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Generalised Abelian Chern-Simons Theories

arXiv:hep-th/9110061v1 21 Oct 1991ISAS/108/91/EPGeneralised Abelian Chern-Simons Theoriesand their Connection to Conformal Field TheoriesMarco A. C. KNEIPP ∗International School for Advanced Studies (SISSA), Strada Costiera 11, 34014 Trieste,ItalyAbstractWe discuss the generalization of Abelian Chern-Simons theories when θ-anglesand magnetic monopoles are included. We map sectors of two dimensional Confor-mal Field Theories into these three dimensional theories.July 1991∗E-mail: KNEIPP@ITSSISSA.bitnet or 38028::KNEIPPWork partially supported by CNPq - Brazil

Three dimensional Chern-Simons(CS) Gauge Theories are interesting for many math-ematical and physical reasons. As shown by Witten [1], CS theories can be used to con-struct new knot invariants.

A mapping between the three dimensional gauge invariantwave-functions and the conformal blocks of some Conformal Field Theories(CFT) in twodimensions [1 - 6] was also estabished. In terms of physical applications, Abelian CStheory is relevant for the description of the Fractional Quantum Hall Effect and maybefor High Tc Supercondutivity [7].

With this last motivation, Iengo and Lechner [8], con-structed the gauge invariant wave-function of the U(1) CS with a magnetic monopole andθ-angles on a torus, using the path-integral approach. A natural question that one canraise is to which CFT this generalised models are related.

The purpose of this letter toconsider this question. Here, however, we will adopt the operatorial formalism to recoverthe results in [8] and consider the more general case of a multidimensional Rd/Λ compactAbelian gauge group, where Λ is a d-dimensional integral lattice.The Chern-Simons action of a multidimensional Rd/Λ compact Abelian gauge groupwith time independent charges is defined asS = 14πZΣ×R⃗A ∧d⃗A+X⃗q⃗q ⃗A0(rq)(1)where the three dimensional manifold has the structure Σ×R, Σ being a two dimensionalcompact Riemann surface without boundary and R corresponding to the time.

The 1-form ⃗A ≡Ai⃗bi is a gauge field with⃗bi as basis vector of a lattice Λ with metric Kij ≡⃗bi⃗bj.The static charges ⃗q ≡qi⃗b∗i , qi ∈Z, belong to the dual lattice Λ∗which is generated bythe basis vectors ⃗b∗i defined by the relation ⃗b∗i⃗bj = δij.Adopting as boundary condition the gauge fields going to zero at t = ±∞and usingcomplex coordinates, the action can be rewritten in the formS =ZΣ×R dx0dzd¯z12π⃗A¯z∂0 ⃗Az + ⃗A012π⃗Fz¯z +X⃗q⃗qδ2(z −zq)(2)where we used the fact that the boundary terms vanish due to the boundary conditionat t = ±∞and since Σ don’t have boundaries. Here ⃗A0 appears as Lagrange multiplierwith the Gauss law,⃗G ≡12π⃗Fz¯z +X⃗q⃗qδ2(z −zq),(3)as a constraint and ⃗A¯z and ⃗Az are obviously conjugate variables.

It follows that thecomponents of ⃗A¯z and ⃗Az will satisfy the commutation relationh ⃗Az(t, z, ¯z), ⃗A¯z(t, w, ¯w)i= 2πiδ2(z −w)(4)Choosing the ⃗A0 = 0 gauge, we obtain an action that is invariant under time indepen-dent gauge transformations which are generated by the Gauss law. In order to recover atthe quantum level the equation of motion following to the variation of A0, the physicalstates, |Ψph >, must satisfy ⃗G|Ψph >= 0.1

The class of allowed gauge transformations depends on the topology of Σ. If Σ hasnon trivial cycles, one has discrete large gauge transformations.

We have to establish theaction of them on |Ψph >. In particular if Σ = T 2 with modular parameter τ = τ1 +iτ2, atime independent element of the gauge group can be written as g(z, ¯z) = ei⃗φ(z,¯z) ⃗H, where⃗H ∈Λ∗and the gauge parameter ⃗φ(z, ¯z) has the form⃗φ(z, ¯z) = ⃗φ0 + 2πz−¯τ ⃗φ1 + ⃗φ22iτ2+ ¯zτ ⃗φ1 −⃗φ22iτ2++X{n1,n2}̸=0⃗φn1n2e2πihz−¯τn1+n22iτ2+¯zτn1−n22iτ2i(5)where ⃗φ1, ⃗φ2 ∈Λ to g be single-valued and ⃗φ0 are continuos parameters with ⃗φ0 and⃗φ0 + ⃗λ being identified for an arbitrary ⃗λ ∈Λ to g be compact.

The parameters ⃗φ0 and⃗φn1n2 correspond to the small and ⃗φ1 and ⃗φ2 to the large gauge transformations.We are interested in finding the unitary operator U(g) that produces the finite timeindependent gauge transformations.The small part is obtained by a straightforwardexponentiation. However, if we want an unambiguos expression for all g we should defineU(g) by its properties:U(g) ⃗AzU†(g)=⃗Az + ∂⃗φ(6)U(g) ⃗A¯zU†(g)=⃗A¯z + ¯∂⃗φ(7)From these properties we can verify that U(g) generally is a projective representationhaving the composition law:U(gb)U(ga) = eiπ(⃗φ1a⃗φ2b−⃗φ2a⃗φ1b)U(gagb)(8)Only if Λ is an integral lattice, that is ⃗v⃗u ∈Z for ⃗v, ⃗u ∈Λ, U(g) will be a faithfulrepresentation.

One can define a consistent theory also when the lattice is not integral[9], however in the present letter we will limit ourselves to integral lattices.From (6) and (7) we obtain that, in a basis which ⃗Az is diagonalized, U(g) acts on ageneric state Ψ[ ⃗Az] in the following way:U(g)Ψ[ ⃗Az] = e−i2π[RΣ12 ∂⃗φ¯∂⃗φ+ ⃗Az ¯∂⃗φ]−2πi(⃗θ1⃗φ1+⃗θ2⃗φ2)Ψ[ ⃗Az + ∂⃗φ](9)where ⃗θ1 and ⃗θ2 are arbitrary. The origin of the terms with these parameters are duethe fact that (6), (7) and the condition of a faithful representation determine the form ofU(g) up to a 1-cocycle.

They are analogous to the θ-angle of QCD.Due to the fact that ⃗GΨph[ ⃗Az] = 0, the physical states satisfy the relationU(g)Ψph[ ⃗Az] = e−iP⃗q ⃗q⃗φ(zq, ¯zq)Ψph[ ⃗Az](10)It is easily verified that a solution of this equation can be constructed as2

Ψph[ ⃗Az]=ZD⃗φ eiP⃗q ⃗q⃗φ(zq, ¯zq)U(g)Ψ[ ⃗Az]=ZD⃗φ eiP⃗q ⃗q⃗φ(zq, ¯zq)e−i2π[RΣ12∂⃗φ¯∂⃗φ+ ⃗Az ¯∂⃗φ]−2πi(⃗θ1⃗φ1+⃗θ2⃗φ2)Ψ[ ⃗Az + ∂⃗φ](11)where Ψ[ ⃗Az] is a arbitrary state. Here we see that Ψph[ ⃗Az] corresponds to the correlatorof the vertex operators ei⃗q⃗φ(zq, ¯zq) of a free scalar CFT on Σ, compactified on the lattice Λand with an external gauge field ⃗Az.Comparing (9) and (10)we see that the physical states fulfil the relation:Ψph[ ⃗Az + ∂⃗φ] = ei2π[RΣ12 ∂⃗φ¯∂⃗φ+ ⃗Az ¯∂⃗φ]+2πi(⃗θ1⃗φ1+⃗θ2⃗φ2)−iP⃗q ⃗q⃗φ(zq, ¯zq)Ψphh ⃗Azi(12)Using an arbitrary constant gauge transformation, ⃗φ(z, ¯z) = ⃗φ0, in the last relation weobtain the condition thatP⃗q ⃗q = 0 for the physical states.On the torus, the ⃗Az component can be decomposed in the form⃗Az(z, ¯z)=iπτ2⃗a +X{n1,n2}̸=0⃗an1n2e2πihz−¯τn1+n22iτ2+¯zτn1−n22iτ2i=iπτ2⃗a + ∂⃗φsm(13)where the last term can be eliminated by a small gauge transformation ⃗φsm.

Therefore,to calculate a generic physical state Ψph[ ⃗Az], it is enough to find Ψphiπτ2⃗aand then,from (12), we arriveΨphh ⃗Az(z, ¯z)i= ei2π[RΣ12 ∂⃗φsm ¯∂⃗φsm]−iP⃗q ⃗q⃗φsm(zq, ¯zq)Ψphiπτ2⃗a(14)To determine the form of Ψphiπτ2⃗a, we use another time (12) and obtain the quasi-periodicity relation:Ψphiπτ2⃗a −⃗φ2 + ¯τ ⃗φ1=(15)= eπτ2[⃗a(τ ⃗φ1−⃗φ2)+ 12(τ ⃗φ1−⃗φ2)(¯τ ⃗φ1−⃗φ2)]+2πi(⃗θ1⃗φ1+⃗θ2⃗φ2)+iP⃗q 2π⃗qhz−¯τ ⃗φ1+⃗φ22iτ2+¯zτ ⃗φ1−⃗φ22iτ2iΨphiπτ2⃗aFor the case without charges, this relation has the independent (non-normalized) solutionsΨ⃗β iπ⃗aτ2!= exp π⃗a22τ2! X⃗α∈Λe−2πi⃗θ1⃗α exp−iπ¯τ⃗α + ⃗β + ⃗θ22 −2πi⃗α + ⃗β + ⃗θ2⃗a(16)where ⃗β ∈Λ∗/Λ .3

Since in the A0 = 0 gauge the action is invariant under modular transformations,we have to guarantee that these physical states form a representation of the modulartransformationsT :(x′1 = x1 + x2x′2 = x2S :(x′1 = x2x′2 = −x1(17)The unitary operators which implement these modular transformations are given by [10]T = η exp i4π⃗a21S = η′ exp i8πh⃗a21 + ⃗a22i(18)where η and η′ are some phases and ⃗a1 + ¯τ⃗a2 = ⃗a. From the T transformation we obtainthe condition that⃗λ⃗λ2 + ⃗λ⃗θ2 ∈Z∀⃗λ ∈Λ(19)To solve it, we use the fact that for an arbitrary Λ integral, the vector basis can bedecomposed as ⃗bi = {⃗o1, .

. .

, ⃗on,⃗en+1, . .

. , ⃗ed} with⃗ei⃗ei∈2Z(20)⃗oi⃗oi∈2Z + 1(21)We verify that the general solution of (19) is⃗θ2 =Xj⃗oj∗2(mod Λ∗)(22)The covariance of the wave-function under S transformation impose that⃗θ1 = ⃗θ2(mod Λ∗)(23)For the case of a even lattice (that is, a lattice generated by a basis composed onlyof vectors with even norm), we will have that ⃗θ1 = ⃗θ2 = 0 and we recover the results in[2, 3].

However, the important consequence of considering the θ-angles is that it opensthe possibility to consider a Rd/Λ CS with an arbitrary Λ. We don’t have any more theconstraint that Λ is even as normally considered.Now we will see that the physical wave-functions (16) are the conformal blocks ofsectors of two differents CFT’s defined on T 2.

The first one is a CFT defined by a chiralalgebra generated by Gi ≡z⃗b2i /2 : e(i⃗bi⃗φ(¯z)) : and ⃗G0 ≡i∂¯z⃗φ(¯z) with the chiral scalarfield ⃗φ(¯z) = ⃗q −i⃗p ln ¯z + iPn̸=0 ⃗αn¯zn. The generators Gi have scaling dimensions ⃗bi2/2that can be integer or half-integer and we will call them as ”bosonic” and ”fermionic”respectivaly.

The fermionic set has two more features with respect to the bosonic one.The first feature is that the chiral algebra is compatible which Ramond(periodic) orNeveu-Schwarz(anti-periodic) boundary conditions. The requirement that the Gi’s areperiodic, selects for the Ramond sector the momenta ⃗λ∗+Pi ⃗o∗i /2 and for the the Neveu-Schwarz sector the momenta to ⃗λ∗, ∀⃗λ∗∈Λ∗.

The second feature comes from the fact4

that since the fermionic operators have semi-integer dimensions, they would producemonodromies inside the representations. Therefore we decompose the representationsthrough a projection operator.

This projection should separate the states connected bya even number of applications of the fermionic operators. Let’s consider a generic state|⃗α + ⃗β > produced from the highest weight state |⃗β > with ⃗α = αoddi⃗oi + αeveni⃗ei.

Wecan verify that the operators 1±(−1)F2with F ≡Pi αoddi= 2⃗α⃗θ, produces the wantedprojection.It is known that modular transfomations generaly mix the different boundaries con-ditions and only the Ramond-Ramond sector by itself is invariant under modular trans-formations. To obtain the corresponding characters of the Ramond-Ramond sector forour chiral algebra, we proceed exactly in the same way as for the ordinary fermionic case:the path integral with periodic boundary conditions in both directions corresponds tothe trace of the operator (−1)F exp(−2πi¯τL0 −⃗a ⃗G0) over the states in a representationwith momenta ⃗λ∗+Pi ⃗o∗i /2.

This will give as the result for a representation built froma highest weight state |⃗β >, β ∈Λ∗/Λχ⃗β =X⃗α∈Λ(−1)F exp−iπ¯τ ⃗α + ⃗β +Xi⃗o∗i /2!2−2πi ⃗α + ⃗β +Xi⃗o∗i /2!⃗a(24)which corresponds to difference of the conformal blocks that comes from the sectors1+(−1)F2and 1−(−1)F2with momenta to ⃗λ∗+ Pi ⃗o∗i /2. Comparing this last result with thewave-functions (16) and using our definition of F, we can conclude that our physicalwave-funtions are the characters of the Ramond-Ramond sector of the above CFT.The second CFT is defined from a even lattice that we will denote by ˜Λ.

In this latticewe substitute the odd vector basis ⃗oi of our original Λ by ⃗fi ≡2⃗oi and leave unchangedthe even vector basis ⃗ei. Correspondly, in the new dual lattice, ˜Λ∗, we substitute ⃗o∗i by⃗f ∗i ≡⃗o∗i /2 in order to preserve the relation ⃗bi⃗b∗j = δij, ⃗bi =n⃗f1, .

. .

, ⃗fn,⃗en+1, . .

. , ⃗edo.

Thisis a special kind of even lattice since more than the condition ⃗bi⃗bi ∈2Z, we have thatalso ⃗bi ⃗fj ∈2Z.From this lattice we define as before a chiral algebra, but now containig only bosonicoperators.We can verify that for this lattice we have the discrete global symmetryproperty that the Gi’s that are connected by shifts of ⃗fj/2, have the same scaling dimen-sion (modulo an integer). It is not difficult to prove that the following combination ofconformal blocks form a representation of modular group:χ⃗β = ˜χ⃗β −nXi=1˜χ⃗β+ 12 ⃗fi +jXi=1nXj=2˜χ⃗β+ 12( ⃗fi+ ⃗fj) −· · · + (−1)n ˜χ⃗β+ 12( ⃗f1+ ⃗f2+···+ ⃗fn)(25)where˜χ⃗γ≡X⃗α∈˜Λexph−iπ¯τ (⃗α + ⃗γ)2 −2πi (⃗α + ⃗γ)⃗ai⃗β∈Xmi,ni∈Zh(2mi + 1) ⃗fi∗+ ni⃗ei∗i/Xpi,qi∈Zpi2⃗fi + qi⃗ei5

We can put it in a compact form:χ⃗β =X⃗g(−1)2⃗g Pi ⃗fi∗˜χ⃗β+⃗g(26)where ⃗g are the different shift produced by the combination of ⃗fi/2. If we rewrite thewave-function (16) using the vector basis of ˜Λ, we obtain the same modular matrices.We discuss now the quantisation of our theory in the presence of a magnetic monopole.Following the procedure of t’Hooft [11], we can construct a configuration with magneticmonopole on T 2 by using the boundary conditions:⃗Ai(x1, 1) = ⃗Ai(x1, 0) + ∂i⃗r(x1, 0)i = 1, 2⃗Ai(1, x2) = ⃗Ai(0, x2) + ∂i⃗s(0, x2)i = 1, 2(27)⃗s(0, 1) −⃗s(0, 0) −⃗r(1, 0) + ⃗r(0, 0) ∈2πΛwhere the last condition guarantee that the Dirac string stay invisible.

Therefore, we canconsider that ⃗r = 2π⃗µx1 and ⃗s = 0 where ⃗µ ∈Λ. Passing to complex coordinates, thegauge fields and the field strength⃗Az=⃗Apz + ¯τπ2τ 22(z −¯z) ⃗µ⃗A¯z=⃗Ap¯z −τπ2τ 22(z −¯z) ⃗µ(28)⃗Fz¯z=⃗F pz¯z + πiτ2⃗µfulfil the above boundary conditions, where ⃗Apz, ⃗Ap¯z and ⃗F pz¯z correspond to the field con-figuration with periodic boundary conditions.

Substituting (28) in (1), and doing someintegration by parts, we obtainS =ZΣ×R dx0dzd¯z12π⃗Ap¯z∂0 ⃗Apz + ⃗A012π⃗F pz¯z +X⃗q⃗qδ2(z −zq) +12iτ2⃗µ(29)plus some non periodic terms. Using as guiding principle to consider only truely periodicquantities in the action, we will consider (29) as a new definition of the action, replacing(1).

This action has a vanishing hamiltonian as the previous one. Comparing with (2),we see that the only effect of the inclusion of a magnetic monopole was a change in theGauss law.

Therefore, the physical states will satisfy the new condition:U(g)Ψph[ ⃗Az] = e−iP⃗q ⃗q⃗φ(zq, ¯zq)+i⃗µ ⃗φ0Ψph[ ⃗Az](30)Here we put only the constant part of ⃗φ for the magnetic monopole, since the partthat comes from the large gauge transformation can be absorved by a redefinition of theθ-angles. A general solution to this equation will be:6

Ψph[ ⃗Az]=ZD⃗φ eiP⃗q ⃗q⃗φ(zq, ¯zq)−i⃗µ ⃗φ0U(g)Ψ[ ⃗Az]=ZD⃗φ eiP⃗q ⃗q⃗φ(zq, ¯zq)−i⃗µ ⃗φ0e−i2π[RΣ12∂⃗φ¯∂⃗φ+ ⃗Az ¯∂⃗φ]−2πi(⃗θ1⃗φ1+⃗θ2⃗φ2)Ψ[ ⃗Az + ∂⃗φ] (31)From (30) and (9) we arrive that Ψph will by constrained by:Ψph[ ⃗Az + ∂φ] = ei2π[RΣ12 ∂⃗φ¯∂⃗φ+ ⃗Az ¯∂⃗φ]+2πi(⃗θ1⃗φ1+⃗θ2⃗φ2)−iP⃗q ⃗q⃗φ(zq, ¯zq)+i⃗µ⃗φ0Ψphh ⃗Azi(32)Using a constant gauge transformation we obtain the new conditionP⃗q ⃗q = ⃗µ for thephysical states. Therefore, when there is a magnetic monopole the charge distributionmust have a total charge different from zero.

Moreover, since ⃗q ∈Λ∗and ⃗µ ∈Λ thisrelation impose that Λ ⊂Λ∗, which means that in the presence of a magnetic monopoleΛ is necessarily an integral lattice.From (32) we arrive to the same quasi-periodicity relation (15) but with the conditionP⃗q ⃗q = ⃗µ. This relation will have the solutionsΨ⃗β iπ⃗˜aτ2!= expπ⃗˜a22τ2X⃗α∈Λ(−1)F exp−iπ¯τ⃗α + ⃗β + ⃗θ2 −2πi⃗α + ⃗β + ⃗θ⃗a −X⃗q⃗q¯zq(33)where ⃗β ∈Λ∗/Λ and ⃗˜a ≡⃗a + P⃗q ⃗q (zq −¯zq).

Therefore, for the one dimensional lattice,we recover the results in [8]. We can conclute that the only consequence of the inclusionof the monopole is that now P⃗q ⃗q = ⃗µ.The condition of a total charge different from zero appears already at the classicallevel: as ⃗q couples directly to the magnetic field in the equations of motion (3), theintroduction of a magnetic monopole changes the charge conservation law from P⃗q ⃗q = 0toP⃗q ⃗q −⃗µ = 0.

This result can be extented to other Riemann surfaces. In particular wecan do the same for the sphere.

Therefore we can recognize the condition P⃗q ⃗q ̸= 0 as thesame considered by Dotsenko and Fateev [12] for the Feigin-Fuchs construction on thesphere. There, through the introduction of a termR R√gφ in the action and using aftercomplex coordinates, they arrive in a correlator similar to (31).

It seems therefore thatthere is exist a connection between the CS theory with a magnetic monopole and minimalmodels. However, the screening operators necessary for the Feigin-Fuchs constructiondon’t have a clear interpretation from the CS point of view.AcknowledgementsI would like to thank A. Schwimmer for the suggestion of this problem and for theenlightening and patient conversations.References[1] E. Witten, Comm.

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