes we may think ofLGC as analytic maps of an annulus into the group GC. The relative normalization of thekinetic energy term of (2.3) and the Wess-Zumino term was forced on us by demandingthat we choose a lagrangian which admits an LG × LG symmetry4.We now explain the phrase “Wess-Zumino” term which is liberally used throughoutthis article.

A Wess-Zumino term is a special case of the following general set up. AssumeM is a connected, simply connected manifold with a line bundle with connection A. Thelagrangian describing the motion of a particle (on the base M) moving in the presenceof the connection A will generically have three types of terms: kinetic energy terms,potential energy terms and a gauge coupling term.We are interested in the gaugecoupling term.

Assume a path γ begins at a point u0 ∈M and ends at u1. The gaugecoupling term contribution to the path integral is simplyexpZγ A ,(2.4)i.e., parallel transport from u0 to u1 along γ.

Note that this object transforms “bi-locally”under a gauge transformation. When γ is a loop, the simply connected nature of M tellsus that γ = ∂D where D is a disk.

In this case we see thatexpZγ A = expZD F(2.5)where F = dA is the curvature. Thus for loops, the lagrangian may be formulated interms of curvature.

Note that expRD F is independent of choice of disk because the firstChern class of a line bundleR iF/2π is integral. In the case where M = LG the termRD F is called a Wess-Zumino term.Although we have to consider open paths, we will be able to formulate the problemin terms of curvature which simplifies calculations.Our path integral calculation isdominated by the critical points of the steepest descent approximation.

Since the resultis given exactly by the quadratic approximation all we have to do is understand whathappens in a neighborhood of a critical point uc. Pick a family {Γ(u)} of fiducial paths4Throughout this article we will follow the physics convention of referring to this symmetry as LG ×LG.

A further discussion of LG versus LGC will be given shortly.5

connecting the origin uc to a point u in the neighborhood. If γ is a path connecting theinitial point u0 to the final point u1 then the gauge coupling term may be written asexpZγ A = exp(ZD(γ) F −ZΓ(u0) A +ZΓ(u1) A),(2.6)Where D(γ) is a disk with boundary given by the loop Γ(u0) ◦γ ◦Γ(u1)−1.In thesteepest descent approximation to the path integral, we have to sum over all paths γ inthe neighborhood of the critical point uc.

As the path γ varies, the only term in (2.6)which changes is the curvature term. The line integrals along Γ(u0) and Γ(u1) are thereto enforce the gauge transformation properties of parallel transport.

Remember that thecurvature term is gauge invariant. The situation is actually a bit better as we will seelater in this section.Even though the correct coupling is (2.4) we will abuse the situation and write it as aWess-Zumino curvature term.

The justification is two fold. First, we have the discussionof the previous paragraph.

Second, in the case of a loop group, it is very easy to writethe curvature yet the expression for the connection is neither nice nor illuminating. Fromnow on we will blur the distinction between a Wess-Zumino term and the correct gaugecoupling.

We interpret a Wess-Zumino term as parallel transport when necessary.The connection to Borel-Weil theory will require a supersymmetric model and in an-ticipation we have included a (0, 1/2) fermion γ in (2.3). The fermion γ(z, ¯z) is a leftinvariant element of the Lie algebra of G and is the superpartner of g (see Section 3).Equation (2.3) is a chiral (0, 1) supersymmetric extension of the ordinary WZW action5.Notice that a term of the form (g−1∂zg)γγ does not appear in (2.3) due to a cancellationbetween the contribution in the curly braces and the one in the square brackets.

Thecurly braces expression and the square brackets expression are each independently su-persymmetric. In fact, the term in curly braces is the generalization of equation (4.15)of I to field theory.

The term in square brackets is the supersymmetric version of theWess-Zumino term. It is of the form A ˙x + Fψψ discussed in I (actually the A ˙x term iswritten as a curvature term).

Later in this section we will see that the curvature is givenby (2.28) and thus (g−1∂zg)γγ is of the Fψψ type.The classical equations of motion are∂zg−1∂¯zg=0 ,(2.7)∂zγ=0 . (2.8)5A non-chiral (1, 1) supersymmetric WZW was first studied in [21].

The model discussed here is quitedifferent.6

Action (2.3) is invariant under the supersymmetry transformations:g−1δsg=εγ ,(2.9)δsγ=ε(g−1∂¯zg −γγ) ,(2.10)where ε is an anticommuting parameter.The associated supercurrent has conformalweight (0, 3/2) and is defined byS ∝Trhg−1∂¯zgγ + γγγi. (2.11)The two current algebras associated with LG × LG are given byJ¯z=2kK(hψ, hψ)g−1∂¯zg + γγ,(2.12)Jz=−2kK(hψ, hψ) (∂zg) g−1 .

(2.13)Note that J¯z generates the right group action and Jz generates the left group action. Toeach element X ∈Lg, the Lie algebra of LG, we associate the operatorJX=−Idz2πiK(X(z), J(z))(2.14)=Idz2πiXa(z)Ja(z)(2.15)=2kIdz2πiK(∂zgg−1, X)K(hψ, hψ).

(2.16)The affine algebra is given by the commutation relations:[JX, JY ] = J[X,Y ] −2kIdz2πiKX(z), dY (z)dzK(hψ, hψ),(2.17)or in an orthonormal basis where K(ea, eb) = −δab and [ea, eb] = fabcec define the structureconstants, the associated operator product expansion isJa(z)Jb(w) ∼−δabK(hψ, hψ)2k(z −w)2 + fabcJc(w)(z −w) . (2.18)We now return to the discussion of LG and LGC.

For simplicity we will temporarilyassume that the worldsheet Σ is either Minkowski or Euclidean space and only providea “local description”. The confusion in whether to write LG or LGC arises in the Wickrotation from Minkowski space to Euclidean space.

The physical world sheet is Minkowskispace. The analogues of complex coordinates are the light cone coordinates x± = x ± t7

where x is the spatial coordinate and t is the temporal coordinate. In terms of thesecoordinates, the symmetry of the WZW model is g(x, t) 7→hL(x−)g(x, t)hR(x+).

Weimmediately see that hL and hR are functions of a single real variable. If we take theworldsheet to be a Minkowski cylinder then we will have a legitimate LG×LG symmetry.When we Wick rotate to Euclidean space x−→z and x+ →¯z, so hL(x−) and hR(x+)become functions of z and ¯z respectively; thus one has to complexify and look at analyticand anti-analytic maps into GC.

The following observation illustrates the nature of hL(z).Consider the standard mode expansion of the current operators in conformal field theoryJa(z) =∞Xn=−∞Ja,nzn+1 . (2.19)The hermiticity of the currents in Minkowski space translates into the operator relationsJ†a,n = Ja,−n in the conformal field theory.

The operatorexp JX = expIdz2πi Xa(z)Ja(z)(2.20)can be a unitary operator on the Hilbert space if X(z) is chosen appropriately. If in themode expansion Xa(z) =P∞−∞Xan/zn we require Xa−n = −Xan then JX is antihermitianand one formally gets a unitary operator exp JX on the Hilbert space.

It is in this sensethat one has a map into LG, more precisely, a unitary representation of the centrallyextended loop group. Such a X(z), which in general is not an analytic function, maybe formally considered a map of the annulus into g. Note that on |z| = 1, Xa(z) ispure imaginary and thus define via exponentiation a map into LG.

Often in physics oneconcentrates collectively on the basis {Ja,n} and thus the distinction of whether one isworking on Lg or LgC is blurred.For our purposes we will need a different interpretation of the Wess-Zumino termin (2.3). Notice that this term is first order in the time derivatives and thus is of theAµ(x) ˙xµ form previously mentioned and also described in detail in I. Equivalently, thecentrally extended loop group gLG may be interpreted as a U(1) bundle over LG, seefor example [3].

The WZW action describes the motion of a superparticle on LG in thepresence of a U(1) gauge potential; therefore the quantum mechanical wavefunction forthis system is a section of a line bundle over LG with first Chern class k.In summary, we found a supersymmetric action IWZW for a superparticle moving onLG which admits LG×LG as a symmetry group — this is still too large a group since theHilbert space would decompose into representations of LG×LG [22]; we need only one LGsymmetry factor. Right now we are at the same developmental stage as equation (4.15)of I where we had a Lagrangian for a superparticle moving on G with symmetry groupG×G.

Now we can exploit the full machinery developed in that paper: in particular, we8

can use the horizontal supersymmetry construction to build a supersymmetric σ-modelfor a particle moving on LG/T. Our construction guarantees that the model remainssupersymmetric and only admits a left action by the loop group LG as a symmetry; inprojecting down from LG to LG/T we lose the right action of LG as a symmetry.

Ifwe write L(λ,k) as L(0,k) ⊗L(λ,0) then we are still missing the implementation of the linebundle L(λ,0) over LG/T, a problem we address in Section 3.The supersymmetric LG/T model we schematically described above admits the fol-lowing maximal set of commuting operators:• P0: the hamiltonian which generates time translations;• P1: the momentum which generates spatial translations;• (−1)F: the fermion parity operator;• {Hi}: a basis for the Cartan subalgebra corresponding to the left T action onLG/T.It is important to notice that the holomorphic sector (right moving) and antiholomor-phic sector (left moving) of the σ-model are not identical. Our σ-model has a (0, 1)-supersymmetry which acts only on the left moving sector.

Also, the Virasoro centralextensions c and ¯c of the left and right moving sector do not coincide. It is convenientto introduce the operators L0 and ¯L0 defined by:P0=(L0 −c24) + (¯L0 −¯c24) ,(2.21)P1=(L0 −c24) −(¯L0 −¯c24) .

(2.22)The supersymmetry generator ¯G0 is related to ¯L0 by¯G20 = ¯L0 −¯c24 . (2.23)Of fundamental importance in our work is the quantum mechanical partition functionZ(θ, τ1, τ2)=Tr(−1)Feiθe2πiτ1P1e−2πτ2P0=Tr(−1)FeiθqL0−c/24 (¯q)¯L0−¯c/24(2.24)where q = exp(2πiτ), θ =Pj θjHj ∈t, and τ = τ1 + iτ2.Using {(−1)F, ¯G0} = 0and the usual pairing of states argument (implied by 2.23) one concludes that the fulltrace reduces to a trace only over the kernel of ¯G0.

This kernel consists of precisely the9

supersymmetric states of the theory, namely those states Ψ of the Hilbert space whichsatisfy ¯G0Ψ = 0. The partition function may be written asZ(θ, τ1, τ2) = TrSUSY(−1)FeiθqL0−c/24 .

(2.25)In the aboveTrSUSY means the trace only over the kernel of ¯G0. Note that Z(θ, τ1, τ2) isan analytic function of τ and that it is the character index of ¯G0.

The analyticity of thepartition function in τ plays a crucial role in our path integral computations. We willstudy the path integral in the τ2 →0 limit.

In this limit, the path integral is dominatedby critical points and we show that the quadratic approximation near the critical pointsleads to an analytic function of τ. The corrections to the quadratic approximation are apower series in √τ 2 and thus will not be analytic.

Supersymmetry tells us that all theseterms must vanish. Thus the path integral in the τ2 →0 limit may be used to calculatethe index.At the risk of repeating ourselves, perhaps a more mathematical synopsis of thispaper would be useful.

We learn from examining the elliptic genus that there are twoways of computing the S1–index of the Dirac operator ∂/ on LM (in the weak couplinglimit [23]). One can use a fixed point formula or one can use path integrals generalizingthe supersymmetric quantum mechanics derivation of the index formula for the Diracoperator [24, 25, 26].In [3] one finds a heuristic sketch deriving the Weyl-Kˇac character formula via thefixed point method extending Atiyah and Bott for the Weyl character formula.

As a warmup exercise, we derived the Weyl formula via path integrals in I. Here we “complete thesquare” by using path integrals to obtain the Weyl-Kˇac formula.One expects that the extension from G to LG should be routine but there are severalobstacles.

First, the standard supersymmetric non-linear sigma model Lagrangian (2.2) isnot invariant under left or right translation by elements of LG (because of the derivativein the S1 direction). Adding a Wess-Zumino term restores LG invariance, and has ageometric interpretation as parallel transport for a line bundle with connection over LG.Now the Lagrangian for paths on LG/T is simple: the usual kinetic term for thecurve and its fermionic partner (a tangent vector field along the loop), potential energyterms associated with the natural vector field on LG, plus a Wess-Zumino term we havejust described.

Although the Lagrangian is conceptually simple, it is not amenable tocomputation. We need to lift curves in LG/T to curves in LG which are, of course, mapsof a cylinder (or torus) into G. An essential step is the lifting of supercurves on LG/Tto superhorizontal curves on LG.

For simplicity we discuss the nonsupersymmetric case(the reader can verify by using concepts developed in Section 3 that all the arguments10

we shall give go through in the supersymmetric case). We can then express the originallagrangian in terms of a lagrangian on the lifts.

That is done locally by a local splittingof LG into eg(U) × T using a section eg : U ⊂LG/T →LG. One is finally in a positionto compute the path integral by the steepest descent approximation at the fixed pointsof the action of T on LG/T.We now present the geometric background in a little more detail (see [3, Chapter 4]).The space LG has a natural bi-invariant inner product which at the identity element isthe inner product on the Lie algebra of LG:⟨X, Y ⟩=Z 10 dx K(X(x), Y (x)) .

(2.26)Hence LG/T has an inherited inner product, and LG is a principal bundle with group Tand has a natural connection ω — the orthogonal complement of T-orbits.The evaluation map e : S1 × LG →G gives a closed left invariant 2–form Λ on LG,given by the formulaΛ=−2πiZS1 e∗σ ,(2.27)=12πiK(hψ, hψ)Z 10 dx Tr g−1 dgdx g−1δg ∧g−1δg!,(2.28)where σ is the basic integral 3–form on G generating H3(G, Z).Now iΛ/2π is inH2(LG, Z) and so defines a line bundle LΛ over LG with connection whose curvatureis Λ. But Λ is not the pull back of a 2–form on LG/T.

It appears as if the standardWess-Zumino term on LG cannot be used to describe motion on LG/T since it does notdescend. We will see that this is not so.We could instead have used the 2–form Ωon LG withΩ(X, Y ) =iπK(hψ, hψ)*dXdx , Y+.

(2.29)For conceptual6 use Ωis much better than Λ because it is left invariant under LG. Itis easy to see that iΩ/2π is the pull back of a closed 2–form ieeΩ/2π on LG/T which isintegral so that L is the pullback of a line bundle eeL with connection eeB.We want to be as close to the WZW model as possible for practical reason, i.e., wewould like to use Λ.

But Λ = Ω+ dµ where µ is the 1–form on LG:µ(X) =12πiK(hψ, hψ)*g−1 dgdx, X+(2.30)6Stone [27] has studied the WZW action using the form Ωfrom a geometric quantization viewpoint.Alekseev and Shatashvili [28] have also discussed loop groups and their representations from the pointof view of geometric quantization.11

at g(x). Although µ does not come from LG/T, it is right invariant under T. Split µinto µv + µh, its vertical and horizontal pieces, so thatµv(W) =12πiK(hψ, hψ)*g−1 dgdx, ω(W)+.

(2.31)Now µh is the pullback of eeµh and we can modify the connection eeB to eeB + eeµh, withcurvature eeΩ+ deeµh. We use this connection in a Wess-Zumino term; when we lift tohorizontal curves, we get the same Wess-Zumino term as using Λ and its connectionA.

That is, Λ = dA, Ω= dB and Λ −Ω= d(A −B) = dµ = dµv + dµh. HenceA −(B + µh) = µv + df for some function f since LG is simply connected.

The functionf may be absorbed into the choice of A by letting A →A −df. This does not changethe curvature Λ.

On horizontal lifts µv is zero soZC A =ZC(B + µh) ,(2.32)where C is the horizontal lift of the path γ on LG/T up to the bundle. Formula (2.32)is very important from the practical viewpoint because it means that we can use theWess-Zumino termR Λ on LG in our calculations.The path integral for the motion of a particle on LG/T which we have to evaluate toget the Weyl-Kˇac character formula is a supersymmetric variant of the following:ZP(ℓ) ρ(u, ℓ) exp−IK[γ(u, ℓ)] −IV [γ(u, ℓ)] −kIP[γ(u, ℓ)] −IT[γ(u, ℓ)].

(2.33)P(ℓ) is the set of all paths γ(u, ℓ) with initial point u ∈LG/T and endpoint ℓ· ubeing the translate of u by the induced action of ℓ∈T on LG/T. The kinetic energycontribution to the action IK[γ(u, ℓ)] is simply the square of the velocity integrated alongthe curve.

The potential energy term IV [γ(u, ℓ)] is the square of the natural S1 vectorfield on LG/T (induced from the natural S1 action on LG) integrated along the curve.The parallel transport term, kIP[γ(u, ℓ)], is parallel transport on the k-th power of Lvia the connection k( eeB + eeµh). Finally we need to select a T character and for this weuse the induced natural T-connection ω on an associated homogeneous line bundle withinfinitesimal T-character λ.

IT[γ(u, ℓ)] is parallel transport on this line bundle. Thus wesee that we have a quantum mechanical system whose wave function is a section of ahomogeneous line bundle (with connection) over LG/T which we shall denote by L(λ,k).Now ℓ∈T acts on this line bundle and maps the fiber over u into the fiber over ℓ· u viaa map ρ(u, ℓ).

Putting all this together we see that (2.33) is gauge invariant.It is possible to write down the full supersymmetric action on LG/T, but it is cum-bersome to do so. It is expressed most easily on LG.12

3The LagrangianLet us summarize briefly what we have done so far. At the one loop level, the Wess-Zumino-Witten model can be seen either as a modified σ–model on a torus with targetspace G or the quantum mechanics for a particle moving on LG, the loop group ofG.

In the former approach one knows that the Wess-Zumino term renders the theoryconformally invariant and that there exists an infinite number of conservation laws corre-sponding to the generators of an affine Lie algebra at level k and the associated Virasoroalgebra. In the latter approach which better corresponds to the geometrical intuitionwe have tried to convey, the Wess-Zumino term corresponds exactly to a coupling ofthe particle to a U(1) gauge field.

This coupling, linear in the time derivative, is of theform Aµ ˙xµ and the gauge field comes from the U(1) central extension gLG of the loopgroup. It was explained previously why we have to build the operator ¯∂⊗IL(λ,k) andhow it corresponds to the generator of a chiral (0, 1) supersymmetry.

We then built thesupersymmetric extension of this model but we still need its projection to the coset spacegLG/(T × U(1)) = LG/T.The trick to constructing a supersymmetric lagrangian on LG/T is to exploit the dis-cussion of the previous section on the supersymmetric WZW model. Let us temporarilyforget about supersymmetry and review how one would construct a bosonic lagrangianon LG/T given lagrangian (2.2) on LG as a starting point.

For pedagogical reasonswe begin by discussing the example of I. The lagrangian for a particle moving on G is(g−1(y)∂yg(y))2, where g(y) is the curve on G. How does one construct the lagrangianfor the motion of the particle on G/T?

One notices that G is a principal T-bundle overG/T with a bi-invariant metric and a natural T connection (g−1dg)t. Thus G/T hasa natural metric ⟨·, ·⟩induced by the horizontal spaces of the T-connection. A curveu(y) on G/T has a unique horizontal lift to G (after specifying the starting point) whichwe will call gh(y).

From the geometry it is clear that the natural lagrangian on G/T:⟨∂yu(y), ∂yu(y)⟩is the same asTrg−1h (y)∂ygh(y)2 . (3.1)We remark that the right invariance of (3.1) under the action of T shows that (3.1) isindependent of the starting point for the lift.

This invariant description suffers at thepractical level. Namely, gh(y) is a complicated solution to a differential equation and thusgh(y) is not very useful in a path integral computation.

The solution to our dilemmais to give a local reformulation of the invariant description by exploiting the principalT-bundle structure in such a way that everything will patch smoothly. Let eg : G/T →Gbe a local section.

We can lift the curve u(y) on G/T to G as eg(u(y)). We know that13

gh(y) and eg(u(y)) are related by an element of T: gh(y) = eg(u(y))t−1(y). By using theT-connection we see that locallyTrg−1h (y)∂ygh(y)2 = Treg−1(u(y))∂yeg(u(y))2m .

(3.2)We leave it as an exercise to the reader to verify that the local description patchestogether in a natural way. Thus a section can be used to locally describe the Lagrangianin a way which as we shall see is amenable for efficient path integral use.

For example, aconvenient section near the identity of G is to write eg = exp eϕm where eϕm has values inm, and a convenient section near any other point is the left translate of exp eϕm.Let us introduce some notation for discussing the LG/T case. An element in LG willbe written g(x), x ∈[0, 1], and an element in T will simply be written t. Curves onthese spaces will also depend on the time variable y ∈[0, τ2].

From a two dimensionalviewpoint we will have fields g(x, y) and t(y) together with their respective supersym-metric partners7 γ(x, y) and bτ(y). The variables (x, y) parametrize the two dimensionaltorus.

We also define for later use the complex variables z = x + iy and ¯z = x −iy. Thegenerator of supersymmetry is given by Q = ∂θ −θ∂¯z, where θ is a grassmann variableof weight (0, −12).

We use δ to denote the differential on the infinite dimensional spaceof fields.To commence our discussion of the LG/T case we forget about supersymmetry. Thenonlinear sigma model (2.2) describes the evolution of a curve g(x, y) in LG.Thislagrangian has both a kinetic energy termZdx Tr(g−1(x, y)∂yg(x, y))2and a potential energy termZdx Tr(g−1(x, y)∂xg(x, y))2 .To construct a natural lagrangian on LG/T induced from (2.2) we exploit that LG isa principal T-bundle over LG/T with a bi-invariant metric and a natural T-connection.The T–connection on the bundle is defined as follows.

The connection 1-form ω maps atangent vector to LG at g(x) into an element of t. The tangent vector translated to theidentity in LG is an element of the Lie algebra of LG, namely Lg, and denoted by X(x).Project for each x onto t and integrate over S1:ω(X) =ZS1 dx X(x)t . (3.3)7Please note that the modular parameter of the torus is denoted by τ while the supersymmetricpartner of t is denoted by bτ.14

In terms of the left invariant differential forms on LG this may be written asω =Z 10 dx (g−1(x)δg(x))t . (3.4)More geometrically, ω is orthogonal projection of the tangent space to LG onto thetangent space to the orbit of T relative to the bi-invariant metric on LG which at theidentity isRS1 K(· , ·).

It follows that LG/T has a natural metric ⟨·, ·⟩induced by thehorizontal spaces of the connection. A curve u(y) on LG/T has a unique horizontal lift(after specifying the initial point) to LG which we will call gh(x, y).

From the geometryit is clear that the natural kinetic energy term on LG/T: ⟨∂yu(y), ∂yu(y)⟩may be writtenasZ 10 dx Trg−1h (x, y)∂ygh(x, y)2 . (3.5)Note that the potential energy term on LG descends to a function V [u] on LG/T whichis defined byV [u] =Z 10 dx Trg−1h (x, y)∂xgh(x, y)2 .

(3.6)We find ourselves in much the same situations as discussed in the G/T case. Althoughwe have an invariant formulation it turns out that working with gh is impractical.

Wegive a local description which patches together nicely. Let eg : LG/T →LG be a localsection.

We can lift the curve u(y) on LG/T to LG as eg(u(y)). We know that gh(x, y)and eg(u(y)) are related by an element of T: gh(x, y) = eg(u(y))t−1(y).

By using theconnection ω we see that the horizontal condition on gh(x, y) requires t(y) to satisfy thedifferential equationZ 10 dxeg−1(u(y))∂yeg(u(y))t −∂yt(y) t−1(y) = 0 . (3.7)If we define Fourier modeseg−1(u(y))∂yeg(u(y))t =∞Xn=−∞Hy,n(y)e2πinx(3.8)then one can see that the kinetic energy term may be written asZ 10 dx Treg−1(x, y)∂yeg(x, y)2m +Xn̸=0Tr Hy,n(y)Hy,−n(y) .

(3.9)One can verify that the kinetic energy term above patches nicely. We leave the potentialenergy term as an exercise to the reader.We now return to the supersymmetric discussion associated with the LG/T case.In what follows we will often suppress the coordinate dependence of the fields but it isimportant to remember that since t and bτ belong to T and its tangent space and not toLT, they do not depend on the spatial coordinate x.15

We will now use the natural T-connection (3.4) on LG to induce supersymmetry onLG/T from a naturally formulated supersymmetry on LG. This is precisely analogousto using a connection to define the horizontal tangent spaces on the bundle and relatingthese to tangent spaces on the base.

The importance of our construction is that it allowsus to express the supersymmetric model on LG/T in terms of quantities defined on LGsuitable for path integral use. Firstly we must define supersymmetry on LG.

Considera supercurve which may be expressed in superfield notation as G(x, y) = g(x, y)eθγ(x,y)(see I). The supersymmetric variation of G is given byδsG = εQG ,(3.10)where ε is the anticommuting parameter of the transformation.

In terms of componentsthe supersymmetry transformations are given byg−1δsg=εγ ,(3.11)δsγ=εg−1∂¯zg −γγ. (3.12)Note that the supersymmetry transformations are equivariant under the right T-actionon LG.How do we lift a supercurve on LG/T to a superhorizontal curve on LG?Thecondition that a curve in LG is the horizontal lift of a curve in LG/T is that the global 1–form ω =R 10 dx (g−1δg)t vanish when evaluated along the curve.

We generalize this to thesupersymmetric case by noticing that one can interpret (3.11) as a tangent vector; thusit is natural to impose the vanishing of ωs =R 10 dx (g−1δsg)t as the first superhorizontalcondition. Using (3.11) we see that this condition is simplyZ 10 dx γt(x, y) = 0 .

(3.13)For consistency we must also impose that the supersymmetric transform of (3.13) alsovanish:Z 10 dxg−1(x, y)∂¯zg(x, y) −γ(x, y)γ(x, y)t = 0 . (3.14)If one forgets about the fermions then the above is almost the condition that the lift behorizontal in the ordinary sense8.

The additional term is a Pauli type coupling (see I).Note that the formulation of superhorizontal has been done in a global way.The equivariance of the superhorizontality conditions tells us that arguments con-cerning the lagrangian we gave in the bosonic case will go through in the supersymmetric8It would be the standard condition if it was a derivative with respect to y, see (3.7).16

case. For example, the supersymmetric kinetic energy term on LG/T may be formulatedon LG by the use of superhorizontal lifts.We now turn to the local parametrization of supersymmetry and superhorizontal lifts.We parametrize a loop g(x) in LG by a local section eg and an element of T as g(x) = egt−1.Using the decomposition of the Lie algebra of G into g = t ⊕m leads to the equationsg−1δgt=eg−1δegt −dtt−1 ,(3.15)g−1δgm=teg−1δegm t−1 .

(3.16)To find the equivalent relations for γ it is best to reintroduce a superfield notationG = geθγ. We have the local parametrization G = fGT−1 given by the local supersectionsfG = egeθeγ and superfiber variables T = teθbτ.

This givesγt=(eγ)t −bτ ,(3.17)γm=(teγt−1)m . (3.18)In terms of the section, the T-connection in local coordinates may be written asω = A −dtt−1 whereA =Z 10 dxeg−1(x)δeg(x)t .

(3.19)This is the connection we will use to get local formulas.The supersymmetry transformations of the fiber T are given by (defining t = exp f)δstt−1=δsf = εbτ ,(3.20)δsbτ=ε∂¯zf . (3.21)We now have enough information to formulate the supersymmetry transformations ofthe local sections:eg−1δsegm (x, y)=εeγm(x, y) ,(3.22)eg−1δsegt (x, y)=δsf(y) + ε (eγt(x, y) −bτ(y))=εeγt(x, y) .

(3.23)In the last equation we have used (3.20).Using similar algebraic manipulations will give the supersymmetric transformation ofthe fermionic partner of eg. Expressing γ in terms of the section we haveδsγ=δsteγt−1 −bτ=δsteγt−1 + tδseγt−1 −teγt−1δstt−1 −δsbτ=εbτteγt−1 + εtδseγt−1 + εteγt−1bτ −ε∂¯ztt−1 .

(3.24)17

The same variation can be written by the use of (3.12) which in terms of the sectionreadsδsγ = εtheg−1∂¯zeg −t−1∂¯zt −eγeγ + eγ bτ + bτ eγit−1 . (3.25)Comparing the two expressions we findδseγ = εeg−1∂¯zeg −eγeγ(3.26)which can be decomposed asδseγm=ε(eg−1∂¯zeg)m −(eγeγ)m,δseγt=ε(eg−1∂¯zeg)t −(eγmeγm)t.(3.27)We are now ready to express the superhorizontality conditions in terms of the localsection.

We denote the superhorizontal lifts of the supercurve on LG/T by gh and γh.From (3.17) we find the first condition0 =Z 10 dx (eγt(x, y) −bτ(y)) . (3.28)Applying a supersymmetry transformation to this equation we find the second horizon-tality condition0=Z 10 dx (δseγt(x, y) −δsbτ(y))=Z 10 dxh(eg−1∂¯zeg)t −(eγmeγm)ti−∂¯ztt−1 .

(3.29)Using the mode expansioneγt(x, y) =∞Xn=−∞eγt,n(y)e2πinx ,(3.30)the first condition (3.28) giveseγt,0 −bτ = 0 . (3.31)From the expressions (3.17) and (3.18) we find γh = teγmt−1 + eγt −bτ.

Equivalently, using(3.30) and (3.31), the final form of the superhorizontal lift isγh(x, y) = t(y)eγm(x, y)t−1(y) +Xn̸=0eγt,n(y)e2πinx . (3.32)Note the important fact that the absence of zero modes implies that all dependence onbτ has disappeared.

Similar algebraic manipulations and the mode expansioneg−1∂aegt =∞Xn=−∞Ha,n(y)e2πinxfor (a = z, ¯z). (3.33)18

given ̸= 0 :g−1h ∂aght,n=Ha,n ,(3.34)n = 0 :g−1h ∂aght,0=Ha,0 −∂af . (3.35)From (3.19) we see that the T-connection in local coordinates is given byA(y) =Z 10 dx Hy(x, y) = Hy,0(y)(3.36)Note that Hx,0 is gauge invariant with respect to T gauge transformations.It is now a matter of algebra to project the kinetic part of the SUSY-WZW lagrangianon LG to LG/T:Zd2z Trg−1h ∂zghg−1h ∂¯zgh+Zd2z Tr γh∂zγh=Zd2z Treg−1∂zegmeg−1∂¯zegm+Xn̸=0Zdy Tr Hz,nH¯z,−n+Zdy Tr(Hz,0 −∂zf)(H¯z,0 −∂¯zf)+Zd2z Tr eγm(∂zeγm + [∂zf, eγm])+Zd2z ξt∂zξt .

(3.37)In the above, ξt is defined byξt =Xn̸=0eγt,n(y)e2πinx . (3.38)The Wess-Zumino term on LG restricted to a superhorizontal lift becomesZTrg−1h dgh3 =ZTreg−1deg3 −6iZdy Tr (Hz,0∂¯zf −H¯z,0∂zf) .

(3.39)For completeness we list below the expression of the superhorizontal lift in terms ofthe local section for each term in the WZW action. The bosonic kinetic energy term isgiven byZd2z Trg−1h ∂zghg−1h ∂¯zgh=Zd2z Treg−1∂zegmeg−1∂¯zegm+Xn̸=0Zdy Tr Hn,zH−n,¯z+Zdy Tr Hx,0 (eγmeγm)t,0−Zdy Tr (eγmeγm)2t,0 .

(3.40)19

The fermionic kinetic energy term isZd2z Tr γh∂zγh=Zd2z Tr ξt∂zξt + 12Zd2z Tr eγm∂xeγm−i2Zd2z Tr eγm (∂yeγm + [A, eγm])+Zdy Tr Hx,0 (eγmeγm)t,0 −2Zdy Tr (eγmeγm)2t,0(3.41)The WZ term is given byZTr(g−1h dgh)3=ZTr(eg−1deg)3 + 3Zdy Tr AHx,0−3iZdy Tr H2x,0 + 6iZdy Tr Hx,0 (eγmeγm)t,0(3.42)One can verify that the quantity in curly braces is T-gauge invariant.Collating all the terms together we find the following action for the supersymmetricmodel on LG/T:iπK(hψ, hψ)2kILG/T=−i2Zd2z Tr(eg−1∂zeg)m(eg−1∂¯zeg)m+i2Zd2z Tr eγm(∂zeγm + [Hz,0, eγm])+112ZTr(eg−1deg)3 + 3Zdy Tr AHx,0−i2Xn̸=0Zdy Tr Hz,nH¯z,−n + i2Zd2z Tr ξt∂zξt−i4Zdy Tr Hx,0Hx,0+iZdy Tr Hx,0 (eγmeγm)t,0−i2Zdy Tr (eγmeγm)t,0 (eγmeγm)t,0 . (3.43)It is not clear to us which is the best way of writing the above.

The reason is that theLG/T lagrangian is not Lorentz invariant and therefore there is no obvious way to groupthe terms. We decided on the above grouping because it makes certain features clear.The first two lines are the kinetic energy terms for bosons and fermions on L(G/T).

Thethird line is the Wess-Zumino term on LG/T. The fourth line is the kinetic energy termsfor bosons and fermions on LT/T.

The last three lines are respectively T gauge invariantpotential energy, Yukawa and curvature terms which are required by supersymmetry.It is important to observe that the lagrangian we just derived solves one of our mainproblems which was to find a good way of handling curves on LG/T. The main tool20

used in this respect was the implementation of horizontal supersymmetry which allowsus to lift supercurves on LG/T to superhorizontal curves on LG, a space which is moreamenable to field theoretic methods.Our next task is to describe the appropriate modifications of this basic lagrangianwhich will give the coupling to the different irreducible representations of LG. In whatfollows we will often refer to it as the matter coupling, adopting the traditional fieldtheoretic language.

Firstly we construct the line bundles over LG/T and study the U(1)and T action on them. We have explained above why these bundles play such a crucialrole in the construction of the irreducible representations of LG.

A line bundle L(ν,0) overLG/T is determined by an appropriate one dimensional representation of the group Twith infinitesimal character ν. A section of L(ν,0) is the same as a function on the entireprincipal bundle π : LG →LG/T satisfyingF(gt−1) = ρ(t)F(g) ,(3.44)where ρ(t) is the irreducible representation of T with infinitesimal character ν.

We willoften indicate explicitly the dependence on the point u in the base LG/T; for exampleeg(u) will stand for a given local section of the bundle LG.It is important to keep in mind the difference between the right and the left action ofT. The right action lets us move up and down the fiber and tells us that the function Ftransforms under the representation given by ρ.

A local section eF of L(ν,0) must then beparametrized by the coordinates u of LG/T and is defined byeF(u) = F(eg(u)) . (3.45)This determines the left action of T on eF:LℓeF(u) = F(ℓeg(u)) ,(3.46)where ℓ∈T.Note that ℓeg(u) is a new element of LG in the fiber above the pointℓ·u = π(ℓeg(u)) in the base.

We can use right multiplication to relate ℓeg(u) to the sectionby introducing t(u, ℓ) ∈T as follows ℓeg(u) = eg(ℓ· u) t−1(u, ℓ) We now can rewrite theleft T-action on a section eF as follows:LℓeF(u)=Feg(ℓ· u)t−1(u, ℓ)=ρ(t(u, ℓ))F (eg(ℓ· u))=ρ(t(u, ℓ)) eF(ℓ· u) . (3.47)We see that the transformation law (3.47) for the sections has both an orbital part anda “spin” part.

Because of the presence of the spin part there are some phases which willhave to be accounted for in the path integral computation.21

We now have all the necessary ingredients to construct the matter coupling at thelagrangian level. This will be done by using a minimal coupling scheme, i.e., by writing aterm of the form qR dtAµ ˙xµ.

Notice that such an abelian coupling term in the lagrangiankeeps track of the change in angle along the path between the initial and final pointmultiplied by the appropriate “charge”. This fixes its normalization uniquely.

In thecase at hand we just have to compute the T-connection which is determined by (∂¯zt)t−1.Notice that t = t(y) depends only on the “time” variable y. We write t = exp f, and withthe help of (3.33), rewrite the usual T-connection in Eq.

(3.19) as A(y) =R 10 dxHy(x, y).Notice that A does not depend on x since this is a T connection and not an LT connection.Using this and the mode expansion introduced in (3.30), we rewrite the supersymmetrichorizontality condition (3.29):∂yf = A(y) −iZ 10 dxHx,0 + 2iZ 10 dx(eγmeγm)t.(3.48)Notice that the last two terms are specifically supersymmetric contributions. This equa-tion gives us the required matter action:IT = 2iZd2z ν (H¯z −(eγmeγm)t) .

(3.49)This last equation exhibits the (0, 1) nature of the coupling as required by our chiralsupersymmetry. This action gives the desired modification of the generator of sypersym-metry.

In other words it produces the appropriate twisting of the Dirac operator by theholomorphic T-bundle associated with the infinitesimal T-character ν as required by theBorel-Weil construction.4The Weyl-Kˇac Character Formula4.1The Quadratic Approximation Around the Fixed PointsThe partition function (2.24) may be computed in the τ2 →0 limit as was mentionedat the end of Section 2. We remind the reader that the partition function is actually ananalytic function of q = exp(2πiτ) and we can exploit this fact to compute it exactly.In this limit, any time dependent9 field configuration will lead to an action that behavesas 1/τ2 and thus such configurations will be suppressed.

The dominant contributions tothe path integral will arise from static field configurations which satisfy the appropriateboundary conditions. Supersymmetry requires the fields to be periodic under z →z +1.

However, it is clear from (2.24) and the discussion in I that we must use twisted9The reader is reminded that y plays the role of time.22

boundary conditions on the fields in the time direction. On the bosonic fields eg(x, y), theappropriate twisted boundary conditions are given byeg(x + τ1, τ2)T = ℓeg(x, 0)T(4.1)where ℓ= exp(iθ) is the element of T in expression (2.24) and τ1 enters because of therotation induced by the momentum operator P1.

We now use the fact that the dominantconfigurations must be static and conclude that the saddle points are described byeg(x + τ1, 0)T = ℓeg(x, 0)T . (4.2)We remark that in the above, two elements of LG are identified if they differ by anelement of T and a translation in x.

This equation has to be true for all ℓ∈T and thusthe above may be rewritten in the equivalent formeg(x + τ1, 0)−1ℓeg(x, 0) ∈T . (4.3)The solution to the above is the groupN =nn exp (2πiˇµx) | n ∈N(T : G), ˇµ ∈ˇTo,(4.4)where N(T : G) is the normalizer of T in G, and ˇT is the coroot lattice (see the Appendixfor a definition).

If we observe that the momentum operator P1 generates a circle group S1of symmetries of the lagrangian by translating the loop parameter then we can recast N ina more group theoretical setting. Consider the group S1⋉LG and note that the maximallycommuting subgroup is S1 × T where S1 is the circle group associated with translatingthe loop parameter.

Our collection N is actually the group N(S1 × T : S1 ⋉LG) andthe quotient Waff= N /(S1 × T) is a group called the affine Weyl group of LG. Fromthe definition we see that the affine Weyl group Waffis the semidirect product of W, theordinary Weyl group of G, and ˇT, the coroot lattice.

Its elements arewaff= (w, e2πiˇµx) with w ∈W . (4.5)Each element of Waffis associated with a fixed point of the S1 × T action in LG/T.

Theevaluation of the path integral by steepest descent will require a sum over the infinite setof Weyl points described above.Since we only have to study the path integral in the τ2 →0 limit, it is clear from theabove discussion that it will suffice to consider the fluctuations around the Weyl points.A supersection of LG near the Weyl point represented by n exp(2πiˇµx) may be writtenasfG(x, y) = ne2πiˇµxeeϕ(x,y)eeγ(x,y)θ ,(4.6)23

where eϕ and eγ parametrize the fluctuations. The superfield is periodic under z →z + 1and under z →z + τ it satisfies10fG(x + τ1, τ2)T = ℓfG(x, 0)T .

(4.7)If we defineκ = e−2πiˇµτ1(n−1ℓn) ∈T(4.8)then the boundary conditions (4.7) may be formulated aseϕ(x + τ1, τ2)=κ eϕ(x, 0)κ−1 ,(4.9)eγ(x + τ1, τ2)=κeγ(x, 0)κ−1 . (4.10)It is important to remember that, since we are working on the coset space LG/T, thereis no “translationally invariant” mode11 in eϕt and eγt.

We also note that ℓw ≡n−1ℓn onlydepends on the choice of coset w = nT (see I). We shall often, by abuse of notation,write w−1ℓw to remind the reader that it only depends on the choice of an element ofthe Weyl group of G. The above boundary conditions may be equivalently written aseϕm(x + τ1, τ2)=κ eϕm(x, 0)κ−1 ,(4.11)eγm(x + τ1, τ2)=κeγm(x, 0)κ−1 ,(4.12)eϕt(x + τ1, τ2)=eϕt(x, 0) ,(4.13)eγt(x + τ1, τ2)=eγt(x, 0) .

(4.14)It is easy to verify that to quadratic order near a Weyl point we have:A(y)=−12Z 10 dx [ eϕm, ∂y eϕm]t + O( eϕ3) ,(4.15)Hx,0=2πiˇµ + 12Z 10 dx [∂x eϕm + [2πiˇµ, eϕm] , eϕm]t + O( eϕ3) . (4.16)These expressions enter in the perturbative expansion of the full lagrangian near the fixedpoints.

After a considerable amount of algebra one finds the following relatively simpleform for the lagrangian to quadratic order near a fixed point:I(2)total=k4πhg(2π2τ2 Tr(ˇµˇµ) −8π2hgkτ2 Tr(νˇµ)−Zd2z Tr ∂z eϕm −12"2πiˇµ −8πihgkν, eϕm#! ∂¯z eϕm + 12 [2πiˇµ, eϕm]!10A more detailed explanation can be found in Section 4.2.11Note thatR 10 dx eϕt(x, y) = 0 and similarly for eγt.24

+Zd2z Tr eγm ∂zeγm −12"2πiˇµ −8πihgkν, eγm#!−Zd2z Tr ∂z eϕt∂¯z eϕt+Zd2z Tr eγt∂zeγt). (4.17)Formula (4.17) is the final form of the quadratic part of the lagrangian we will use; itsderivation is nontrivial and involves subtle and delicate cancellations which reflect theunderlying geometry.

The bosonic degrees of freedom which appear in the above are arepresentation of the following geometrical fact. There is a local equivalence betweenLG/T and the spaceLGLT × LTT.

(4.18)More precisely, LG/T is a principal LT/T bundle over LG/LT. Notice that LG/LTis the configuration space for an “ordinary” sigma model since one can also show thatLG/LT = L(G/T).

Locally, the fields in our model may be thought as an ordinary sigmamodel on G/T represented by eϕm and some extra abelian excitations, eϕt associated withLT/T. We have previously emphasized that the abelian excitations do not contain aconstant mode.

The gaussian integration of the above quadratic action yieldsexp"−2πkτ2K(ˇµ, ˇµ)K(hψ, hψ) + 2πτ2ν(ˇµ)#×" Yα≻0det∂¯z + 122πiα(ˇµ)#−1 "ddet(∂¯z)#−l/2,(4.19)where ddet indicates the omission of the x translationally invariant modes and l is therank of the group G.As was discussed after Eq. (3.47), we also have to take into account the prefactorscoming form the “spin” part of the transformation law arising from both the circle actionand the T action on the wave functions.

We will see that these prefactors are crucial inturning the above into an analytic function of τ as required by (2.25).4.2Group Action Around the Weyl PointsIn the partition function Z(θ, τ1, τ2) defined in Eq. (2.24), the operators eiθ and e2πiτ1P1act on the wave functions at the end of the paths or more precisely on the sections of thematter line bundles.

We saw in Section 3 that this action induces both a spin and anorbital spin part in our discussion of the line bundle L(ν,0). As discussed at the beginning25

of Section 2 we actually need to work with the line bundle L(λ,k). By mimicking thederivation at the end of Section 3, we will compute the action of S1 × T × U(1) on asection of L(λ, k) and find both a “spin” and an orbital part.

The former will appear asa prefactor in the computation of the path integral while the latter also determines theproper boundary conditions to use in the evaluation of the determinants.The general form of the left action on the sections was given in Eq. (3.47).

Using thesame notations we see that we have to determine the representation matrix ρ(t) where tis the solution of ℓg(x) = g(x)(g−1(x)ℓg(x)) = g(x)t−1 with ℓ∈T and g(x) ∈LG. Theprefactor will simply be given by the computation of ρ(t) on the line bundle L(λ,k).

Toperform this calculation we require the Lie algebra relation[(X, r), (Y, s)] = ([X, Y ], φ(X, Y )) ,(4.20)where (X, r) ∈Lg ⊕R (the infinitesimal version of the gLG multiplication law). Thealgebra cocycle is explicitly given byφ(X, Y ) = iπZ 10 dxK(X(x), ddxY (x))K(hψ, hψ).

(4.21)From the quantum field theory viewpoint it is useful to reexpress the above relationsdirectly in terms of the currents JX which were defined in (2.14). Note that we view JXas an operator acting on a Hilbert space.

In the same spirit we can often view the groupelement (g, u) as an operator of the form ueJX in the case k = 1. We then recover thegroup law as a relation between operators:ueJX veJY=uveJX+JY + 12[JX,JY ]+···(4.22)=uveJX+JY + 12(J[X,Y ]+φ(X,Y ))+···(4.23)=uve12φ(X,Y )+···eJX+JY + 12J[X,Y ]+··· ,(4.24)where we have used the current algebra (2.17).

In computing the prefactor, rememberthat the character index is localized on the fixed points of S1 × T in LG/T, so we needonly to know the behavior of the sections only around the elements waffof the affineWeyl group Waff. In a neighborhood of the fixed points we can parametrize an elementof LG/T by egw,ϕ(x) = w exp (2πiˇµx) eeϕ(x).

It is just a matter of algebra to findeiθe−2πiτ1P1 egw,ϕ(x)=we2πiˇµxeeϕ′(x−τ1)ei(θw−2πτ1 ˇµ)×exp"−2i K(ˇµ, θw)K(hψ, hψ)#×exp"2πiτ1K(ˇµ, ˇµ)K(hψ, hψ)#e−2πiτ1P1 ,(4.25)26

where we have defined θw = w−1θw and eϕ′ = e−2πiˇµxeiθw eϕe−iθwe2πiˇµx. From the abovewe deduce two results.

Firstly we see that the boundary conditions on the fluctuationsaround the fixed points are given byeϕ(x + τ1, τ2) = eϕ′(x, 0) . (4.26)Secondly the prefactor (or spin part) for a bundle with weight (λ, k) is:eiλ(θw−2πτ1 ˇµ) exp"k 2i K(ˇµ, θw)K(hψ, hψ) −2πiτ1K(ˇµ, ˇµ)K(hψ, hψ)!#.

(4.27)It is worth mentioning that this factor is independent of the scale of the scalar product.The inclusion of the superpartner does not modify the discussion above.In more mathematical terms, the line bundle L(λ,k) is induced from the representation(0, λ, k) on S1×T×U(1) for the principal bundle S1× gLG over (S1× gLG)/(S1×T×U(1)) =LG/T. Left translation by S1 × T × U(1) has an induced action on L(λ,k).

At a fixedpoint of the action of S1×T given by the affine Weyl coset waffT = we2πiˇµxT, the inducedaction is multiplication by a complex number of modulus one on the line L(λ,k) at thecoset waffT. That number in terms of λ, k and waffis given by formula (4.27).

In otherwords, the spin part of the action is the lift of left translation to the line bundle.4.3DeterminantsThe evaluation of the determinants will follow the discussion given in [7, 6]. Becausewe are working on a torus one can associate eigenvalues with each of the first orderdifferential operators which appear in (4.17).

In any reasonable regularization schemeone will have a term by term cancellation between the fermionic modes and the “anti-holomorphic” bosonic modes. This is guaranteed by the existence of the supersymmetry.Thus only the “holomorphic” bosonic sector contributes in a non trivial way.

We wouldlike to regulate the determinants in such a way that holomorphicity is preserved.The determinant of ∂¯z is easily evaluated with the resultddet∂¯z = η(τ)2 ,(4.28)where the Dedekind η-function is defined byη(τ) = q1/24∞Yn=1(1 −qn) . (4.29)Since our lagrangian describes particle motion on LG/T we note that there are no “point-like particle” modes12 associated with T ensuring the absence of any dependence on τ2 in12A pointlike particle mode would be the time evolution of a constant loop.27

the determinant ddet∂¯z. Had the pointlike particle modes associated with T been presentthen we would have found det′ ∂¯z = 2τ2η(τ)2 as the result of the gaussian integration13.We should carefully keep track of all such extraneous factors because equation (2.25) tellsus that the final answer must be an analytic function of τ.We will now carefully definedet∂¯z + 122πiα(ˇµ)(4.30)in a way which preserves holomorphicity in τ and guarantees the correct periodicity inT.

By using the boundary condition one easily determines that the eigenvalues are givenbyπτ2(m + nτ −ζ)(4.31)where m and n are integers, andζ = α(θw)2π−α(ˇµ)τ . (4.32)It is useful to study the following formal ratio of determinantsdet∂¯z + 122πiα(ˇµ)det′ (∂¯z)=Y πτ2(m + nτ −ζ)Y′ πτ2(m + nτ)(4.33)where the prime again means to eliminate the m = 0, n = 0 mode.

Note that the righthand side is formally an odd function of ζ. The above ratio may be written as−2πζ2τ2Y′ 1 −ζm + nτ!,(4.34)which is only formal since the product is divergent.

We proceed in two different ways.Firstly we note that (4.34) is essentially the definition of the Weierstrass σ-function. Ifwe employ some unspecified cutoffthen it may be rewritten as:det∂¯z + 122πiα(ˇµ)=−2πη(τ)2σ(ζ; τ)×exp−X′ζm + nτ −12X′ ζm + nτ!2.

(4.35)The entire issue revolves on how we handle the divergent sums. Expression (4.35) alreadytells us that the ambiguity in different regularizations is a quadratic polynomial in ζ.13The prime means the omission only of the zero eigenvalue mode.28

Secondly, this may also be seen differently by noticing that formally∂3∂ζ3 log det∂¯z + 122πiα(ˇµ)(4.36)is finite without need for regularization. We can define the regularized determinant bythe differential equation∂3∂ζ3 log det∂¯z + 122πiα(ˇµ)= −X2(m + nτ −ζ)3 .

(4.37)The right hand side is an elliptic function, the derivative of the Weierstrass ℘function,because the sum is uniformly convergent. Integrating (4.37) leads to an expression whichwill be ambiguous by a quadratic polynomial in ζ.

In conclusion we have a holomorphicregularization scheme which leads to a polynomial ambiguity as in all renormalizablequantum field theories. For our purposes it is more convenient to express the answer interms of ϑ-functions14.

The following identity−2πη(τ)2σ(ζ; τ) = ϑ11(ζ; τ)η(τ)eη1ζ2(4.38)allows us to express the determinant asdet∂¯z + 122πiα(ˇµ)= ϑ11(ζ; τ)η(τ)eP(ζ) ,(4.39)where P(ζ) is a quadratic polynomial in ζ. It is easy to resolve the ambiguity in thedefinition of the determinant.

We note that as we go around a cycle in the maximal torusT the variable ζ shifts by an integer, see (4.32). For example, in SU(2) the change is 2 aswe go around the cycle in T. Since we are interested in studying integral representationsof gLG, it follows that the determinant should be periodic under ζ →ζ + r, where theinteger r is the greatest common factor of the shifts when all cycles of T are considered.The function ϑ11(ζ; τ) has the aforementioned periodicity; hence the polynomial mustbe of the form P(ζ) = 2πipζ/r + constant, where p is an integer.The integer p iszero because it is natural to require the determinant to be odd under ζ →−ζ as waspreviously remarked.

Thus we conclude that p = 0. The value of the constant may befixed by requiring that the n = 0 mode of the determinant reproduces the correspondingone for a point particle (up to the standard central charge correction).

In summary,we can choose P(ζ) to be an appropriate constant and all properties we require will besatisfied. Thus we summarize by saying that the value of the determinant in question isdet∂¯z + 122πiα(ˇµ)=−iϑ11 α(θw)2π−α(ˇµ)τ; τ!η(τ).

(4.40)14We use the ϑ function conventions of Mumford [29]. The function ϑ11(ζ, τ) is odd in ζ. Definitionsof the Weierstrass functions and the constant η1 may be found in [30].29

The mathematical interpretation of (4.40) is as follows. Let J(Σ) be the Jacobian ofthe complex torus Σ.

Each point of J(Σ) gives an elliptic operator ¯∂χ where ¯∂: Λ0(Σ) →Λ(0,1)(Σ) and ¯∂χ is ¯∂coupled to the flat line bundle determined by the character χ ofπ1(Σ) →C\{0}. The operators ¯∂χ have index zero so we get a holomorphic map φ fromJ(Σ) to F0, the space of Fredholm operators of index zero15.

But F0 is a complex manifoldwith a natural holomorphic line bundle, the determinant line bundle16 DET(F0), and anatural section s [31, 32]. Now DET(¯∂) = φ∗(DET(F0)) is a holomorphic line bundleover J(Σ) with holomorphic section φ∗(s).

If we take the simply connected covering ofJ(Σ) with coordinates (ζ, τ), then φ∗(s) pulled up to the covering is a section of a trivialbundle; it is the ϑ-function−iϑ11 (ζ; τ)η(τ)(4.41)with transformation law (4.45) below. We are restricting the family of ¯∂operators tothose with ζ = α(θw)/2π −α(ˇµ)τ.

Hence we get the function−iϑ11 α(θw)2π−α(ˇµ)τ; τ!η(τ). (4.42)4.4The Character Index FormulaWe are now in a position to put the prefactors and the determinants together in a conciseexpression for the character index I(ν,k)(θ, τ) = Z(θ, τ) of the Dirac-Ramond operator.The reader is reminded that one has to sum over Waff, the fixed points of the S1 × Taction and that Waff= W ⋉ˇT.

Collating all the information developed in this sectionwe haveI(ν,k)(θ, τ)=Xw∈WXˇµ∈ˇTqk K(ˇµ, ˇµ)/K(hψ, hψ)q−ν(ˇµ)×exp iν(θw) −2ik K(ˇµ, θw)K(hψ, hψ)!×" Yα≻0det∂¯z + 122πiα(ˇµ)#−1 "ddet(∂¯z)#−l/2. (4.43)Before expressing (4.43) in terms of classical functions we make several remarks aboutthe abstract structure.

As we have noted, the troublesome determinant we discussed is15Strictly speaking, F0 is the space of index zero Fredholm operators from a Sobolev space H1(Σ) toL2(Σ).16The determinant line bundle of an appropriate space S will be denoted by DET(S).30

not a function but a holomorphic section of a line bundle over the Jacobian variety of Σ.The nontriviality of this section is closely related to the necessity for regularization. Wewill presently see that the shift by the Coxeter number arises because we have a sectionand not a function.

Had the determinant been finite (which of course it cannot) the shiftby the Coxeter number would not be there. Formula (4.43) may be rewritten as,I(ν,k)(θ, τ)=Xw∈WXˇµ∈ˇTqk K(ˇµ, ˇµ)/K(hψ, hψ)q−ν(ˇµ)×exp iν(θw) −2ik K(ˇµ, θw)K(hψ, hψ)!×Yα≻0−iϑ11 α(θw)2π−α(ˇµ)τ; τ!η(τ)−1η(τ)−l ,(4.44)and is the central result of this article.

It is the natural form for the character indexfrom the path integral viewpoint. All other forms are derived from this one by ϑ-functionidentities and algebraic manipulation.

There are several important remarks which shouldbe made before proceeding. As was strongly advertised, (4.44) is an analytic function ofq which involves ϑ-functions on the Jacobian of Σ and not Θ-functions on the torus T.As expected, the expression is independent of the choice of scale for the inner product.Formula (4.44) is almost the Weyl-Kˇac character formula; it is the index of the Dirac-Ramond operator on LG/T instead of the ¯∂operator on LG/T.

As explained in I, ¯∂andthe Dirac operator are related by twisting. If we are interested in the character associatedto a representation with highest G weight λ then we should choose λ to differ from ν bythe Weyl weight ρ.To transform (4.44) into a more conventional form of the character formula and tosee the shift by the Coxeter number we need the ϑ-function identityϑ11(ζ + mτ; τ) = (−1)mqm2/2e−2πimζϑ11(ζ; τ)(4.45)where m is an integer and formula (A.19) for the Coxeter number hg.

Thus the indexmay be rewritten asI(ν,k)(θ, τ)=Xw∈WXˇµ∈ˇTq−ν(ˇµ)q(k + hg)K(ˇµ, ˇµ)/K(hψ, hψ)×exp iν(θw) −2i(k + hg) K(ˇµ, θw)K(hψ, hψ)!31

×η(τ)−lYα≻0−iϑ11 α(θw)2π; τ!η(τ)−1. (4.46)We used the fact that, since ρ is a weight and ˇµ is in the coroot lattice, then ρ(ˇµ) ∈Z.Also, if we use the Killing form to identify t with t∗, one can easily see that 2ˇµ/K(hψ, hψ)is a weight.To write (4.46) in a more recognizable form, one proceeds in two different ways.

Eitherwe do the W sum first or we do the ˇT one. These two alternatives lead to very differentlooking formulas.

We recall from I that if ν is a weight of G then the T-index of theDirac operator coupled to the ν-line bundle isIν(θ)=Xw∈Weiν(θw) Yα≻012i sin 12α(θw)(4.47)=Xw∈W(−1)ℓ(w)eiν(θw) Yα≻012i sin 12α(θ) ,(4.48)where ℓ(w) is defined as the number of positive roots turned into negative roots by w.Note the ordinary index has only a single subscript while the loop one has a doublesubscript.It is convenient to define the Weyl denominator byDW(θ) =Yα≻02i sin 12α(θ)(4.49)and the Kˇac denominator byDK(θ) =Yn>0(1 −qn)l Yα≻0Yn>01 −qneiα(θ) 1 −qne−iα(θ). (4.50)The Weyl and the Kˇac denominators are closely related to our index formulas becauseof the identity−iϑ11(ζ; τ)η(τ)= q1/12 2i sin πζYn>01 −qne2πiζ 1 −qne−2πiζ.

(4.51)It is now a matter of algebra to transform (4.46) into one of the standard forms forthe Weyl-Kˇac character formula. Define the sublattice ˇT ∗= {2ˇµ/K(hψ, hψ) | ˇµ ∈ˇT}of the weight lattice, and the dilated-translated lattice ˇT ∗(ν, a) = ν + a ˇT ∗.

Now let usexpress (4.46) in a different way by first summing over the Weyl group W. This organizes32

the elements of the expansion in terms of the ordinary Dirac index:I(ν,k)(θ, τ)=q−(dim g)/24DK(θ)q−(ν, ν)/[(k + hg)(ψ, ψ)]×Xω∈ˇT ∗(ν,k+hg)Iω(θ)q(ω, ω)/[(k + hg)(ψ, ψ)] . (4.52)Next, we could have first summed over the coroot lattice generating a Θ-function.

Con-sider the lattice ˇT ∗(ν, a) and the associated Θ-functionΘ(ν,a)(z, τ) =Xω∈ˇT ∗(ν,a)exp"2πiτa(ω, ω)(ψ, ψ) + 2πiω(z)#. (4.53)The index may be written asI(ν,k)(θ, τ)=q−(dim g)/24DW(θ)DK(θ) q−(ν, ν)/[(k + hg)(ψ, ψ)]×Xw∈W(−1)ℓ(w)Θ(ν,k+hg) θw2π, τ!.

(4.54)In order to incorporate the twist that turns the Dirac operator into ¯∂we remind youthat in the Weyl character case the highest weight λ is related to ν by ν = λ + ρ andthat the character index and group character for G are related byIν(θ) = χλ(θ) . (4.55)Thus we see that the Weyl character formula for the highest weight representation λ ofthe group G isχλ(θ)=Xw∈Wei(λ+ρ)(θw) Yα≻012i sin 12α(θw)(4.56)=Xw∈W(−1)ℓ(w)ei(λ+ρ)(θw) Yα≻012i sin 12α(θ) .

(4.57)In what follows we will write χλ even if λ is not a highest weight because every weight λis conjugate via an element of the Weyl group to a highest weight.In the same way, the loop index and the associated character for gLG are related byI(ν,k)(θ, τ) = χ(λ,k)(θ, τ) . (4.58)A little algebra leads to the following two formulas for the characterχ(λ,k)(θ, τ)=q−(dim g)/24DK(θ)q−[(λ + ρ, λ + ρ) −(ρ, ρ)]/[(k + hg)(ψ, ψ)]33

×Xω∈ˇT ∗(λ,k+hg)χω(θ) q[(ω + ρ, ω + ρ) −(ρ, ρ)]/[(k + hg)(ψ, ψ)] , (4.59)=q−(dim g)/24DW(θ)DK(θ) q−(λ + ρ, λ + ρ)/[(k + hg)(ψ, ψ)]×Xw∈W(−1)ℓ(w)Θ(λ+ρ,k+hg) θw2π, τ!. (4.60)Equation (4.59) is the same as equation (14.3.10) of [3] with the proviso that we useL0 −c/24 in our trace while they use L0.

It is important to realize that in this contextc = dim g, see (4.17), and that c is not the Sugawara value. Equation (4.60) may be putin a more useful form by mimicking the following computation with the Weyl characterformula.

If one considers the trivial representation (highest weight λ = 0 with χ0(θ) = 1)then one easily sees that the Weyl denominator may be written asDW(θ) =Xw∈W(−1)ℓ(w)eiρ(θw)(4.61)and thus the Weyl character formula may be rewritten asχλ(θ) =Xw∈W(−1)ℓ(w)ei(λ+ρ)(θw)Xw∈W(−1)ℓ(w)eiρ(θw). (4.62)The analogous equation in the loop group case exploits the fact that the trivial repre-sentation has λ = 0, k = 0, and χ(0,0)(θ, τ) = q−(dim g)/24.

Thus we conclude that thedenominators satisfyDW(θ)DK(θ) = q−(ρ, ρ)/[hg(ψ, ψ)] Xw∈W(−1)ℓ(w)Θ(ρ,hg) θw2π, τ!,(4.63)leading to the following form for the character formulaχ(λ,k)(θ, τ)=q−(λ + ρ, λ + ρ)/[(k + hg)(ψ, ψ)]×Xw∈W(−1)ℓ(w)Θ(λ+ρ,k+hg) θw2π, τ!Xw∈W(−1)ℓ(w)Θ(ρ,hg) θw2π, τ! (4.64)which may explicitly obtained from the formulas in Chapter 12 of [2] as discussed in thevicinity of equation (A.25) in reference [22].It is well known that the affine characters have modular transformation properties[2, 3].

The origin of these properties was originally considered very mysterious but the34

connection of affine Lie algebras to conformal field theory demystified the issue. In [22],the authors discussed the modular invariance of the WZW model’s partition function,a sum of the modulus squared of characters.We can use our results to discuss theorigin of the modular properties of individual characters.

The key observation is thatthe quadratic action (4.17) is a non-chiral conformal field theory. One should view thedeterminants in (4.43) as short hand for the path integral over the quadratic action.The modular transformations properties of this chiral conformal field theory explains themodular properties of the characters.AcknowledgementsWe would like to thank C. Itzykson for insisting that the “proof of the pudding is inthe writing”.

Each author would like to thank the home institutions of the other twoauthors for visits while the research was in progress.ANotational ConventionsLet G be a connected, simply connected, compact simple Lie group with Lie algebra g.Let T be its maximal torus with Lie algebra t. The adjoint actions by the element x ∈gis denoted by ad x and is defined by (ad x)y = [x, y] for y ∈g. The (negative definite)Killing form is defined byK(x, y) = Tr(ad x ad y) .

(A.1)The Killing form guarantees an orthogonal decomposition g = t ⊕m. Here and in whatfollows a subscript t or m indicates the projection along t or m. A root α is an element ofthe vector space t∗dual to t which satisfies (ad t)eα = α(t)eα where t ∈t and eα are the“raising” and “lowering” generators.

The set of roots will be denoted by ∆. If the rootα is positive then we will write α ≻0.

We are implicitly working in the complexificationof the Lie algebra. We use the Killing form to associate elements of t with elements oft∗.

Our notation is as follows: given β ∈t∗one associates tβ ∈t by the standard relationβ(h) = K(h, tβ),∀h ∈t . (A.2)The Lie algebra commutation relations may be written as[h, eα]=α(h)eα ,(A.3)[eα, e−α]=K(eα, e−α)tα ,(A.4)35

for h ∈t. It is convenient to find the “standard” su(2) subalgebras.

Define the innerproduct (·, ·) on t∗by (α, β) = K(tα, tβ). If one defines coroots hα byhα =2tα(α, α)(A.5)then[hα, e±α]= ±2e±α ,(A.6)[eα, e−α]= hα .

(A.7)The coroot lattice, ˇT, is the integral lattice spanned by the hα. Elements of the corootlattice will usually be denoted by a “check” accent, e.g.

ˇµ. For future reference noticethatK(eα, e−α)=2/(α, α) ,(A.8)K(hα, hα)=4/(α, α) ,(A.9)exp(2πihα)=I .

(A.10)Casimir operators and other quadratic objects will constantly appear in our formulasand for this reason we offer a compendium of formulas. Let {ea} be an arbitrary basisfor g and let Kab = K(ea, eb) and Kab be the inverse matrix then:1.

If k1, k2 ∈t thenK(k1, k2) =Xα∈∆α(k1)α(k2) ,(A.11)see also (A.20).2. The quadratic Casimir operator for an irreducible representation R of g is definedbyC(R) = KabR(ea)R(eb) .

(A.12)3. The trace normalization T(R) for an irreducible representation is defined byTr R(ea)R(eb) = T(R)Kab .

(A.13)In our conventions we have T(ad) = 1.4. The following well known relation existsT(R) = dim Rdim g C(R) .

(A.14)It follows that C(ad) = 1.36

5. Let {µ} be the set of weights of a representation included in the set according tomultiplicity17 thenC(R) = dim gdim R1rank gX{µ}(µ, µ) .

(A.15)Applying this to the adjoint representation leads to the formularank g =Xα∈∆(α, α) . (A.16)6.

If λ is the highest weight of R thenC(R)(λ, λ) = 1 + 2 (λ, ρ)(λ, λ)= (λ + ρ, λ + ρ) −(ρ, ρ)(λ, λ),(A.17)where the Weyl weight ρ is defined by 2ρ =Pα>0 α. Note that both the left handside and the right hand side of the above are independent of the scale of the innerproduct.

If one applies the above to the adjoint representation with highest rootψ, which we will take to be a long root, then one finds that the (dual) Coxeternumber hg is given byhg ≡C(ad)(ψ, ψ)=1 + 2 (ψ, ρ)(ψ, ψ)(A.18)=1(ψ, ψ)=K(hψ, hψ)/4 . (A.19)Note that 2 (ψ, ρ) / (ψ, ψ) = ρ(hψ) and thus the Coxeter number is an integer sinceρ is a weight.Also notice that some of the above formulas are independent ofthe scale chosen for the inner product.

Other formulas just express these “scaleinvariant” quantities in a specific choice of inner product.7. The above observation leads to a scale invariant way of writing (A.11).

Note thatthe right hand side of (A.11) is independent of the scale of the inner product whilethe left hand side is not. We may rewrite (A.11) as4hgK(k1, k2)K(hψ, hψ) =Xα∈∆α(k1)α(k2) .

(A.20)8. Note also that (A.16) may also be written in a scale invariant wayrank g = 1hgPα∈∆(α, α)(ψ, ψ).

(A.21)17If a weight has multiplicity two then it appears twice in the set.37

9. Another useful formula is the “strange formula” of Freudenthal which states that(ρ, ρ) = (dim g)/24 and may be written in a scale invariant way asdim g24= 1hg(ρ, ρ)(ψ, ψ) .

(A.22)10. The generator of H3(G, Z) isσ = (ψ, ψ)48π2Trad(g−1dg)3 =112π2K(hψ, hψ) Trad(g−1dg)3 ,(A.23)where the trace is taken in the adjoint representation of g. The last expressiondemonstrates that the generator is independent of the normalization chosen for theinner product as expected.References[1] H. Weyl.

The Classical Groups. Princeton University Press, second edition, 1946.

[2] V. Kˇac. Infinite Dimensional Lie Algebras.

Cambridge University Press, third edition,1990. [3] A.M. Pressley and G.B.

Segal. Loop Groups.

Oxford University Press, 1986. [4] I. Frenkel, J. Lepowsky, and A. Meurman.

Vertex Operator Algebras and the Monster.Academic Press, 1988. [5] O. Alvarez, I.M.

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