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NON-COMPACT WZW CONFORMAL FIELD THEORIES1,2

arXiv:hep-th/9110076v1 31 Oct 1991NON-COMPACT WZW CONFORMAL FIELD THEORIES1,2Krzysztof Gaw¸edzkiC.N.R.S., I.H.E.S.91440 Bures-sur-Yvette, FranceABSTRACTWe discuss non-compact WZW sigma models, especially the ones with symmetricspace HC/H as the target, for H a compact Lie group. They offer examples of non-rational conformal field theories.

We remind their relation to the compact WZW modelsbut stress their distinctive features like the continuous spectrum of conformal weights,diverging partition functions and the presence of two types of operators analogous tothe local and non-local insertions recently discussed in the Liouville theory. Gaugingnon-compact abelian subgroups of HC leads to non-rational coset theories.

In partic-ular, gauging one-parameter boosts in the SL(2, C)/SU(2) model gives an alternative,explicitly stable construction of a conformal sigma model with the euclidean 2D blackhole target. We compute the (regularized) toroidal partition function and discuss thespectrum of the theory.

A comparison is made with more standard approach based onthe U(1) coset of the SU(1, 1) WZW theory where stability is not evident but whereunitarity becomes more transparent.1. INTRODUCTIONThe four years which passed since the previous Carg`ese Institute of the series havebrought a marked progress in the understanding of rational Conformal Field Theories(CFT’s), a class of 2D massless quantum field models, see e.g.

[1]. The simplest ofthose theories is the free field with values in a circle of rational radius, more complicatedexamples are provided by the Wess-Zumino-Witten (WZW) sigma models with a generalcompact Lie group G as the target [2],[3][4] or by the coset theories obtained by gauginga subgroup of G in the WZW theory [5],[6].

The characteristic property of the rationalCFT’s is1. decomposition of the euclidean Green functions into a finite sum of products of1extended version of lectures read at the Summer Institute “New Symmetry Principles in QuantumField Theory”, Carg`ese, July 16-27, 19912based on joint work in progress with Antti Kupiainen; these notes cover also some preliminarymaterial far from being completely understood, reflecting there the present author’s point of viewwhich has been changing with time and has not yet reached the final form1

holomorphic and antiholomorphic “conformal blocks”.This is accompanied by other simplifying features, which are more or less special fromthe point of view of the general quantum field theory, like2. discreteness of finite volume energy spectrum,3.

one-to-one correspondence between states and operators,4. simple structure of the operator product expansions,5.

finiteness of the partition functions,6. simple factorization properties.Our knowledge about the rational WZW and coset theories seems rather satisfactorytoday (although one might argue that a subtle cleaning of some fine mathematical pointsremains to be done; also the basic problem of classification of the rational CFT’s hasnot been solved).

It contains the exact solution for the spectrum and for the low genusGreen functions (see e.g. [7]).It seems then reasonable to go beyond the study ofrelatively simple conformal theories where properties 1-6 hold, especially since examplesof conformal models without those properties appear rather naturally.

The best knowninstance is the Liouville theory describing the conformal mode of 2D gravity [8]. It isin this model, essential for the treatment of non-critical string theory, where the newfeatures related to the failure of 1-6 where first discussed, see inspiring lectures [9].Mathematically, the passage from rational to irrational CFT’s involves a shift frompurely algebraic treatment to more analysis.

The parallel might be the passage fromrepresentation theory of compact Lie groups to the non-compact case.Indeed, thecanonical WZW example of rational CFT is related to the representation theory of loopgroups of compact groups [10] and it is expected that (largely non-existent, see however[11],[12]) theory of representations of loop groups of non-compact type will underliean interesting class of irrational CFT’s. The first candidates which come to mind arethe WZW theories with non-compact groups as targets.

These however, if quantizedas in the compact case, have unbounded below energy and, consequently, no stabilityand no euclidean picture. A possible solution is to pass to their coset models wherein some cases one may expect to recover stability, see Sec.

5 below. In the presentcourse, we shall start instead from a different series of non-rational CFT’s which havebounded below energy and stable euclidean picture but are non-unitary.

These are theWZW-type sigma models with non-compact target spaces HC/H where H is a compact(simple, connected, simply connected) Lie group. We shall call them shortly HC/HWZW models.

It should be stressed that, contrary to what the name might suggest,this is a different class of models than the coset G/H theories in the CFT sense. Thelatter are obtained by gauging a subgroup H ⊂G (or more generally H ⊂Gleft ×Gright)in the group G WZW model and correspond rather to conformal sigma models withorbit space of the left-right action of H on G as the target.

To avoid the terminologicalconfusion, we shall label them as G mod H coset theories3. In fact, a coset theoryG mod H factorizes into the group G WZW theory times the HC/H one, decoupledin the planar topology, and, in general, coupled only via zero modes.

This is how the3we are fully aware that this arrogant attempt to change accepted terminology is bound to be futile2

general HC/H WZW theories manifested themselves for the first time [13],[14]. TheSL(2, C)/SU(2) model has been discussed earlier in [15].

The Green functions of theHC/H models which appear in this context have also a 3D interpretation: they computethe scalar product of Schr¨odinger picture states of the 3D Chern-Simons [16],[17],[7] fieldtheory with gauge group H. In more geometric terms, they give the hermitian structurewhich pairs conformal blocks of the (rational) group H WZW model into its Greenfunctions. In this guise, the HC/H theories may be thought of as models dual to theones with the compact group H as the target.

All this is briefly recalled in Sec. 2.In Sec.

3, we discuss free field representations of the HC/H WZW models on thesimplest example with H = SU(2).We compute explicitly (the finite part of) thepartition function of the model and discuss its spectrum and the relation between thespace of states and the operators of the theory.The coset scenario for producing new CFT’s may also work in the case of the HC/HWZW models if one gauges out a non-compact abelian subgroup N ∈HC (the resultwill be called an HC/H mod N theory). In Sec.

4, using free field representations, weshow that the SL(2, C)/SU(2) mod R model where R is embedded into SL(2, C) byt 7−→etσ3 gives a conformal sigma model with the recently found [18],[19] 2D euclideanblack hole as the target. We discuss the partition functions and the spectrum of thismodel.A comparison is made between the SL(2, C)/SU(2) mod R theory and therational parafermionic SU(2) mod U(1) model.Finally, in Sec.

5, we contrast our approach to the black hole conformal sigma modelwith Witten’s original proposal [18] based on the SU(1, 1) mod U(1) coset theory, seealso [20]-[29].The free field calculation of the partition functions of the black holemodel may be also repeated within Witten’s scenario giving the same result but itrequires complex shifts and rotations of the fields in the functional integral. One mayreasonably expect that both models coincide, the two approaches being complementary:the SL(2, C)/SU(2) mod R picture provides an explicitly stable construction whereasthe SU(1, 1) mod U(1) approach should be more useful for demonstrating unitarity ofthe theory.2.

ORIGIN OF THE HC/H WZW MODELS2.1. From the coset G mod H theoriesLet us start by recalling the formulation of a coset G mod H theory as a partiallygauged, group G WZW model (with compact G).

The basic fields on the closed Riemannsurface Σ are GC-valued functions g and gauge fields A = Azdz + A¯zd¯z with values inthe complexified Lie algebra HC of a group H ⊂G. H is supposed to be embedded intoG in two possibly different ways: ιl,r : H ֒→G.

We shall denote by “tr” the invariantform on the Lie algebra G (H) of G (H) normalized to give 2 as the length squared ofthe longest roots. We assume that via embeddings ιl,r, tr on G induces a single invariantform on H equal η times tr on H. The euclidean action of the coset model takes the3

form [14]kS(g, A) = kS(g) + ikπZΣtr[Arz(g−1∂¯zg) + (g∂zg−1)Al¯z+iAdg(Arz)Al¯z −iηAzA¯z d2z(1)where the superscripts “l, r” refer to the embeddings of H into G. kS(g) is the pureWZW actionS(g) = −12πZΣtr(g−1∂zg)(g−1∂¯zg) d2z +124πiZΣd−1tr(g−1dg)∧3(2)where we have used a shorthand notation for the Wess-Zumino topological term [2].Coupling constant k (“level”) is a positive integer. Under the complex (HC-valued)chiral gauge transformationsg 7−→hl1ghr†2A¯z 7−→h1A¯z ≡Adh1(A¯z) −ih1∂¯zh−11,Az 7−→h2Az ≡Adh†2−1(Az) −ih†2−1∂zh†2 ,action (1) transforms like follows:S(hl1ghr†2 , h2Azdz + h1A¯zd¯z) = S(g, A) + ηS(h1h†2, h2Azdz + h1A¯zd¯z) .

(3)In particular, it is invariant under the unitary gauge transformations with H-valuedh1 = h2 = h .The Green functions of the coset model are formally given by the functional integralZ−e−kS(g,A)Dg DA(4)over G-valued fields g and real (i.e. H-valued) gauge fields A.

As the insertion, we shouldtake an expression invariant under the unitary gauge transformations. An example isprovided byYαtrRα g(ξα)nα(5)where “trR” stands for the trace in representation R of G (in vector space VR) andnα ∈G satisfyulnαur† = nα(6)for u ∈H.

For example, if ul = ur, we may take nα = 1.On the Riemann sphere, we may parametrize real gauge fields A by HC-valued gaugetransformations by putting A¯z(h) = h−1∂¯zh. Action (1) becomes thenkS(g, A(h)) = kS(hlghr†) −ηkS(hh†) .

(7)4

The Jacobian of the change of variables is (we ignore the zero modes for the moment)∂(A(h))∂(h)= det(¯∂∗h ¯∂h) = e2h∨S(hh†) (det(−∆))dimH(8)where ¯∂h = d¯z(∂¯z + adA¯z(h)), h∨is the dual Coxeter number of H and ∆is the scalarLaplacian. More exactly, the change of variables A 7→h gives the following expressionfor the Green functions (4) with insertion (5):CZ YαtrRα g(ξα)(hlnαhr†)−1(ξα)e−kS(g) e(ηk+2h∨)S(hh†) Dg δ(h(ξ0)) Dh(9)where C = (det′(−∆)/area)dimH with the determinant without the zero mode contri-bution.

Expression (9) combines the Green functions of the compact group G WZWmodelΓ =Z(OαgRα(ξα)) e−kS(g) Dg ∈OαEnd VRα(10)(where gR denotes the representation R matrix of g) with those of a field theory withfields hh†Z(< Γ, ⊗(hlnαhr†)−1Rα(ξα) ) > eκS(hh†) δ(hh†(ξ0)) D(hh†) . (11)In the last expression Γ may be any tensor in ⊗EndVRα such that(⊗γlRα)Γ(⊗γr†Rα) = Γ(12)for γ ∈HC.

This condition guarantees that the integral is independent of point ξ0 inthe δ-function in (11) fixing the global HC invariance. Green functions (10) certainlysatisfy condition (12).

< ·, · > in (11) stands for the scalar product induced from thatof spaces VR. Fields hh† may be viewed as taking values in the non-compact symmetricspace HC/H and functional integral (11) as defining the (euclidean) Green functions ofthe HC/H WZW theory (also in a general world-sheet topology).

The euclidean action−κS(hh†) of the model is unambiguously defined4 and real, non-negative [14]. We shallsee that it leads to functional integrals of type (11) which are stable for any real κ > h∨.On the other hand, the Minkowskian action is not real: the Wess-Zumino term is purelyimaginary so that we should not expect the theory to be unitary.

We shall return tothese issues below.On a higher genus Riemann surface a similar treatment of the coset theory Greenfunctions produces again a combination of the G and HC/H WZW Green functions butthis time both twisted by coupling to an external flat gauge field Aflat and the resultcontains an integral over the moduli of Aflat [14], essentially coinciding with the moduliof complex HC-bundles.4HC/H is topologically trivial5

2.2. From the scalar product of the Chern-Simons theory statesThe Schr¨odinger picture states of the 3D Chern-Simons theory with gauge group Hon manifold Σ × R and in the presence of the Wilson lines {ξα} × R in representationsRα are functionalsψ : A −→OαVRα(13)on space A of real gauge fields A [30].

A has a natural complex structure obtained byidentifying it with the space of forms A¯zd¯z. Functionals ψ are required to be holomorphicand to transform covariantly under the complex gauge transformations:ψ(hA) = ekS(h−1)+π−1ik Rtr (h−1∂zh)A¯z d2z OαhRα(ξα) ψ(A) .

(14)k this time denotes the coupling constant of the Chern-Simons theory. The space ofstates defined as above is finite-dimensional.

The scalar product of the states is formallygiven by the functional integral∥ψ∥2 =Z< ψ(A), ψ(A) > e−π−1k RtrAzA¯z d2z DA . (15)On Σ = CP 1, upon the change of variables A 7→h, eq.

(15) becomes∥ψ∥2 = (det′(−∆)/area)dimH·Z< ψ(0) ⊗ψ(0), ⊗(hh†)−1Rα(ξα) > e(k+2h∨)S(hh†) δ(hh†(ξ0)) D(hh†)(16)which is a Green function of type (11) (for G = H and nα ≡1).2.3. From the hermitian structure coupling conformal blocks of the group H WZWtheoryGreen functions of the group H WZW model in an external H-valued field AZ(Oαg(ξα)Rα) e−kS(g,A) Dg(17)can be expressed asXa,bΩab ψa(A) ⊗ψb(A) e−π−1k Rtr AzA¯z d2z(18)where (ψa) is a basis of the Chern-Simons states considered above and the inverse matrix(Ω−1)ab = (ψa, ψb)(19)in the scalar product of (15), see [7],[31].

In the planar or toroidal geometry, the de-pendence of the basis vectors ψa on the insertion points and the complex structure maybe chosen analytic and such that the scalar products (ψa, ψb) remain constant. Expres-sion (18) gives then the decomposition of the Green functions into sum of combinations6

of conformal blocks demonstrating the rational character of the WZW theories withcompact targets. As we see, scalar product (15) given by the Green functions of theHC/H theory determines the way the conformal blocks of group H WZW theory areput together to build the complete Green functions.The WZW theories with targets H and HC/H may be considered as dual to eachother.

An elegant way to express this duality is to consider the coset H mod H model.This is a topological theory in the sense of [32]: its Green functionsZ(YαtrRα g(ξα)) e−kS(g,A) Dg DA(20)are independent of the location of the insertions and of the complex structure of thesurface [14]. Integrating representation (18) over gauge fields A, one infers [33] thatthey are in fact equal to the dimensions of the spaces of states ψ known explicitly dueto [34].

On the other hand, the coset Green functions factorize, as we have seen, intoa combination of products of those of the group H and of the symmetric space HC/HWZW models. This is the precise expression of the duality between both theories.In the planar case, the HC/H theory with Green functions (11) may be also viewedas an analytic continuation of those of the H theory to negative levels.

This relationbecomes more complicated on higher genera as, for example, a look into the respec-tive partition functions shows. It is not excluded, however, that both models describedifferent aspects of the same structure analytic in k.3.

FREE FIELD REPRESENTATION OF THE HC/H WZW THEORYFunctional integral (2.11) defining the Green functions of the HC/H WZW theorymay be computed by iterative Gaussian integration. This was noticed in [15] for theH = SU(2) case and was implemented in the present context and for general H in[13],[14] for the twisted toroidal partition function and in [7],[35] for the planar Greenfunctions.

One can also compute toroidal Green functions. Free field representation forthe model on a surface of genus > 2 is still an open problem.

Below, we shall stick tothe SU(2) case, for simplicity.Symmetric space SL(2, C)/SU(2) coincides with the upper sheet H+3 of 3D mass hy-perboloid. Convenient global coordinate system on H+3 is provided by the parametriza-tionhh† = eφ(1 + |v|2)1/2v¯ve−φ(1 + |v|2)1/2(1)with φ real and v complex.

The SL(2, C)-invariant measure on H+3 , d(hh†) = dφd2v.In coordinates (1),S(hh†) = −1πZ[(∂z ˜φ)(∂¯z ˜φ) + (∂z + ∂z ˜φ)¯v (∂¯z + ∂¯z ˜φ)v)] d2z(2)where5 ˜φ ≡φ −12log(1 + |v|2).We shall also need a gauged version of the action.If we gauge the U(1) group embedded into SU(2) asymmetrically by ιl(eiθ) = eiθσ3,5it will become clear below why we use φ and not ˜φ in parametrizing H+37

ιr(eiθ) = e−iθσ3, then the transformation law (2.3) implies, for h1 = eλσ3 and h2 = e−λσ3,thatS(eλσ3geλσ3, A + idλ) = S(g, A)(3)for any SL(2, C)-valued g (in particular for g = hh†) and for any complex 1-form A.Consequently, taking A purely imaginary may be interpreted as gauging of subgroupR ֒→{eλσ3 | λ real} in SL(2, C) (which is the global symmetry group of the H+3 WZWmodel). R corresponds to the boosts in the third direction under the standard relationbetween SL(2, C) and the Lorentz group.

A direct computation givesS(hh†, 12iA) = −1πZ[(∂z ˜φ + Az)(∂¯z ˜φ + A¯z)+(∂z + ∂z ˜φ + Az)¯v (∂¯z + ∂¯z ˜φ + A¯z)v] d2z . (4)Invariance (3) becomes obvious in (4) since transformation hh† 7−→eλσ3hh†eλσ3 trans-lates in coordinates (1) into (φ, v) 7−→(φ + 2λ, v).3.1.

Toroidal partition functionFirst, let us describe the calculation [14] of the twisted partition function ZH+3 (τ, U)of the H+3 WZW theory on torus Tτ ≡C/(2πZ+2πτZ), τ = τ1 +iτ2, τ2 > 0. It is givenby the functional integral:ZH+3 (τ, U) =ZeκS(γU hh†γ†U) D(hh†)(5)where γU = exp[−14τ2 U(z −¯z)σ3], U ≡U1 + iU2, satisfiesγU(z + 2π) = γ(z) and γU(z + 2πτ) = e−πiUσ3γU(z)and the action is extended to twisted field configurations [14] by puttingS(γUhh†γ†U) = S(hh† , 14i (τ −12¯Udz + τ −12 Ud¯z)) +πτ2 U21 .

(6)Using the explicit form (4) of the action, we obtainZH+3 (τ, U) = eπκτ −12U21Ze−π−1κR(∂zφ+τ −12¯U/2)(∂¯zφ+τ −12U/2) d2z· eR(∂z+∂zφ+τ −12¯U/2)¯v (∂¯z+∂¯zφ+τ −12U/2)v d2z D(hh†) . (7)where we have shifted ˜φ 7→φ.

The v-integral is gaussian and producesdet(¯∂+ ¯∂φ + 12τ −12 Ud¯z)∗(¯∂+ ¯∂φ + 12τ −12 Ud¯z)−1= e2π−1R(∂zφ)(¯∂¯zφ) d2z +(2πi)−1RφR det(¯∂+ 12τ −12 Ud¯z)∗(¯∂+ 12τ −12 Ud¯z)−1(8)where R denotes the metric curvature form. Rather surprisingly, the resulting effectiveφ theory is the free field with the background charge so that we obtain again a calculable8

functional integral. Eq.

(8) follows by the standard chiral anomaly calculation and doesnot depend on the regularization scheme used to define the determinants, within a largeclass. In particular, the absence of the Liouville ∼R eφ term in the effective action, ofthe type appearing in the conformal anomaly calculation, is here not an artifact of thechoice of the zeta function regularization.The presence of the (generic) twist U breaks the global SL(2, C) symmetry of thetheory to the diagonal U(1)C. The remaining symmetry results, however, in the diver-gence of the φ-integral (and, consequently, of the partition function) due to the zeromode contribution.

This divergence may be extracted in the usual way as the infinitevolume of U(1)C leading to the insertion of δ(φ(0)) fixing the φ zero mode under theintegral. The total central charge of the theory is easily computable from the standarddependence of the resulting determinants on the conformal factor of the metric.

It isequal c−κ ≡3κ/(κ −2) or κdimH/(κ −h∨) for general H. The determinants are wellknown [36]. The final result is (in the flat metric; q ≡e2πiτ ):ZH+3 (τ, U) = Cτ −1/22q¯q −1/8 exph−π(κ −2)U22/τ2i| sin(πU)|−2·∞Yn=1(1 −e2πiUqn)(1 −qn)(1 −e−2πiUqn)−2.(9)3.2.

Quantum-mechanical modelIt will be useful to interpret expression (9) in the hamiltonian language. Let us firstdo it in the simpler quantum-mechanical case obtained from field theory by taking fieldconfigurations independent of the space coordinate (this approximation, ignoring thecontributions of stringy oscillations, has been widely used in 2D gravity where it goesunder the catchy name of “mini superspace”).

The quantum-mechanical system that weobtain here is the geodesic motion on H+3 with the euclidean action−Smini(hh†) = κ4Ztr((hh†)−1∂t(hh†))2 dt . (10)Unlike in the 2D theory, in the “mini” case also the real time action is real and unitarityis recovered.

The space of states is L2(H+3 , d(hh†)) ∼= L2(R × C, dφd2v) and it carriesthe unitary representation of SL(2, C) defined by(gf)(hh†) = f(g−1hh†g†−1) . (11)On the infinitesimal level, this action may be described by generators of sl(2, C)⊕sl(2, C)∼= sl(2, C)C :J1 = 14(1 + |v|2)−1/2(veφ −¯veφ)∂φ −12(1 + |v|2)1/2(eφ∂¯v + e−φ∂v) ,J2 =i4(1 + |v|2)−1/2(veφ + ¯veφ)∂φ −i2(1 + |v|2)1/2(eφ∂¯v −e−φ∂v) ,J3 = −12∂φ −12v∂v + 12 ¯v∂¯v ,satisfying [Ja, Jb] = iǫabcJc and by ¯Ja’s given by the complex-conjugate vector fields.Ja∗= −¯Ja so that Ja −¯Ja and i(Ja + ¯Ja) are the hermitian generators of sl(2, C).

The9

Hamiltonian may be taken as −2κ−1∆where ∆denotes the Laplace-Beltrami operatoron H+3 with the SL(2, C)-invariant metric.,∆= ⃗J2 = ⃗¯J2 = 14∂2φ −14|v|2(1 + |v|2)−1∂2φ+(1 + |v|2)∂v∂¯v + 14(v∂v −¯v∂¯v)2 + 12(v∂v + ¯v∂¯v) . (12)−∆has continuous bounded below spectrum starting from 14 and induces the decompo-sitionL2(H+3 ) ∼=Zρ>0LHρ ρ2dρ(13)into the direct integral of irreducible unitary representations of SL(2, C) from the prin-cipal continuous series [37],[38] on which −∆acts as multiplication by (1 + ρ2)/4.

Hρmay be realized as the space of homogeneous functions of degree −1+iρ on non-negativematrices h′h′† with determinant zero, i.e. on the upper light cone V +3 .

The parametriza-tion by (φ, v) together with all the formulae concerning the action of SL(2, C) pass tothe case of V +3 provided that we replace everywhere6 factor 1 + |v|2 by |v|2. The scalarproduct in Hρ is that of L2(δ(2−trh′h′† )d(h′h′†)).

Operators J3−¯J3 = −v∂v+¯v∂¯v andi(J3+ ¯J3) = −i∂φ may be diagonalized at the same time as ∆and their joint spectrum isZ×R in each Hρ which, consequently, is very different from the highest- or lowest-weightrepresentation spaces of sl(2, C) ⊕sl(2, C): both J3 and ¯J3 have continuous unboundedspectrum here!The heat kernel on H+3 is known explicitly and it has a simple form:et∆(h1h†1, h2h†2) = (πt)−3/2dsinhd e−t/4−d2/t(14)where d is the hyperbolic distance between h1h†1 and h2h†2 or between hh† and 1 whereh = h−12 h1. In the more standard parametrization of H+3hh† = (1 + ⃗x2)1/2 + ⃗x · ⃗σ ,(15)d = sinh−1(|⃗x|).

Operator et∆is certainly not of trace class since −∆has continuousspectrum and moreover of infinite multiplicity. In the formal expressionZet∆(e−πiUσ3hh†eπi ¯Uσ3 , hh†) d(hh†)(16)for tr et∆e2πi(UJ3−¯U ¯J3), the integral diverges due to the U(1)C symmetry of the integratedkernel.

That is the familiar problem which we have encountered already in the two-dimensional theory. We solve it again by fixing the U(1)C invariance in the standardfashion.

This leads to the insertion of δ(φ) under the integral of the right hand side of (16)6this is why formula (12) was written in a clumsy way10

which renders it finite (for U ̸∈Z). The hyperbolic distance between e−πiUσ3hh†eπi ¯Uσ3and hh†d = cosh−1 (1 + |v|2) cosh(2πU2) −|v|2 cos(2πU1)(17)for hh† = (1 + |v|2)1/2v¯v(1 + |v|2)1/2and an easy calculation givestrren e4πτ2κ−1∆e2πi(UJ3−¯U ¯J3) ≡Ze4πτ2κ−1∆(e−πiUσ3hh†eπi ¯Uσ3 , hh†) δ(φ) d(hh†)=κ1/28πτ 1/22e−πτ2/κ−πκU22 /τ2 | sin(πU)|−2 .

(18)On the other hand, the quantum-mechanical partition functionZH+3mini(τ, U) =ZeκSmini(hh†) δ(φ(t0)) D(hh†)over twisted paths on [0, 2πτ2] satisfying hh†(2πτ2) = e−πiUσ3hh†(0)eπi ¯Uσ3 may be againcomputed by iterative gaussian integration. Not too surprisingly, one findsZH+3mini(τ, U) = Cτ −1/22e−πκU22 /τ2 | sin(πU)|−2 .

(19)Comparing eqs. (18) and (19), we find thatZH+3mini(τ, U) = C trren e4πτ2κ−1(∆+1/4) e2πi(UJ3−¯U ¯J3)(20)which establishes a Feynman-Kac type formula for the hyperbolic space H+3 .

Similarformulae may be produced for other symmetric spaces HC/H.3.3. Space of statesLet us return now to the interpretation of expression (9) for the 2D partition functionwhich becomes now straightforward.

Using eq. (19) and (20), we obtainZH+3 (τ, U) = C q¯q −c−κ/24 trren e4πτ2(κ−2)−1∆e2πi(UJ3−¯U ¯J3)·∞Yn=1(1 −e2πiUqn)(1 −qn)(1 −e−2πiUqn)−2.

(21)The first term on the right is the familiar prefactor with the central charge. Next comesessentially the mini-space contribution with κ 7→κ −2 and then, multiplicatively, thecontribution of the oscillatory degrees of freedom.

By studying the canonical quanti-zation of the H+3 WZW theory, one may infer that its space of states should carry arepresentation of the affine algebra ˆsl(2, C) ⊕ˆsl(2, C) of level −κ, extending the mini-space representation of sl(2, C)⊕sl(2, C). Let ˆb± (ˆn±) denote the subalgebras of ˆsl(2, C)generated by Jan with ±n ≥0 (±n > 0).

The action of sl(2, C)⊕sl(2, C) in L2(H+3 ) maybe extended to a representation of ˆb+ ⊕ˆb+ by making Jan and ¯Jan for n > 0 act trivially(the bar refers to the second copy). Let us choose a dense invariant subdomain in L2(H+3 )11

like the space S(H+3 ) of fast decreasing functions (in ⃗x of (15)). ˆsl(2, C) ⊕ˆsl(2, C) actsthen in the spaceˆHH+3=U(ˆsl(2, C)) ⊗U(ˆsl(2, C))⊗U(ˆb+)⊗U(ˆb+) S(H+3 )(22)where U denotes the enveloping algebra.

This gives the representation of ˆsl(2, C) ⊕ˆsl(2, C) induced from the action of sl(2, C) ⊕sl(2, C) in L2(H+3 ). In plain English,space ˆHH+3 is spanned by S(H+3 ) and by the descendents obtained by repeated action ofJan and ¯Jbn with n < 0 on the states in S(H+3 ).

As a vector space,ˆHH+3 ∼= Sym(ˆn−) ⊗Sym(ˆn−) ⊗S(H+3 )where Sym denotes the symmetric algebra. As usually, the Sugawara construction allowsto define the action in ˆHH+3 of two commuting Virasoro algebras (of central charge c−κ):Ln = −1κ−2Xm,a: JamJan−m :(23)and similarly for ¯Ln.

It is then the standard result that the contribution of the descendentstates to tr qL0 ¯q ¯L0e4πi(UJ30 −¯U ¯J30 ) is the infinite product factor in (21). SinceqL0¯q¯L0L2(H+3 ) = e4πτ2(κ−2)−1∆,(24)also the (renormalized) zero-level states contribution is recovered in (21).The hamiltonian interpretation of the field-theoretic partition function may be thensummarized in the following (Feynman-Kac type) formula:ZH+3 (τ, U) = q¯q −c−κ/24 trren qL0¯q¯L0e2πi(UJ30 −¯U ¯J30)(25)where on the right hand side the (renormalized) trace is taken over the space ˆHH+3carrying the representation of ˆsl(2, C) ⊕ˆsl(2, C) induced from L2(H+3 ).

The structureof the partition function of (21) and of the space of states appears to be much simplerhere than in the case of compact WZW models. The probable reason is that ˆHH+3 may bedecomposed into a direct integral of representations induced from Hρ, which we expect tobe irreducible, at least in a suitable sense and for almost all ρ.

Similar decomposition inthe compact case (into a finite direct sum) yields representations which should be furtherreduced.ˆHH+3 carries a natural hermitian form ( , ) extending the scalar product ofL2(H+3 ). It may be characterized by the conjugacy relation Jan∗= −¯Ja−n.

It is certainlynon-positive since for χ ∈L2(H+3 )((J1−1 −¯J1−1)χ), (J1−1 −¯J1−1)χ) = −κ2 (χ, χ) . (26)We expect however that ( , ) is non-degenerate.12

3.4. Green functionsIn Sec.2.1 and 2.2, we have seen that the matrix elements hh†(ξ)j of spin j =0, 12, 1, ... representations appear as natural insertions in the SL(2, C)/SU(2) WZWtheory, provided that they are arranged into combinations invariant under the globalSL(2, C) symmetry hh† 7→γhh†γ† (this is like the neutrality condition in the 2DCoulomb gas correlations).

The corresponding Green functions are calculable by theiterative gaussian integration in parametrization (1). Let us explain how this works onthe simplest example of the planar spin12 two-point function [7]Z tr1/2 hh†(ξ1)(hh†)−1(ξ2)eκRS(hh†) δ(hh†(ξ0)) D(hh†)=Z(|(eφv)(ξ1) −(eφv)(ξ2)|2 + eφ(ξ1)−φ(ξ2) + eφ(ξ2)−φ(ξ1))· e−π−1κR[ (∂zφ)(∂¯zφ)+(∂z+∂zφ)¯v (∂¯z+∂¯zφ)v) ] d2z· δ(φ(ξ0)) δ2(v(ξ0)) Dφ Dv(27)where we have already shifted ˜φ 7→φ.The v-integral is gaussian.It produces thepartition functione2π−1R(∂zφ)(¯∂¯zφ) d2z +(2πi)−1RφR det′(¯∂∗¯∂)/area−1(28)(which changes the coupling constant of the effective φ-integral from κ to κ−2, compareeq.

(8)) and the normalized expectation< |(eφv)(ξ1) −(eφ)(ξ2)|2 >= (πκ)−1|ξ1 −ξ2|2 e−φ(ξ1)−φ(ξ2)Ze2φ(ζ)|ξ1 −ζ|−2|ξ2 −ζ|−2d2ζ . (29)Notice the appearance of the linear term ∼∫φR in the effective φ-action and of thee2φ(ζ) insertion corresponding, respectively, to the background and screening charges inthe Coulomb gas interpretation of the resulting φ-field theory.

The integral over φ isagain gaussian but requires a renormalization of the polynomial in e±φ(ξα) and e2φ(ζ) torender it finite. If we extract the most divergent factor multiplicatively, the terms withmilder divergences will not survive the renormalization.

In the case at hand, these areterms eφ(ξ1)−φ(ξ2) + eφ(ξ2)−φ(ξ1) on the right hand side of (27). They drop out leaving uswith the resultconst.

|ξ1 −ξ2|2−1/(κ−2)Z|(ξ1 −ζ)(ξ2 −ζ)|−2+2/(κ−2) d2ζ= const. |ξ1 −ξ2|3/(κ−2)(30)(in the flat metric).

Replacing tr 12 in (27) by trj for higher spins, we obtain a φ-integralwith 2j screening charges and finallyconst. |ξ1 −ξ2|4j (j+1)/(κ−2)(31)13

provided that 2j + 1 < κ −2. Otherwise, the integral over the positions of the screeningcharges diverges7.

Higher Green functions may be computed similarly [7],[39], also forthe general HC/H theories [35].From the form of the general Green functions (also with the current and energy-momentum insertions) one infers that fields hh†(ξ) are primary, both for the ˆsl(2, C) ⊕ˆsl(2, C) and Vir⊕Vir algebras. Their conformal weights ∆j = ¯∆j are, as read from eq.

(31), −j(j+1)κ−2< 0. Occurrence of fields with negative dimensions, so with Green func-tions growing with the distance, might seem incompatible with the stability althoughnot necessarily in a non-unitary theory as ours.

The point, however, lies elsewhere. Suchfields (:eαφ: for α real) are clearly present for the massless free (uncompactified) fieldφ which gives a stable unitary theory and are also expected in the Liouville theory [9],believed to be stable and unitary (there, they correspond to the local operators in ter-minology of [9]).

These operators escape the standard relation between the spectrum ofenergy and of conformal weights since they correspond to eigenfunctions of the Hamilto-nian outside the generalized eigenspaces. This may be seen already in the “mini-space”quantum-mechanical picture which is stable and unitary for the H+3 theory: although−1κ−2∆hh†j = −j(j+1)κ−2 hh†j ,the matrix elements of hh†j are not the generalized eigenfunctions of −∆due to theirtoo rapid growth at infinity.

Appearance of operators with negative conformal dimen-sions may be typical for irrational theories with continuous spectrum of L0, ¯L0. Noticenevertheless that in the HC/H WZW model they come in a finite number whereas forthe massless free field and for the Liouville theory, there is a continuous family of suchfields.Besides fields with negative dimensions which do not correspond neither to truenor to generalized states of the theory, it is natural to expect existence of fields withpositive dimensions corresponding to the states in the spectrum of L0, ¯L0.

The naturalcandidates for such fields are given by fρ,ml,mr(hh†(ξ)) where fρ,ml,mr is a joint generalizedeigenfunction of −∆, J3, ¯J3 corresponding to eigenvalues 14(1+ρ2), ml = 12(n+iω), mr =12(−n + iω) where ρ ≥0, n ∈Z and ω ∈R. In the space Hρ (of homogenous functionson V +3 ), the corresponding eigenfunction ise−iωφ−inarg(v) |v|−1+iρ .

(32)Eigenfunction fρ,ml,mr on H+3 is obtained by applying to (32) the Gelfand-Graev integraltransformation [37] realizing the isomorphism (13):fρ,ml,mr(φ, v) = e−iωφ−inarg(v) (1 + |v|2)iω/2·2πZ0dθ∞Z0dr einθ riρ+iω [1 + 2|v|rcosθ + (1 + |v|2)r2]−1−iρ . (33)7 this is the dual manifestation of the restriction to spins j ≤k/2 in the SU(2) WZW model or,more generally, of its fusion rules, see [7]14

For example, for ρ = ml = mr = 0, we obtain the elliptic integralf0,0,0(v) = π2πZ0(1 + |v|2sin2θ)−1/2 dθ . (34)Unfortunately, we were not able to compute the Green functions of fields fρ,ml,mr exactly.It remains then to be seen if they indeed give rise, upon multiplicative renormalization,to primary fields with conformal weights ∆ρ = ¯∆ρ =1+ρ24(κ−2).4.

SL(2, C)/SU(2) mod R COSET THEORY4.1 2D black hole sigma modelIn Sec. 3, we have coupled the SL(2, C)/SU(2) ≡H+3 WZW model to an abeliangauge field A in the way which rendered the action invariant under the non-compactgauge transformations:S(eλσ3/2hh†eλσ3/2, 12i (A −dλ)) = S(hh†, 12iA) ,(1)see (3.3).

Following the scenario for producing coset theories from compact WZW mod-els, let us consider the functional integralZ−e κS(hh†,(2i)−1A) D(hh†)DA=Z−e−π−1κR[(∂z ˜φ+Az)(∂¯z ˜φ+A¯z)+(∂z+∂z ˜φ+Az)¯v (∂¯z+∂¯z ˜φ+A¯z)v]d2z DφDvDA(2)with gauge invariant insertions. First notice that, by the gauge invariance, the integralover φ factors as the (infinite) volume of the gauge group.

Since A enters quadraticallyinto the action, it may be integrated out (for appropriate, e.g. A-independent, insertions)givingCZ−e−π−1κR(1+|v|2)−1(∂z¯v)(∂¯zv)d2z Yξd2v(ξ)1+|v(ξ)|2 .

(3)The effective action for v:Seff(v) ≡κπZ(1 + |v|2)−1(∂z¯v)(∂¯zv)d2z=κπXa=1,2Z(1 + |v|2)−1(∂zva)(∂¯zva)d2zif we integrate by parts. v = v1 +iv2.

It is the action of a sigma model with the complexplane with metric(1 + |v|2)−1(dz ⊗d¯z + d¯z ⊗dz)(4)as the target. It was noticed recently [18],[19] that this target metric (together withthe dilaton field Φ = log(1 + |v|2)) forms a euclidean black hole solution of equations15

of 2D gravity (with unit mass). It describes an infinite cigar becoming asymptoticallya cylinder (the scalar curvature goes down as |v|−2 as v →∞).The Minkowskiancounterpart of this solution is the metric(1 −v+v−)−1(dv+dv−+ dv−dv+)(5)with the asymptotically flat region ±v± > 0 with future horizon v−= 0, v+ > 0 andpast horizon v+ = 0, v−< 0, another such region for v+ ↔v−, and future and pastsingularities at v+v−= 1.4.2 Toroidal partition functionAs it stands, functional integral (3) for the black hole target is difficult to computedirectly.

Instead, we may go back to expression (2) and integrate first over hh† and thenover A. Let us illustrate this on the example of the twisted toroidal partition functionZbh(τ, U) =Ze κS(γU hh†γ†U, (2i)−1A) D(hh†)DA(6)where the action for the twisted field configurations is coupled to the gauge field byputtingS(γUhh†γ†U, 12i A) = S(hh† , 12i (A + 12τ −12¯Udz + 12τ −12 Ud¯z))+12πτ2 U1Z(Az + A¯z)d2z +πτ2 U21 .

(7)The parametrization of A by the Hodge decompositionA = dµ + ∗dν + τ −12 (¯udz + ud¯z)/2(8)(µ, ν real functions, u = u1 + iu2) gives for the volumesDA = Cτ −22det′(¯∂∗¯∂) δ(µ(ξ0)) δ(ν(ξ0)) d2u Dµ Dν .Due to the gauge invariance of the action, the integral over µ factors out as the (infinite)volume of the gauge group. The ν-integral also factors out after unitary rotation v 7→e−iνv so that the v- and φ-integrals produce the twisted partition function ZH+3 (τ, u) ofthe H+3 WZW theory.

As the result, we obtainZbh(τ, U) = Cτ −22det′(¯∂∗¯∂)Ze−π−1κR(∂zν)(∂¯zν)d2z −πκτ −12(U1−u1)2· ZH+3 (τ, u) d2u δ(ν(ξ0)) Dν . (9)The ν-integral is straightforward and for ZH+3 (τ, u) we have expression (3.9).

HenceZbh(τ, U) = Cτ −1/22Ze−πκτ −12(U1−u1)2 Z(τ, u) |η(τ)|2 d2u= Cτ −12q¯q −1/12Ze −πκτ −12(U1−u1)2−π(κ−2)τ −12u22 | sin(πu)|−2·∞Yn=1(1 −e2πiuqn)(1 −e−2πiuqn)−2d2u(10)16

where η(τ) ≡q1/24 Qn≥1(1 −qn) is the Dedekind function. The u-integral diverges log-arithmically due to the singularity ∼|u|−2 at zero.This singularity is repeated onthe lattice Z + τZ ,as follows immediately from the bi-periodicity of expressione2πτ −12u22 |sin(πu)|−2∞Qn=1(1 −e2πiuqn)(1 −e−2πiuqn)−2.Let us explain this divergenceof a relatively simple nature.4.3 Mini-space partition functionIt is instructive to start with the mini-space case (we remind that this means takingfields hh† and Az, A¯z independent of the space variable).

ForZbhmini(τ, U) =Ze κSmini(γU hh†γ†U , (2i)−1A) D(hh†)DA ,(11)we may also proceed as before integrating first over A to get the twisted partitionfunction for the quantum-mechanical particle moving on the euclidean black hole:Zbhmini(τ, U) = CZe−(κ/2)2πτ2R0(1+|v|2)−1 |(∂t−iτ −12U1)v|2 dt Yξd2v(ξ)1+|v(ξ)|2 . (12)On the other hand, integrating first over hh† and then over A, we obtain:Zbhmini(τ, U) = Cτ −12Ze −πκτ −12((U1−u1)2+u22) | sin(πu)|−2 d2u .

(13)The right hand side of eq. (13) may be rewritten, with the use of eqs.

(3.18)-(3.20), asCτ −1/22Ztrrene 4πτ2κ−1(∆+1/4) e2πi(uJ3−¯u ¯J3)e −πκτ −12(U1−u1)2 d2u= Cτ −1/22Ze 4πτ2κ−1(∆+1/4)(2πu2, e−2πiu1v; 0, v) e−πκτ −12(U1−u1)2 d2u d2v . (14)Notice thatZet∆(2πu2, v; 0, v′) du2 =12π e t∆ω=0 (v; v′)(15)where ∆ω=0 is the restriction of Laplacian ∆to the generalized eigensubspace of operatori(J3 + ¯J3) = −i∂φ corresponding to eigenvalue 0.

From the expression (3.12) for ∆, weinfer that∆ω=0 = (1 + |v|2)∂v∂¯v + 14(v∂v −¯v∂¯v)2 + 12(v∂v + ¯v ¯∂¯v)(16)and is a selfadjoint operator in L2(d2v). Moreover,(κ/τ2)1/2Ze4πτ2κ−1∆ω=0(e−2πiu1v; v′) e−πκτ −12(U1−u1)2 du1= (κ/τ2)1/2Ze4πτ2κ−1∆ω=0 +2πiu1(J3−¯J3)(v; v′) e−πκτ −12(U1−u1)2 du1= e4πτ2κ−1∆ω=0 −πτ2κ−1(J3−¯J3)2 +2πiU1(J3−¯J3)(v; v′) = e 4πτ2κ−1∆bh(e−2πiU1v; v′)(17)17

where we have introduced−∆ω=0 + (J3)2 = −∆ω=0 + ( ¯J3)2= −12∂v(1 + |v|2)∂¯v −12∂¯v(1 + |v|2)∂v ≡−∆bh . (18)It is a Laplacian quantizing the classical Hamiltonian pvp¯v(1 + |v|2) of the particle onthe (euclidean) black hole, with a specific choice of ordering prescription (different fromthe Laplace-Beltrami operator which would correspond to (1 + |v|2)1/2∂v∂¯v(1 + |v|2)1/2 ).We may finally rewrite the mini-space partition function asZbhmini(τ, U) = CZe 4πτ2κ−1(∆bh+1/4)(e−2πiU1v; v) d2v .

(19)The integral is divergent but the nature of this divergence is quite simple. For v →∞,where the metric becomes cylindrical in variable logv, exp[t∆bh(e−2πiU1v; v)|v|2 ap-proaches a constant (equal to the free heat kernel between the points on the cylinder ofconstant difference).

Hence the divergence due to the infinite volume of the black holecigar. It may be easily regularized by cutting integral over v to |v| ≤R.

Going back tointegral (3.8), it is easy to see that such cutoffresults in the replacemente−πκτ −12u22 7−→e−πκτ −12u22 −e−(4πτ2)−1κd2R(20)in the integrand of (13). Here dR = cosh−1 (cosh(2πu2) + 2R2| sin(πu)|2) stands for thehyperbolic distance between e−πiUσ3hh†eπi ¯Uσ3 and hh† = (1 + R2)1/2RR(1 + R2)1/2.Such a replacement makes the integral in (13) convergent but behaving as O(logR) (ormore generally as O(logMR) where M is the black hole mass; we consider here only thecase M = 1).

We could define the finite part of Zbhmini by subtracting this logarithmicdivergence, i.e. by comparing it to half the partition function of a particle on the cylinder.Let us go back to the interpretation of the result (10).

As compared to expression(13) for the mini-space case, the main differences in (10) are the partial shift κ 7→κ −2and the presence of the big product inherited from the oscillatory modes of the H+3theory. The shift of κ is easy: if we drop the infinite product from the right hand sideof (10) to get the level zero (i.e.

zero mode) contribution, we obtain, proceeding as forthe mini-space case,Zbhlevel 0(τ, U) = Cq¯q −(c−κ−1)/24 tr|ω=0 e4πτ2(κ−2)−1∆−2πτ2κ−1((J3)2 + ( ¯J3)2)+2πiU1(J3−¯J3)= C q¯q −(c−κ−1)/24 tr| level 0ml+mr=0 qLcs0 ¯q¯Lcs0 e2πi(UJ30 −¯UJ30)(21)with the coset Virasoro generatorsLcs0 = L0 + 1κXn: J3nJ3−n : ,¯Lcs0 = ¯L0 + 1κXn: ¯J3n ¯J3−n : . (22)The contribution of the higher level oscillatory modes is, however, less transparent thanone may naively think if we want to interpret it in terms of gauge invariant states.18

4.4 Asymmetric parafermionsLet us compare the situation to a somewhat similar case of a variant of rationalparafermionic theory which may be described as the SU(2) WZW model with the axialgauging of the U(1) subgroup, i.e. with the diagonal U(1) gauged asymmetrically.

Thetwisted toroidal partition function for such parafermions is [40],[14]Z pf(U, τ) =Ze−kS(γUgγ†U ,A) Dg DA . (23)The integration is now over real A. Parametrizing A as before by the Hodge decompo-sition, one arrives at the formulaZ pf(U, τ) = Cτ −1/22ZC/(Z+τZ)eπkτ −12(U1−iu2)2 ZSU(2)(τ, u) |η(τ)|2 d2u(24)where ZSU(2)(τ, u) is the asymmetrically twisted partition function of the rational SU(2)WZW model:ZSU(2)(τ, u) = q¯q −ck/24 tr qL0¯q¯L0 e2πi(uJ30 +¯uJ30 ) .

(25)The trace is taken over the space of statesˆHSU(2) =Lj≤k/2ˆHj ⊗ˆHj(26)where ˆHj carries the irreducible spin j level k representation of the Kac-Moody algebraˆsl(2, C). Notice the sign in front of ¯u ¯J30 in (25).

The integrand on the right hand sideof eq. (24) is a function on C/(Z + τZ) only if U1 ∈k−1Z and only such twists shouldbe allowed.For other twists there is a global gauge anomaly: the ungauged globalU(1) symmetry is broken in the parafermionic theory to Zk.

The spaces ˆHj may bedecomposed into the weight spaces according to the integral or half-integral eigenvaluem of J30 and at the same time with respect to the level k representations of the ˆU(1)affine algebra (similarly for the complex conjugates):ˆHj ∼=Lm ˆHsingj,m ⊗ˆH′m(27)where ˆHsingj,m is the subspace of ˆHj where J30 = m and J3n = 0 for n > 0. H′m is the spaceof the level k J30 = m irreducible representation of the ˆU(1) algebra.

The SugawaraVirasoro generator L0 decomposes into the sum of Lcs0 ≡L0 −1kPn:J3nJ3−n: acting onspaces ˆHsingj,m and L′0 ≡1kPn:J3nJ3−n: acting on ˆH′m (in fact on ˆHsingj,m , Lcs0 = L0 −1κm2).Accordingly, we obtain for the partition function of the SU(2) WZW theory:ZSU(2)(τ, u) = (q¯q)−(ck−1)/24Xml,mrZsingml,mr q m2l /k ¯q m2r/k |η(τ)|−2e2πi(uml+¯umr)(28)whereZsingml,mr = tr| ˆHsingml,mr qLcs0 ¯q¯Lcs0(29)19

withˆHsingml,mr ≡LjˆHsingj,ml ⊗ˆHsingj,mr . (30)Zsingml,mr depends only on ml and mr mod k/2 [40] (essentially due to the compact natureof the gauged symmetry).

More exactly,tr| ˆHsingj,m qLcs0 = tr| ˆHsingj,m+k qLcs0 = tr| ˆHsingk−j,−m qLcs0 ,see [40]. Upon the insertion of (28) into the right hand side of (24), the u1-integral willenforce equality ml = −mr ≡m.

The sum over m may be reduced mod k/2, with thesum over the integral part of 2m/k used to extend the integration over u2 to a gaussianone over the entire real line. Finally we getZ pf(τ, U) = C q¯q −(ck−1)/24Xm=0, 12 ,..., k2Zsingm,−m e−4πimU1 .

(31)As we see, the parafermionic partition function is consistent (modulo multiplicity)with the space of states of the coset theory obtained by imposing the gauge conditionsJ30 + ¯J30 = 0,J3n = ¯J3n = 0 for n > 0(32)in the space of states of the ungauged WZW theory with the Virasoro algebra given bythe coset construction. On the other hand, we could replace the first gauge condition byJ30 + ¯J3 = kn for n ∈Z or by J3 = −¯J3 and obtain equivalent theory.

The latter meansthat the asymmetric parafermions are indistinguishable from the symmetric ones.4.5 Space of statesThe level zero contribution (21) to the black hole partition function is fully consistentwith the gauge conditions (32) imposed on states of the H+3 WZW theory (for zero modes,only the first condition of (32) restricts the states). The problem appears on the excitedlevels of the space of states ˆHH+3 of the ungauged theory.

Let us consider, as an example,the first excited level with states of the formXa=±,3(Ja−1ψa + ¯Ja−1 ¯ψa)(33)where ψa, ¯ψa are level zero states, i.e. functions on H+3 .

The J30 + ¯J30 = 0 conditiontranslates into(J30 + ¯J30 ± 1)ψ± = 0 ,(J30 + ¯J30 ± 1) ¯ψ± = 0 ,(J30 + ¯J30)ψ3 = 0 . (34)The other conditions of (32) giveψ3 = 2κ(J+0 ψ+ −J−0 ψ−) ,¯ψ3 = 2κ( ¯J+0 ¯ψ+ −¯J−0 ¯ψ−) .

(35)20

Notice, however, that in L2(H+3 ), J30 + ¯J30 is antihermitian so it has imaginary spectrum.Thus non-trivial solutions of (34) and (35) are not only out of L2(H+3 ) but do not belongto the generalized eigenspaces of J30, ¯J30 (they have e±φ dependence on φ). At best, wehave to change the Hilbert space.

Notice how the situation here differs from the caseof parafermions where no such problems arise. We may understand the above difficultyalso by looking at the level one contribution to the partion function (10) which involvesintegralsτ −12Ze −πκτ −12(U1−u1)2−π(κ−2)τ −12u22 | sin(πu)|−2 e±2πiu d2u= Cτ −1/22Ze 4πτ2(κ−2)−1(∆+1/4)(2πu2, e−2πiu1v; 0, v)· e±2π(iu1−u2) e−πκτ −12(U1−u1)2 d2u d2v .

(36)By spectral analysis, we may decompose operators et∆into the heat kernels acting inthe generalized eigenspaces of J30, ¯J30:et (∆+1/4)(2πu2, e−2πiu1v′ ; 0, v) =XnZKn,ω(t; |v′|, |v|) e2πinu1−2πiωu2 dω . (37)This allows to rewrite integrals (36) asCXnKn,∓i(4πτ2κ−1; |v|, |v|) e−πτ2κ−1(n±1)2 +2πi(n±1)U1 d|v|2(38)involving the analytic continuation of heat kernels Kn,ω to imaginary values of ω. Thequestion is whether such an analytic continuation (which exists) corresponds to a heatkernel in a different Hilbert space.Summarizing.

the gauge conditions (32) do not determine unambiguously the spaceof states. We have to supplement them with regularity conditions specifying domainsof the operators that they involve (the same applies to the BRST definition of gaugeinvariant states).

Ultimately, we should be able to build a Hilbert space of states ateach level and to compute the contribution to the partition function as a trace of a heatkernel in such a space. We shall discuss a candidate solution of this problem in Sec.

5.On top of the above difficulties with the interpretation of the partition function(but not unrelated to them) comes the fact that, as it stands, the integral on the righthand side of eq. (10) diverges.

The source of this divergence is, as in the mini-spaceapproximation, the infinite volume of the target space. This may be regularized forexample by defining˜Zbhreg(τ, U; R) = Cτ −12Ze−πκτ −12|U−u|2 S(τ, u)1 −eR2 S(τ,u)−1d2u(39)whereS(τ, u) ≡q¯q −1/12 e2πτ −12u22 |sin(πu)|−2∞Yn=1(1 −e2πiuqn)(1 −e−2πiuqn)−2.

(40)21

The partition function ˜Zbhreg(τ, U) is finite and when R →∞and for U2 = 0, we recoverthe infinite integral (10) (we have put the twists along both homology cycles in ˜Zbhreg(τ, U)so that in the limit R →∞it corresponds to the black hole functional integral withboundary conditions v(z+2π) = e−2πiΦv(z), v(z+2πτ) = e−2πiΘv(z) where U = Θ−τΦ);for Φ = 0, we recover Zbh(τ, U) with twist only along one cycle). S(τ, U) is invariantunder translations U 7−→U + n + τm for n, m integers and is modular invariant.

As aresult, under SL(2, Z) transformations,˜Zbhreg( aτ+bcτ+d,Ucτ+d; R) =˜Zbhreg(τ, U; R) ,(41)i.e. the regularized partition function is modular covariant.

Again the divergence islogarithmic in R and we could subtract it to define the renormalized partition functionmeasuring the difference between the theories with the black hole and (half-)cylindertargets.4.6 Partition functions at higher generaOn a higher genus Riemann surface Σ with the homology basis (aα, bβ), α, β =1, ...,genus, and with the basic holomorphic forms ωα,Raα ωβ = δαβ,Rbαωβ = τ αβ ≡τ αβ1+ iτ αβ2 , let us define the multivalued field˜γU(P) = eπσ3PRP0(Utτ −12¯ω−¯Utτ −12ω)/2(42)with values in the Cartan subgroup of SU(2). Along the basic cycles˜γU(aαP) = e−πiΦασ3.˜γU(P) ,˜γU(bαP) = e−πiΘασ3˜γU(P)where U = Θ −τΦ.

The twisted partition function on Σ is given by˜Zbh(τ, U) =ZeκS(˜γU ˜γ†U ,(2i)−1A) D(hh†) DA(43)withS(˜γUhh†˜γ†U, (2i)−1A) = S(hh†, 12i (A + π ¯Utτ −12 ω + πUtτ −12 ¯ω))+ 12iZA ∧( ¯Utτ −12 ω −Utτ −12 ¯ω) + πUtτ −12 U . (44)It defines the higher genus partition function for the black hole with twists of the v-fieldby e−2πiΦα facto the aα cycles and by e−2πiΘβ along the bβ ones.

We decompose againthe gauge field according to Hodge:A = dµ + ∗dν + π¯utτ −12 ω + πutτ −12 ¯ω(45)22

and integrate over the v-field (of hh†), ν and µ (the latter integral gives the volume ofthe gauge group). What is left is the φ functional integral and the integral over twistsu:˜Zbh(τ, U) = Cdet′(−¯∂∗¯∂)area1/2 Ze−πκ( ¯U−¯u)tτ −12(U−u) + (2πi)−1κR(∂φ)(¯∂φ)· det¯∂+ ¯∂φ + πutτ −12 ¯ω)∗(¯∂+ ¯∂φ + πutτ −12 ¯ω)−1 δ(φ(ξ0)) Dφ d2genusu .

(46)By the chiral anomaly (compare the genus one formula (3.8)),det¯∂+ ¯∂φ + πutτ −12 ¯ω)∗(¯∂+ ¯∂φ + πutτ −12 ¯ω)−1 = eiπ−1R(∂φ)(¯∂φ) + (2πi)−1RφR·detα,β(∫e2φηuαηuβ) / detα,β(∫ηuαηuβ)−1 det¯∂∗u ¯∂u−1(47)where ¯∂u ≡¯∂+ πutτ −12 ¯ω and ηuα, α = 1, ..., genus −1, form a basis of the 01-forms inthe kernel of ¯∂∗u. Using eq.

(47), we may rewrite the partition function as˜Zbh(τ, U) = Cdet′(−¯∂∗¯∂)area1/2 Ze−πκ( ¯U−¯u)tτ −12(U−u) + (2πi)−1(κ−2)R(∂φ)(¯∂φ) +(2πi)−1RφR· e−R¯ηu exp(2φ) ηu det¯∂∗u ¯∂u−1δ(φ(ξ0)) Dφ dηu du(48)where the gaussian integral over ηu ∈ker ∂∗u was used to express the ηuα determinants.The expression is obviously similar to the Liouville partition function although the realrelation between two theories lies probably deeper. In any way, we expect the φ and ηintegrals to be finite and to lead to an expression regular in u except for the contributionof det¯∂∗u ¯∂u−1 which around u = 0 behaves as |u|−2 which is integrable for genus > 1and diverges logarithmically for genus 1 (ηuα may be chosen regular in u around u = 0).This singularity is repeated around other points of Z + τZ.Thus, similarly as forthe Liouville theory coupled to free bosonic field, see [41],[42], we expect the partitionfunctions at higher genera to be convergent reflecting the finite dimension of the regionin the target space relevant for the stringy interaction.4.7 Green functionsSince the coset theory is an instance of a gauge theory, its Green functions should begiven by functional integral with gauge invariant insertions.

Examples of gauge invariantfields are fρ,ml,mr(v(ξ)) of eq. (3.33) with ml = −mr ≡m whose conformal weights are∆ρ,m = ¯∆ρ,m =1+ρ24(κ−2) + m2κ .

(49)If we instead used fρ,ml,mr(φ(ξ), v(ξ)) with ml + mr ̸= 0 as local fields, we could stillmaintain local gauge invariance by adding compensating currents, i.e. by consideringinsertionsI(hh†, 12iA) =Yαfρα,mlα,mrα(φ(ξα), v(ξα)) e−RcA(50)23

where c is a chain such that δc = Pα (mlα + mrα)ξα. In the planar case, the functionalintegral over the gauge field may be easily done upon parametrization A = dµ + ∗dν.The integral over µ drops out because of gauge invariance and the integral over ν givesexpectation value of chiral vertex operatorsZei Rc+c′∂ν −i Rc−c′¯∂ν −π−1κR(∂zν)(∂¯zν)d2zDν(51)where δc′ = Pα (mlα −mrα)ξα (compare [14] where similar calculation was done for theparafermions).

Altogether, we obtainZI(hh†, 12i A) eκS(hh†, (2i)−1A) D(hh†) DA= const.Yα̸=α′(ξα −ξα′)mlαmlα′/κ (¯ξα −¯ξα′)mrαmrα′/κZI(hh†, 0) eκS(hh†) D(hh†) (52)where theQα̸=α′ factors come from the (properly renormalized) free field integral (51).They modify the conformal dimensions of fields fρ,ml,mr of the H+3 WZW theory to∆ρ,ml =1+ρ24(κ−2) +m2lκ ,¯∆ρ,mr =1+ρ24(κ−2) + m2rκ . (53)producing operators with imaginary spin and hence never local.

It is possible, however,that correlations of fields coming from common eigenfunctions on H+3 of ∆, J3, ¯J3 whichdo not correspond to the spectrum, for example for ω imaginary, may be given sense.If in the left hand side of (52) we integrated out the A-field, we would obtain the blackhole functional integral with insertions which for large values of |v(ξα)| take formYα|v(ξα)|mlα+mrα fρα,mlα,mrα(0, |v(ξα|)e−i Rc+c′∂arg(v) + i Rc−c′¯∂arg(v). (54)We recover then the chiral vertex operators of field arg(v)(ξ) which, for large |v|, becomesa compactified free field.

If fields with real ml+mr existed, they would be mutually localfor ml + mr ∈κZ, as are their asymptotic versions. We shall return to the discussion ofthis possibility in the next section.5.

SU(1, 1) mod U(1) COSET THEORY5.1 Functional integral formulationThe original proposal [18] for the conformal sigma model with 2D black hole targetwas based on a coset construction starting with SU(1, 1) ∼= SL(2, R) WZW model. Theparametrizationg = eiψ(1 + |v|2)1/2v¯ve−iψ(1 + |v|2)1/2(1)24

where ψ is in R/(2πZ) and v is complex gives global coordinates on SU(1, 1). Comparingto parametrization (3.1) of positive elements in SL(2, C), we see that it passes to thepresent one by simple substitution φ 7→iψ.

Consequently, for the WZW action withthe U(1) ⊂SU(1, 1) gauged asymmetrically (i.e. with the axial U(1) gauge), we obtainfrom eq.

(3.4)S(g, 12iA) = −1πZ[(i∂z ˜ψ + Az)(i∂¯z ˜ψ + A¯z)+(∂z + i∂z ˜ψ + Az)¯v (∂¯z + i∂¯z ˜ψ + A¯z)v] d2z . (2)where ˜ψ ≡ψ + 12ilog(1 + |v|2).

The axial gauge invariance isS(eiλσ3g eiλσ3, 12i(A −2idλ)) . (3)The euclidean action ±κS(g) for the SU(1, 1) WZW theory is not bounded below.

Forthe minus sign (and κ positive) this is due to the ˜ψ-field contribution. As a result, thestable euclidean picture is missing for this theory.

In the coset functional integralZ−eκS(g,(2i)−1A) Dg DA ,however, the ˜ψ-field may be gauged out and absorbed by a translation of A. If A istaken real then the A integral is stable and the translation of A is complex (the axialgauge invariance requires imaginary A).

In this case, moreover, after the translation,we recover the same integral as before for the SU(2, C)/SU(2) mod R coset theory.It seems that the two coset theories coincide8. On the quantum-mechanical level, theequivalence of both approaches may be seen clearly.5.2 Particle on SU(1,1)The classical mini-space system which corresponds to the 2D WZW theory withtarget SU(1, 1) is the geodesic motion in the invariant metric on SU(1, 1) of signature,say, (−, +, +).

We may quantize it taking L2(SU(1, 1)) with the Haar measure (equaldψd2v in parametrization (1)) as the space of states in which SU(1, 1)left × SU(1, 1)rightacts unitarily. Infinitesimally, we get the action of sl(2, C) ⊕sl(2, C) generated by Ja’sand ¯Ja’s given by the same formulae as in the case of L2(H+3 ) except for the substitutionφ 7→iψ.

The hermiticity relations change, however, and we obtainJa∗= −Ja ,¯Ja∗= −¯Jafor a = 1, 2 ,J3∗= J3 ,¯J3∗= ¯J3. (4)Also −∆≡−⃗J2 = −⃗¯J2 is no more bounded below.

It is again given explicitly by eq. (3.12) with ∂2φ replaced by −∂2ψ.

In factL2(SU(1, 1)) ∼=Zρ>0ǫ=0,1/2LDρ,ǫ ⊗¯Dρ,ǫ dν(ǫ, σ)MMj=−1,−3/2,...±D±j ⊗¯D±j . (5)8this is the point on which the present author’s opinion has wavered most and might continue to doso with the progress in the understanding of both theories25

Dσ,ǫ carry unitary irreducible representations of SU(1, 1) of the principal continuousseries which may be realized in the space of sections of a spin bundle on the circle(SU(1, 1) acts naturally on S1, ǫ corresponds to two choices of the spin structure).The eigenvalue of ⃗J2 on Dρ,ǫ is −14(1 + ρ2). Spaces D±−j carry the lowest- (highest)-weight representations of sl(2, C) of spin j.

They give the discrete series of unitary,irreducible representations of SU(1, 1) with eigenvalue of ⃗J2 equal to j(j + 1) which is≥0. If, instead of SU(1, 1), we considered its simply-connected coveringgSU(1, 1) (whereψ takes values in the non-compactified real line), the direct sums in decomposition (5)over ǫ and j would be replaced by direct integrals over 0 ≤ǫ < 1 and j < −1/2.

Since−∆plays the role of Hamiltonian, the energy is not bounded below (nor above). Thisproblem with stability renders the above quantization physically not very satisfactory.Indeed, the way we proceeded here is not the one used for example to quantize a particlein Minkowski space where one recovers satisfactory solution of the stability problempassing to the second-quantized level.

Finding a stable quantization of the particle onSU(1, 1) or, more importantly, of the SU(1, 1) WZW field theory remains an open andseemingly very interesting problem9. Here, however, we shall be interested only in thecoset SU(1, 1) mod U(1) theory where coupling to the gauge field removes the unstable˜ψ field.

On the quantum-mechanical level, the gauge condition J3 + ¯J3 = i∂ψ = 0, cutsout from L2(SU(1, 1)) the contribution of the discrete series (and more) making −∆positive. Besides,L2(SU(1, 1))|J3+ ¯J3=0 ∼= L2(d2v) ∼= L2(H+3 )|J3+ ¯J3=0in a natural way and this isomorphism preserves (restrictions of) ∆, J3 and ¯J3.

Thisproves on the mini-space level the identity of the coset theories SU(1, 1) mod U(1)and SL(2, C)/SU(2) mod R. The generalized eigenfunctions fρ,ml,mr of ∆, J3, ¯J3 onSU(1, 1), corresponding to eigenvalues −14(1 + ρ2), ml, mr with ml ± mr ∈Z, are givenby Jacobi functions [38]. For ml +mr = 0 they are independent of ψ and, although givenby different expressions, coincide with similar eigenfunctions on H+3 .

For example, fromthe harmonic analysis on SU(1, 1), we obtainf0,0,0(v) = π2πZ0(1 + 2|v|2 + 2v(1 + |v|2)1/2cosθ)−1/2 dθ(6)which should be compared with eq. (3.34).

For both ml + mr equal and different fromzero, eigenfunctions fρ,ml,mr seem to generate primary fields of dimensions given by eq. (4.53) (for ml + mr ̸= 0, they should be dressed with line integrals of the gauge field,like in (4.50)).

If ml + mr ∈κZ, the corresponding fields are mutually local.5.3 Space of states, unitarity, duality, problemsOn the level of 2D field theories, neither SL(2, C)/SU(2) mod R nor SU(1, 1) modU(1) theory has been shown to exist, least solved completely, so comparison is more9we thank G. Gibbons for attracting our attention to it26

difficult. The computation of the partition function in the first case did not requirecomplex rotations or shifts of the fields so it seems more trustable.

Nevertheless, wehave seen that the interpretation of the excited contributions to it required analyticcontinuation of the heat kernels on the eigensubspaces of J3, ¯J3 in L2(H+3 ) to imaginaryeigenvalues ω of1i (J3 + ¯J3) = i∂φ. But this should be given by the heat kernels in theeigenspaces of J3, ¯J3 in L2(SU(1, 1)), or more generally in L2(gSU(1, 1)) , obtained bythe substitution φ 7→iψ.

It is then possible that the partition function becomes a traceover the gauge-invariant states of the SU(1, 1) WZW theory. Superficially, the U(1)coset of the latter has the same problem as the parafermionic model discussed in Sec.4.4: the ungauged (vector) U(1) symmetry has global anomaly which seems to breakU(1) to Zk.

Here, this is a spurious problem, however: if we start from thegSU(1, 1)WZW theory rather than from the SU(1, 1) one, the coset theory is the same but thecomplete U(1) symmetry is present. The space of states of thegSU(1, 1) WZW theoryshould be a subspace ofZρ>00≤ǫ<1LˆDρ,ǫ ⊗ˆ¯Dρ,ǫ dν(ǫ, σ)MZj<−1/2±LˆD±j ⊗ˆ¯D±j(7)where “ˆ” denotes the representation space of the Kac-Moody algebra ˆsl(2, C) induced(in the sense of Sec.

3.3) from the representations ofgSU(1, 1). What exactly shouldbe the subspace taken does not seem to be clear yet.

A possibility is the appearance ofthe fusion rule −12(κ −1) < j in the discrete series, analogous to the rule j ≤k/2 ofthe SU(2) WZW theory. Spaces ˆD may be provided with the hermitian form for whichJan∗= −Ja−n for a = 1, 2 and J3n∗= J3−n (this agrees at level zero with the scalar productinduced from L2(SU(1, 1)) ).

The encouraging sign is the important result of Dixon-Lykken-Peskin [12] (see also [43]) who proved that the gauge conditions J3n = 0, n > 0,cut out, under the restriction −12κ ≤j on the discrete series, the negative norm statesfrom the induced representations.Notice, that the latter condition disposes of therepresentations with negative eigenvalues of Lcs0 . Their absence should then be assuredby stability if the coincidence with the explicitly stable H+3 mod R model really takesplace.

In that case, the SU(1, 1) mod U(1) approach should allow to show the unitarityof the euclidean black hole CFT. Moreover, we should be able to assemble the calculatedpartition functions from the characters of the induced representations ˆD.

This is notsimple even on the quantum mechanical level where we know that it works.The gauge condition J30 + ¯J30 = 0 leaves us with spin-less, U(1)-charge zero sector ofthe theory. The functional integral for the partition function, as in any gauge theory,should be given by the trace over this subspace of states, as is also clearly indicated bythe U dependence of the result (4.10).

The primary fields fρ,ml,mr with ml + mr = 0correspond to vectors in this sector. On the other hand, gauge conditions J30 + ¯J30 = lκshould give for 0 ̸= l ∈Z sectors with the spin and the U(1) charge different from zero.Fields fρ,ml,mr with ml + mr = lκ should correspond to states in these sectors.

Fromthe point of view of the asymptotic free field with the cylindrical part of the cigar as27

the target, these are the winding sectors, see formula (4.54)10. The partition functionscorresponding to the winding sectors can be also computed, essentially by inserting aPolyakov line with charge lκ into the functional integral.

We plan to return to theseissues elsewhere.Another open problem in the black hole CFT is a relation between the coset modelsSU(1, 1) mod U(1) obtained by gauging the axial and the vector U(1) subgroup. Thevector theory has a more serious stability problem than the axial one since the vectorgauging does not seem to remove completely the unbounded below modes.

On a ratherformal level one can argue that both theories have the same spectrum of mutually localoperators [20]-[22],[44]. It was expected that they give the same CFT.

The vector cosetresults in a sigma model with singular metric on the target. In the asymptotic region,the target also looks like a half-cylinder and the identity of the models would becomethere that of free fields compactified on dual radia [45].

We have not been able, however,to stabilize the functional integral for the vector theory in a sensible way to show thatit has the same partition function as the axial coset. The situation should be contrastedwith the case of parafermions.

There, as we have seen in Sec. 4.4, both gaugings givethe same theory, in fact already on the mini-space level.

In particular, both partitionfunctions coincide.The duality between the two U(1) cosets of the SU(1, 1) WZWtheory requires, in our opinion, further study. It may be that the vector descriptionmay be maintained only in the asymptotically flat region.

The issue is important forunderstanding whether the coupling to dynamical gravity washes out the singularity atv+v−= 1 of the classical Minkowskian metric (4.5) interchanged by the duality withthe non-singular horizon v+v−= 0, see [20]. Even less clear is what sense we can makeof the sigma model which Minkowskian 2D black hole as the target which formallycomes from gauging non-compact subgroup in SU(1, 1) theory [18] and how all thesetheories fit together.

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