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Extra Observables in Gauged WZW Models

arXiv:hep-th/9110071v1 30 Oct 1991UCSBTH-91-55Extra Observables in Gauged WZW ModelsNobuyuki Ishibashi ∗Department of PhysicsUniversity of CaliforniaSanta Barbara, CA 93106AbstractIt is known that Liouville theory can be represented as an SL(2, R) gauged WZWmodel. We study a two dimensional field theory which can be obtained by analyti-cally continuing some of the variables in the SL(2, R) gauged WZW model.

We canderive Liouville theory from the analytically continued model, ( which is a gaugedSL(2, C)/SU(2) model, ) in a similar but more rigorous way than from the originalgauged WZW model. We investigate the observables of this gauged SL(2, C)/SU(2)model.

We find infinitely many extra observables which can not be identified withoperators in Liouville theory. We concentrate on observables which are (1, 1) formsand the correlators of their integrals over two dimensional spacetime.

At a specialvalue of the coupling constant of our model, the correlators of these integrals on thesphere coincide with the results from matrix models.October 1991∗Address after Nov. 1: Department of Physics, University of California, Davis, CA 95616

1IntroductionPolyakov’s discovery of the existence of SL(2) current algebra in two dimensional quan-tum gravity in the light-cone gauge[1] and the subsequent success of the derivation of scalingexponents[2], suggested that two dimensional quantum gravity could be rewritten in such away that the SL(2) current algebra is more apparent. The authors of [3][4] showed that twodimensional quantum gravity in the light cone gauge could be represented by an SL(2, R)WZW model with the constraintJ−= 1.

(1)The “soldering” procedure in [6] and the study of SL(2, R) Chern Simons theory exploredin [7], exposed the relation between two dimensional quantum gravity and SL(2, R) gaugetheory consisting of zweibein and spin connection. These works imply that the SL(2, R)structure is not a peculiarity of the light cone gauge but a more fundamental feature of twodimensional quantum gravity.Although the light cone gauge reveals the SL(2, R) structure, the most convenient gaugeof two dimensional quantum gravity is conformal gauge.

In conformal gauge, two dimen-sional quantum gravity is described by Liouville theory, which is simpler and more usefulthan the complicated light cone gauge action. In [5], it was shown that Liouville theorycould also be represented as a constrained SL(2, R) WZW model classically.

This time theSL(2) currents should be constrained asJ+ = ¯J−= √µ. (2)In [8][9], a gauged WZW model realizing the above constraints was analyzed.

Liouvilletheory was derived from this SL(2, R) gauged WZW model at the quantum level. Therefore,in this formulation, we can see the SL(2, R) structure is also hidden in Liouville theory.However the analysis of [8][9] is somewhat formal because they are dealing with WZWmodel of a noncompact group.

In this paper, we will propose another model which representsLiouville theory. This new model, which is a gauged SL(2, C)/SU(2) model, is obtainedby analytically continuing some of the variables in the SL(2, R) gauged WZW model.

Theanalytic continuation does not spoil the left and right SL(2) current algebras in the WZWmodel.Because of this analytic continuation, the arguments of [8][9] go through morerigorously in this model. We will study the observables and their correlation functions inthis model.The organization of this paper is as follows.

In section 2, we first review the analysis of[8][9]. In order to make their arguments more rigorous, we analytically continue the SL(2, R)gauged WZW model to the gauged SL(2, C)/SU(2) model.

Liouville theory is derived fromthis gauged SL(2, C)/SU(2) model in a more rigorous way than in the derivation in [8][9].In section 3, observables in the gauged SL(2, C)/SU(2) model are discussed. It is natural toexpect that all the observables correspond to Liouville theory operators.

However, we findthat there exist infinitely many extra observables, which cannot be identified with operatorsin Liouville theory. We concentrate on the observables with conformal weight (1, 1) and their1

integrals over two dimensional spacetime. The correlators of such integrals are calculatedon the sphere.

If one chooses the coupling constant of our model so that it corresponds toLiouville theory induced by c = −2 conformal field theory, these correlators coincide withthe correlators of the observables at the first critical point of the one matrix model. Insection 4, we present a brief discussion of our results.

Appendix is devoted to definitionsand some useful formulas about SL(2, C)/SU(2) model.2Gauged WZW Model and Liouville TheoryLet us consider the SL(2, R) gauged WZW model with gauge fields A+z and A−¯z following[5],I = kSW ZW(g)+kπZd2x{A−¯z (tr(t+∂gg−1) −√µ)+A+z (tr(t−g−1¯∂g) −√µ)+A+z A−¯z tr(t+gt−g−1)}. (3)Here g ∈SL(2, R) and SW ZW(g) is the action of the WZW modelSW ZW(g) = −12πZd2xtr∂g ¯∂g−1 +i12πZB d3xtr(g−1dg)3,(4)and t± = t1±t2 are the generators of SL(2, R) with positive and negative roots respectively.The action I is invariant under the gauge transformations δg = −ǫ−t+g −gǫ+t−, δA+z =∂ǫ+, δA−¯z = ¯∂ǫ−.In [8][9], it was shown that Liouville theory can be deduced from this gauged WZWmodel.

Let us review their derivation of Liouville theory. g ∈SL(2, R) can be parametrizedvia the Gauss decompositiong = 1v01!

eφ00e−φ! 10¯v0!.

(5)In these coordinates, the action becomesI=kπZd2x{∂φ¯∂φ + e−2φ ¯∂v∂¯v+A+z (e−2φ ¯∂v −√µ) + A−¯z (e−2φ∂¯v −√µ)(6)+e−2φA+z A−¯z }.The authors in [8][9] derived Liouville theory from this field theory of φ, v, ¯v and A’s. 11 As was stressed in [5], the Gauss decomposition is possible for an element near the identity in the groupmanifold.

In this case, the SL(2, R) matrix g =abcd, with d = 0 cannot be represented by eq. (5).Therefore, there is a subtlety in representing the SL(2, R) WZW model as the field theory of φ, v and ¯v.2

Liouville theory appears if one integrates out v, ¯v and A’s in the partition functionZ =Z [dφdvd¯vdA]V ol.e−I. (7)Here V ol.

denotes the gauge volume of the gauge transformationδ¯v = −ǫ+,δA+z = ∂ǫ+,δv = −ǫ−,δA−¯z = ¯∂ǫ−. (8)Integration over the v’s and A’s was done by choosing the gauge v = ¯v = 0 or equivalentlyshifting the integration variable A’s by A+z →A+z + ∂¯v, A−¯z →A−¯z + ¯∂v.

After doing so, weare left with the following expression for the partition function:Z =Z [dφdvd¯vdA]V ol.exp{−kπZd2x(∂φ¯∂φ + e−2φA+z A−¯z −√µA+z −√µA−¯z )}. (9)The v, ¯v integration merely corresponds to an overall constant.

Since the functional inte-gration measure for φ, v, ¯v is defined by the norm∥δg∥2=Zd2xtr(g−1δg)2=2Zd2x{(δφ)2 + e−2φδvδ¯v},(10)The v, ¯v integration divided by the gauge volume V ol. 2 gives us the factor arising from thedeterminant which we formally write asQx e−2φ.

The measure for A+z and A−¯z is defined bythe norm∥δA∥2 =Zd2xδA+z δA−¯z . (11)The integration over A+z , A−¯z contributes the inverse ofQx e−2φ.

Naively this cancels thedeterminant coming from the v integration. Thus we obtain,Z =Z[dφ] exp{−kπZd2x(∂φ¯∂φ −µe2φ)}.

(12)The partition function therefore is the same as the partition function of Liouville theory.However, notice that we are comparing the determinants of the operator e−2φ acting bymultiplication on v with the same operator acting by multiplication on A. This is a situationanalogous to the one we encounter in the ghost number anomaly in string theory.

In thatcase, one should compare the determinant of an operator acting by multiplication on theghost with that of the same operator acting on the antighost. Since the spins of the ghostand the antighost are different, there is a nontrivial difference between the two determinants.Since the spins of v and A are different, we expect that there is also a nontrivial difference2 V ol.

is defined by the functional integration over the gauge parameter ǫ with the norm ∥ǫ∥2 =Rd2xǫ+ǫ−.3

between the two determinants in our case.3Assuming that the difference of the twodeterminants changes eq. (12) into the form,Z =Z[dφ] exp{−k′πZd2x(∂φ¯∂φ −Qφ∂¯∂σ −µ′e2φ+σ)},(13)where the spacetime metric is ds2 = eσdzd¯z, we can determine the values of k′ and Q `a laDDK[10] as k′ = k −2, Q = k−1k−2.

The Virasoro central charge of the Liouville theory isc =3kk−2 + 6k −2. This is exactly the relation between the level of the SL(2, R) currentalgebra and the Virasoro central charge in [2].In this way, the authors of [8][9] showed that the partition function of the gauged WZWmodel coincides with that of Liouville theory.

Here, the following two remarks are in order.1) In the Gauss decomposition eq. (5) of g ∈SL(2, R), v and ¯v are real.Thereforethe system of bosons v and ¯v involves a negative signature kinetic term and the normeq.

(10) is not positive definite. The origin of such a negative kinetic term and norm is thenoncompactness of SL(2, R) .

Accordingly the norms of A’s and the gauge parameter ǫ failto be positive definite. Therefore, strictly speaking, the functional integral over v’s and A’sdiscussed above is not well defined.

One way to make the functional integral over v and ¯vwell defined is to continue v and ¯v so that the norm and the kinetic term become positivedefinite. If k > 0 ( which we assume in the following ), this amounts to regarding v and¯v ( and accordingly A+z and A−¯z ) as complex conjugate to each other.

As is shown in theappendix, the action eq. (7) with v and ¯v complex conjugate to each other, can be consideredas a gauged version of SL(2,C)/SU(2) model [11].

Therefore, strictly speaking, we shoulddo the analytic continuation in order to make the above calculation rigorous. The analyticcontinuation seems to be legitimate, if what we are dealing with is the field theory of φ, vand ¯v.

However, considering that we are dealing with an SL(2, R) gauged WZW model,this analytic continuation seems subtle, because the negative kinetic term of v’s stems fromthe noncompactness of SL(2, R) which is an essential feature of the group SL(2, R) .2) In the usual SL(2, R) WZW model ( or SL(2,C)/SU(2) model ), the variables v,¯v and φ are all scalars and A is a vector field. However, because of the presence of theterms proportional to √µ in eq.

(7), the action is not even rotationally invariant under suchspin assignments. In order to make the theory conformally invariant, we should change theassignments.

We have to take the left and right conformal weights of φ, v and ¯v so thatthe currents e−2φ∂¯v, e−2φ ¯∂v have the left and right conformal weights (0, 0). This can beachieved by “twisting” the model.

Namely, as was done in [3][4][5], we add to the stresstensor a derivative of the zeroth component of the chiral SL(2, R) current:T ′zz = Tzz + ∂J0z , T ′¯z¯z = T¯z¯z + ¯∂J0¯z . (14)This corresponds to shifting the conformal weights of φ, v and ¯v to (0, 0), (1, 0) and (0, 1)respectively.

The conformal weights of the A’s are taken to be (1, 1). Accordingly we should3 In [8], such a difference was intentionally neglected.4

modify the action asI=kπZd2x{∂φ¯∂φ −φ∂¯∂σ + e−2φ−σ ¯∂v∂¯v+A+z (e−2φ−σ ¯∂v −√µ) + A−¯z (e−2φ−σ∂¯v −√µ)+e−2φ−σA+z A−¯z }. (15)In [8][9], such a twisting was not mentioned at the stage of considering the gauged WZWaction eq.(3).

However, we should start the discussion from this twisted action in order todefine the gauged WZW model to be rotationally invariant. This modified action dependsexplicitly on the conformal factor σ of the metric.

However it is invariant under the Weyltransformation σ →σ + ǫ, φ →φ −12ǫ as in the case of a Feigin-Fuchs boson.The arguments above show that the analysis of [8][9] described in the first part of thissection is somewhat formal. In order to make it more rigorous, we should start from theaction eq.

(15), with v and ¯v complex conjugate to each other, instead of eq.(7). Howeverthe analytic continuation does not seem to be legitimate, considering that we are dealingwith the SL(2, R) WZW model.

Therefore we would rather propose this ( twisted ) gaugedSL(2, C)/SU(2) model eq. (15) as a new model related to the SL(2, R) gauged WZW model.In this model, the analysis of [8][9] goes through more rigorously.

Here, instead, we will usean alternative method to deduce Liouville theory starting from this gauged SL(2,C)/SU(2)model eq.(15). In our method, we can derive eq.

(13) directly without any assumption, andit gives a more rigorous derivation of Liouville theory from the gauged SL(2,C)/SU(2) modeleq. (15).Let us consider the partition function of the gauged SL(2,C)/SU(2) model 4Z =Z [dφdvdA]V ol.e−I.

(16)Now I is the action in eq. (15) and v and ¯v are complex conjugate to each other .

V ol. denotesthe volume of the gauge transformation eq.

(8), with ǫ+ and ǫ−being complex conjugate toeach other. We will integrate out v, ¯v and A in eq.

(16) and obtain Liouville theory. Afterthe twisting mentioned above, the functional integration measures for these variables aredefined by the norm∥δv∥2=Zd2xe−2φδvδ¯v,∥δA∥2=Zd2xe−σδA+z δA−¯z .

(17)In order to integrate out v and A, one should somehow take care of the gauge invariance.Essentially, what we will do here is to fix the gauge as A+z = A−¯z = 0. The gauge fixed action4For notational simplicity, we will discuss this model on the sphere with the conformal metric ds2 =eσdzd¯z.5

then becomesI=kπZd2x{∂φ¯∂φ −φ∂¯∂σ + e−2φ−σ ¯∂v∂¯v}+ 1πZd2x(b¯∂c + ¯b∂¯c),(18)and the theory becomes a system consisting of the twisted SL(2,C)/SU(2) model and ghosts.In this form, our model is solvable using the current algebra technique. However there is onething one has to notice with such a gauge choice.

One cannot choose such a gauge globallyon a compact Riemann surface. Indeed, by expanding the A’s in terms of the eigenfunctionsof the Laplacian on the surface, one can show that A’s can be decomposed asA+z=∂¯Λ + a0eσ,A−¯z=¯∂Λ + ¯a0eσ.

(19)Here Λ and ¯Λ are (1, 0) and (0, 1) forms respectively and a0 and ¯a0 are constants. Thesecond terms in eq.

(19) cannot be gauged away. Therefore the gauge eq.

(18) is possible onlylocally. Of course, such global obstructions do not matter when one is canonically quantizingthe system and calculating commutation relations of operators.

Therefore, quantities suchas anomalous dimensions of operators can be reliably computed using the current algebratechnique available in the gauge eq.(18). In order to derive Liouville theory, we will constructan alternative form of the action depending explicitly on the moduli a0.Let us change variables from A to Λ, ¯Λ, a0 and ¯a0 in the functional integration eq.

(16).The partition function becomesZ =Z [dφdvdΛda0]V ol.det′(∆)e−I. (20)The integration measure for Λ and a0 are defined by the norm∥δΛ∥2=Zd2xδΛδ¯Λ,∥δa0∥2=Zd2xeσδa0δ¯a0.

(21)∆denotes the Laplacian −¯∂e−σ∂on (0, 1) forms. det′∆is the Jacobian for the change ofvariables A →Λ, ¯Λ.

The action I is written asI=kπZd2x{∂φ¯∂φ −φ∂¯∂σ + e−2φ−σ ¯∂(v + Λ)∂(¯v + ¯Λ)+a0(e−2φ ¯∂(v + Λ) −√µeσ) + ¯a0(e−2φ∂(¯v + ¯Λ) −√µeσ)+e−2φ+σa0¯a0}. (22)Since the functional integration measure for v defined by eq.

(17) is invariant under thetransformation v →v + Λ, ¯v →¯v + ¯Λ, we can factorize the Λ integration in eq. (20), which6

cancels the gauge volume V ol..Eventually, one obtains the following expression of thepartition functionZ =Z[dφdvda0]det′(∆)e−If,(23)If=kπZd2x{∂φ¯∂φ −φ∂¯∂σ + e−2φ−σ ¯∂v∂¯v+a0(e−2φ ¯∂v −√µeσ) + ¯a0(e−2φ∂¯v −√µeσ)+e−2φ+σa0¯a0}. (24)det′(∆) can be expressed by a ghost c ( (1, 0) form ) and an antighost b ( (0, 0) form ) andtheir complex conjugates as usual.

The sum of If and the ghost action Igh = 1πR d2x(b¯∂c +¯b∂¯c) gives us a gauge fixed action with explicit moduli dependence. Locally it is possible togauge away the moduli a0 in If to obtain eq.(18).

We can construct the BRST chargesQ=IdzJBRST =Idzc(J+z −k√µ)¯Q=Id¯z ¯JBRST =Id¯z¯c(J−¯z −k√µ),(25)whereJ+z=k(e−2φ−σ∂¯v + a0e−2φ),J−¯z=k(e−2φ−σ ¯∂v + ¯a0e−2φ). (26)The functional integration over v, a0 and ghosts in eq.

(23) will be done using the followingtrick. Let us further decompose the moduli a0eσ and ¯a0eσ asa0eσ=∂¯f + f0e2φ+σ,¯a0eσ=¯∂f + ¯f0e2φ+σ.

(27)This can be done by considering the nondegenerate bilinear form∥ω∥2 =Zd2xe−2φ−σω¯ω,(28)on (1, 1) forms and a Laplacian −¯∂∂e−2φ−σ, which is self-adjoint with respect to this bilinearform. Eqs.

(27) amount to the orthogonal decomposition of the (1, 1) form eσ into the zeromode and nonzero modes of this Laplacian and its complex conjugate. Nonzero mode partsare written as derivatives of a (1, 0) form f and (0, 1) form ¯f.

The coefficients of the zeromode e2φ+σ aref0 = a0R d2xeσR d2xe2φ+σ , ¯f0 = ¯a0R d2xeσR d2xe2φ+σ . (29)7

Inserting this decomposition into eq. (24), we obtainIf=kπZd2x{∂φ¯∂φ −φ∂¯∂σ + e−2φ−σ ¯∂v′∂¯v′ −µe2φ+σ}+kπ(R d2xeσ)2R d2xe2φ+σ a′0¯a′0,(30)wherev′ = v + f,a′0 = a0 −√µR d2xe2φ+σR d2xeσ,¯v′ = ¯v + ¯f,¯a′0 = ¯a0 −√µR d2xe2φ+σR d2xeσ.

(31)A good thing about this form of the action is that ¯a′0 and v′ decouple from each other.We can do the integration over v and a0 separately. The a0 integration is just a simpleGaussian integrationZ[da0] exp{−kπ(R d2xeσ)2R d2xe2φ+σ a′0¯a′0} = const.

×R d2xe2φ+σR d2xeσ. (32)The integration over v can be evaluated by the standard anomaly calculation [11]:Z[dv] exp{−kπZd2xe−2φ−σ∂¯v′ ¯∂v′}=R d2xeσR d2xe2φ+σ det′(∆)−1 exp{ 2πZd2x∂φ¯∂φ −1πZd2xφ∂¯∂σ}.

(33)Putting all these together, we obtainZ=Z[dφdvda0]det′(∆)e−If=const.Z[dφ]e−ILiou.,ILiou. = k −2πZd2x∂φ¯∂φ −k −1πZd2xφ∂¯∂σ + µ′Zd2xe2φ+σ.(34)Eq.

(34) is the partition function of Liouville theory with the cosmological constant µ′ = −kµπ .Thus, the partition function of the SL(2, C)/SU(2) model coincides with that of Liouvilletheory.We would like to conclude this section by several comments.For k > 0, µ should be negative for the functional integral to be well defined. Thisimplies that the action in eq.

(3) has an imaginary part proportional to õ. It does notcause a serious problem in our analysis, because eventually the õ term is relevant only inthe gaussian integration eq.

(32).8

The Liouville theory obtained in eq. (34) is not always relevant to two dimensional quan-tum gravity.

In two dimensional quantum gravity the cosmological constant µ′ should becoupled to the lowest dimensional operator in the matter theory dressed by gravity. There-fore, eq.

(34) is relevant to quantum gravity for only special values of k.It is intriguing to observe that in the above derivation, the existence of the moduli a0 isessential to generate the cosmological termR d2xe2φ+σ. This moduli also play an essentialrole in the next section.3Extra ObservablesIn this section, we would like to discuss the observables and their correlation functionsin the gauged SL(2, C)/SU(2) model proposed in the previous section.In the light ofits relation to Liouville theory, it is natural to expect that every observable in this modelcorresponds to an operator in Liouville theory.

However we will find that there exist infinitelymany extra observables which do not correspond to Liouville theory operators.Let us consider the gauged SL(2, C)/SU(2) model in the gauge fixed form eq.(24). Theobservables in this gauge are determined by the usual BRST procedure using the BRSTcharges in eq.(25).

The only singular operator product expansions of J+z and J−¯z with φ, v,and ¯v are given byJ+z (z)v(w, ¯w)∼1z −w,J−¯z (¯z)¯v(w, ¯w)∼1¯z −¯w. (35)Therefore, operators made out of only φ are BRST closed.

It can be proved that correlationfunctions of such operators Vi ( i = 1, · · · , n ) in the gauged SL(2, C)/SU(2) model reduceto correlation functions in Liouville theory< V1 · · · Vn >=Z[dφdvda0]det′(∆)e−IfV1 · · · Vn=const.Z[dφ]e−ILiou.V1 · · · Vn,(36)following the same procedure in the previous section.The operators of the form e−2lφ are important in the application of Liouville theory totwo dimensional quantum gravity. In the gauged SL(2, C)/SU(2) model such operators arehighest weight operators of the SL(2, R) current algebra:J+z (z)e−2lφ(w)∼regular,J0z (z)e−2lφ(w)∼lz −we−2lφ.

(37)9

The conformal weights of such highest weight operators are −l(l+1)k−2 −l 5 which of coursecoincide with the values evaluated in Liouville theory. When l = −1, the conformal weightis (1, 1), which is consistent with the fact that this operator corresponds to the volumeelement in quantum gravity.Therefore, every operator in Liouville theory can be represented as a BRST invariantoperator in the gauged SL(2, C)/SU(2) model.

If such an operator is a null observable in thegauged SL(2, C)/SU(2) model, it will decouple from the other operators in Liouville theoryas can be seen in eq.(36). Hence, if all the observables in the gauged SL(2, C)/SU(2) modelare made out of φ, we can have a complete correspondence between the nontrivial operatorsin Liouville theory and the observables in the gauged SL(2, C)/SU(2) model.

However,there exist observables which consist not only of φ but also of other fields in the gaugedSL(2, C)/SU(2) model.We can construct such extra observables starting from the following observation. Theghost fields b, c in the gauge fixed action are used to express the determinant det′(∆) ( onthe sphere, for example, )det′(∆) =Z[dbdc]b¯b(z0)e−Igh.

(38)Here, a pair of antighosts is inserted to soak up the zero mode in the ghost path integral.The insertion point z0 can be taken arbitrarily. Since the antighosts are (0, 0) forms, theyhave one zero mode on a surface of any genus.

Such a zero mode does not appear in theaction Igh or in the BRST charge eq.(25). Therefore we eliminate it from the theory byinserting b¯b as above.

This situation is analogous to the treatment of the ξ zero mode ofthe superghost bosonization in superstring theory[12]. This leads an analogue of “picturechanging”O −→{Q, bO},(39)for the left moving sector along with the right moving one.By this operation, we canconstruct a new observable {Q, bO} from an observable O, if O contains no c. The newobservable {Q, bO} is in the form of a BRST exact operator.

However since it is an anti-commutator of BRST operator with an operator including the antighost zero mode, it doesnot necessarily decouple from the other observables as in the case of the picture changingin superstring theory.Notice that this picture changing operation does not change the conformal weight of theobservable, because b is a (0, 0) form field and Q commutes with the Virasoro operators.Therefore we obtain a new observable with the same conformal weight by this operation.In superstring theory, the picture changing operation generates infinitely many equivalentexpressions of one observable. However, as we will see, in our case the picture changingoperation generates an infinite number of distinct observables.

By applying this picturechanging operation to the observables like e−2lφ, we can obtain infinitely many observableswhich contain not only φ but also v’s and a0’s. We are not sure if the observables constructedin such a way exhaust the observables of our model.5 To be precise, this suggests that e−l(2φ+σ) is a (−l(l+1)k−2 −l, −l(l+1)k−2 −l) form.10

Let us consider the correlation functions of such observables on the sphere:< V1 · · · VN >=Z[dφdvda0dbdc]e−If −Ighb¯b(z0)V1 · · · VN(40)Such correlation functions can be calculated as follows. As in the previous section, it isconvenient to rewrite everything in terms of a′0 and v′ in eq.(31).

While a′0 and v′ decouplefrom each other in the action,If = kπZd2x{∂φ¯∂φ −φ∂¯∂σ + e−2φ−σ ¯∂v′∂¯v′ −µe2φ+σ} + kπ(R d2xeσ)2R d2xe2φ+σ a′0¯a′0,(41)there appears the nontrivial interaction termR d2xe2φ+σ. This term can be taken care of bythe method employed in [13][14], namely, by integrating over the φ zero mode first.

Sincethe φ zero mode is coupled to v′ and a′0 in our model, we will proceed as follows. Let usintroduce a spacetime independent integration variable φ0 and couple it to If asI′f = kπZd2x{∂φ¯∂φ −(φ + φ0)∂¯∂σ + e−2φ−σ ¯∂v′∂¯v′ −µe2φ+2φ0+σ} + kπ(R d2xeσ)2R d2xe2φ+σ a′0¯a′0 −2φ0.

(42)The transformation δφ = ǫ, δv = ǫv, δ¯v = ǫ¯v, δa′0 = ǫa′0, δ¯a′0 = ǫ¯a′0, δφ0 = −ǫ leaves[dφdvda0dφ0]e−I′f(43)invariant. Suppose there exists αi such that e−2αiφ0Vi is invariant under the above transfor-mation.

Then the correlation function can be written as< V1 · · · VN >=Z [dφdvda0dφ0dbdc]Ve−I′f −Ighb¯b(z0)Yie−2αiφ0Vi,(44)where V is the volume of the above continuous symmetry of the path integral. This formulais easily proved, if one fixes the symmetry by setting φ0 = 0.

Here we will fix the symmetryby the conditionR d2xeσφ = 0, which kills the zero mode of φ. φ0 plays the role of theφ zero mode. Then, after integrating over φ0, one obtains the following expression for thecorrelation function< V1 · · · VN >=const.

× (µ′)k−1+ΣαiΓ(−(k −1 + Σαi))×Z[dφdvda0dbdc]δ(Zeσφ)e−I0−Ighb¯b(z0)V1 · · · VN(Zd2xe2φ+σ)k−1+Σαi,(45)whereI0 = kπZd2x{∂φ¯∂φ −φ∂¯∂σ + e−2φ−σ ¯∂v′∂¯v′} + kπ(R d2xeσ)2R d2xe2φ+σ a′0¯a′0. (46)11

The αi are the analogs of the scaling dimensions in Liouville theory. SinceQ=kIc(e−2φ−σ∂¯v′ +R d2xeσR d2xe2φ+σ a′0)¯Q=kI¯c(e−2φ−σ ¯∂v′ +R d2xeσR d2xe2φ+σ ¯a′0),(47)the picture changing operation eq.

(39) changes the scaling dimension α by −12. The scalingdimension is one of the physical quantum numbers in our theory, on which the correlationfunctions crucially depend.

Therefore the picture changing operation produces an infinitenumber of distinct observables.If k−1+Σαi is not a positive integer, eq. (45) is not well defined.

One needs the analyticcontinuation as was done in [14] to define it.Since we are not sure if there exist anyjustifications for that in our case, we will restrict ourselves to the case when k −1+Σαi is apositive integer. If k−1+Σαi is a positive integer, the factor Γ(−(k−1+Σαi)) is divergent.This divergence comes from the volume of φ0.

We will replace (µ′)k−1+ΣαiΓ(−(k −1+Σαi))by (−µ′)k−1+Σαi(k−1+Σαi)! log 1µ′ and interpret log 1µ′ as the volume of φ0 as was suggested in [15].In principle one can compute any correlation function of observables by performing thefunctional integral in eq.

(45), which amounts to successive Gaussian integrals. Here we willconcentrate on the following extra observables.

Starting from O0 = e2φ+σ, let us define Oninductively asOn+1(w, ¯w)=1kπ{ ¯Q, [Q, b¯bOn(w, ¯w)]}=1kπI¯w d¯z ¯JBRSTIw dzJBRST b¯bOn(w, ¯w). (48)On ∝(J+z −k√µ)n(J−¯z −k√µ)ne2φ+σ and On does not contain c. Therefore eq.

(48) is a welldefined picture changing operation. Since the conformal weight of O0 = e2φ+σ is (1, 1), allof the On are (1, 1) operators.

In the rest of this section, we will show that it is possible tocompute explicitly correlation functions of σn =R d2xOn,< σn1 · · · σnN >=Z[dφdvda0dbdc]e−If −Ighb¯b(z0)Zd2xOn1 · · ·Zd2xOnN. (49)One should drop the integrals of three of On’s in order to fix the SL(2, C) invariance in theabove correlation function.

We are interested in these observables, because σn are in a sense“descendants” ofR d2xe2φ+σ. The area of spacetimeR d2xe2φ+σ is one of the most importantobservables, in the application of Liouville theory to two dimensional gravity.

Also σn canbe added to the action as a marginal perturbation. Hence if one knows all the correlatorsof σn’s, one is able to solve such perturbed field theories exactly.Since O0 = e−2φ+σ has scaling dimension α = −1, On has α = n −1.

In order fork −1 + Σαi to be a positive integer, k should be an integer. For later convenience, we willrestrict k to be an integer satisfying k ≥4.12

Correlation functions of On’s have a remarkable property which originates from theirdefinition eq.(48). Namely they satisfy< On(x)Om(y) · · · >=< On−1(x)Om+1(y) · · · > (n > 0).

(50)The proof is given using the same argument as one uses for the demonstration of Bose seaequivalence in superstring theory[12]. Writing On(x) =1kπH ¯JBRSTH JBRST b¯bOn−1(x), theleft hand side of the above formula becomesZ[dφdvda0dbdc]e−If−Ighb¯b(z0) 1kπIx¯JBRSTIx JBRSTb¯bOn−1(x)Om(y) · · ·.

(51)Since the point z0 at which the antighost is inserted is arbitrary, we will take it to coincidewith y. Then by using the BRST invariance of the other observables, we move the integrationcontours of BRST currents so that they surround only y.

Thus we obtain the right handside of eq.(50).Eq. (50) is useful in reducing correlation functions of σn’s to a form in which they areeasily calculated.

Eq. (45) and the interpretation of the divergent gamma function suggestthat the following equation between the correlation functions holds,=(−µ′)k−1+Σ−l(k −1 + Σ −l)!

,(52)where Σ =P ni. Eq.

(50) implies=< (Zd2xO1)Σ(Zd2xO0)k−1 > . (53)Therefore the computation of correlation functions of σn’s is reduced to that of the correla-tion functions of σ1’s and σ0’s only.Now we are going to evaluate the right hand side of eq.(53).

We should fix the SL(2, C)invariance to define this correlation function. Suppose we fix the positions of three of theO0’s.

This is possible, since k ≥4. O1 has the form1kπId¯z ¯JBRSTIdzJBRST e2φ+σ = 1π{ ¯Q,¯b(∂¯v′ +R d2xeσR d2xe2φ+σ a′0e2φ+σ)}.

(54)HenceR d2xO1 is written asZd2xO1 = 1π{ ¯Q,Zd2x¯b∂¯v′} + kπ(R d2xeσ)2R d2xe2φ+σ a′0¯a′0. (55)The first term is a commutator of the BRST operator and an operator which does notcontain the antighost zero mode.

It decouples from the other observables in the correlationfunction. Hence the right hand side of eq.

(53) is equal to< (kπ(R d2xeσ)2R d2xe2φ+σ a′0¯a′0)Σ(Zd2xe2φ+σ)k−1 > . (56)13

It is annoying to haveR d2xe2φ+σ in the denominator, but they all disappear after the a0integration. The integration over a0 is easily done.

We find< (kπ(R d2xeσ)2R d2xe2φ+σ a′0¯a′0)Σ(Zd2xe2φ+σ)k−1 >= Σ! < (Zd2xe2φ+σ)k−1 > .

(57)Using eq. (52), one finally obtains< σn1 · · · σnl >= (−µ′)Σ−lΣ!

(k −1)! (k −1 + Σ −l)!Z.

(58)Since the σn’s can be added to the original action as a perturbation, this result makes itpossible to calculate various quantities exactly in such a perturbed theory.The k = 4 case is extremely interesting. Since Z ∝(−µ′)3 log 1µ′,< σn1 · · · σnl >∼(−µ′)3+Σ−lΣ!

(3 + Σ −l)! .

log 1µ′(59)Up to a constant and the Liouville volume log 1µ′, these correlation functions precisely co-incide with the correlation functions of the first critical point of one matrix model on thesphere [16].This coincidence is very suggestive. k = 4 is exactly the point that our SL(2, C)/SU(2)model realizes the Liouville theory which is induced by the matter theory with c = −2.The first critical point of the one matrix model is supposed to correspond to the quantumgravity coupled to c = −2 matter theory.

k = 4 is the only integer greater than three, forwhich the corresponding Liouville theory is relevant to two dimensional quantum gravity.For other values of k, µ′ does not couple toR d2xe2φ+σ in two dimensional quantum gravity.Unfortunately it seems that such coincidence does not exist for the correlators ofR d2xOnon higher genus surfaces.In general, matrix model correlators are not compatible witheq. (50).4Conclusions and DiscussionsIn this paper, we have studied the relationship between the gauged SL(2, C)/SU(2)model and Liouville theory.

One of the crucial points in our analysis is that there existsmoduli a0 on any compact Riemann surfaces. This is because the gauge field in our modelis a (1, 1) form.

The existence of the moduli a0 is essential to obtain the cosmological termin Liouville theory. Then we investigated the observables in the gauged SL(2, C)/SU(2)model.

Although the partition function of this model coincides with that of Liouville theory,the gauged SL(2, C)/SU(2) model possesses more observables than Liouville theory. Theexistence of the zero modes of antighosts ( which is of course deeply related to the existenceof a0, ) was important in the construction of such observables.

We have calculated the14

correlators of some of these extra observables. They look quite similar to correlators in thematrix models and coincide with them when k = 4.We are not sure if our results have any implications for two dimensional quantum gravity.The extra observables studied in section 3 do not exist in Liouville theory.And theircorrelators do not appear to coincide with the matrix model results on higher genus surfaces.Still it is possible to conceive that some modified version of the gauged SL(2, C)/SU(2)model would reproduce the matrix model results completely and elucidate the importanceof the SL(2, R) structure in two dimensional quantum gravity.It is straightforward to extend our analysis to the case of more general constrained WZWmodels.

For example, SL(N, R) WZW model with constraints similar to SL(2, R) case wasshown to be relevant to Toda field theories[5]. Gauged WZW models based on certain superLie groups yield super Toda field theories[17].In both of these cases, there exist (1, 1)gauge fields, and (0, 0) antighosts when one fixes the gauge.

It is possible to consider theconstruction of observables by the “picture changing operation” using these antighost fields.The generalizations to these cases will be reported elsewhere.5AppendixIn this appendix we will give the definition and some useful properties and formulas ofthe SL(2, C)/SU(2) model and its gauged version.The SL(2, C)/SU(2) model[11] is defined by the actionI = kSW ZW(gg†),(60)where g ∈SL(2, C) and SW ZW is the WZW action eq.(4). This model describes the in-duced gauge theory which is obtained by integrating the matter part in G/H gauged WZWmodel [11], when H is SU(2).

The gauge transformation of such an induced gauge theorycorresponds tog −→gh, h ∈SU(2). (61)In order to define the functional integral, we should fix this invariance.

This can be donemost conveniently by taking g to beg = 1v01! eφ200e−φ2,(62)andgg† = 1v01!

eφ00e−φ! 10¯v0!.

(63)Here φ is real, v is complex, and ¯v is the complex conjugate of v. Eq.

(63) is similar to theGauss decomposition eq.(5). Contrary to the Gauss decomposition eq.

(5), however, gg† for15

any g ∈SL(2, C) can be represented as in eq.(63). Inserting this parametrization of g intoeq.

(60), one obtains the action,I=kπZd2x{∂φ¯∂φ + e−2φ ¯∂v∂¯v},(64)with v and ¯v complex conjugate to each other.The SL(2, C)/SU(2) model is exactlydescribed by this action.This theory possesses SL(2) chiral currents:J+z = k√2e−2φ∂¯vJ0z = k(∂φ + e−2φv∂¯v)J−z = k√2(∂v −2∂φv −e−2φv2∂¯v),andJ−¯z = k√2e−2φ ¯∂vJ0¯z = k(¯∂φ + e−2φ¯v ¯∂v)J+¯z = k√2(¯∂¯v −2¯∂φ¯v −e−2φ¯v2 ¯∂v). (65)These currents correspond to the transformationg −→hg, h ∈SL(2, C),(66)and satisfy the left and right SL(2) current algebras.

They are important in constructingthe BRST charge in G/SU(2) gauged WZW models. The form of the currents in terms of φ,v and ¯v are the same as the chiral SL(2) currents in the SL(2, R) WZW model.

However,since v and ¯v are complex conjugate to each other, the left and right currents are relatedvia complex conjugation in a different way in this model. The stress tensor can be writtenin terms of these SL(2) currents in the Sugawara form.We will define the functional integral measure for φ and v by the normZd2xeσ{(δφ)2 + e−2φδvδ¯v},(67)which is invariant under eq.(66).

If one performs the v integration first in the φ backgroundand then do the φ integration, the functional integral 6Z[dφdv]e−I,(68)6Since this system has a global symmetry g →hg, one should divide by the volume of such a globalsymmetry to define this functional integral.16

is evaluated essentially by successive Gaussian integrals.The v integration in the φ background yields the following partition function [11]Z[dv] exp{−kπZd2xe−2φ∂¯v ¯∂v} = (det′(∆)R d2xeσ )−1, exp{ 2πZd2x∂φ¯∂φ + 1πZd2xφ∂¯∂σ}. (69)where the measure [dv] is defined by the norm ∥δv∥2 =R d2xe−2φ+σδvδ¯v.Correlationfunctions of v’s are calculable using Wick’s theorem.

After the v integration, the functionalintegral eq. (68) becomes a theory of boson φ with Feigin-Fuchs type action.Hence, inprinciple, one can calculate any correlation function in this model.

Indeed, computation ofsome of the correlation functions in this model in this way was done in [11], and the resultswere consistent with the Knizhnik-Zamolodchikov equations derived from the SL(2) currentalgebras satisfied by eq. (65).We can gauge SL(2, C)/SU(2) model without any problem in a similar way to theSL(2, R) case in eq.(3).

We obtain exactly the action eq. (7) with v and ¯v complex conjugateto each other.

Also one can twist the SL(2, C)/SU(2) model as was done in section two. Inthis case, the action should be modified as followsI = kπZd2x{∂φ¯∂φ −φ∂¯∂σ + e−2φ−σ ¯∂v∂¯v},(70)where now v ( ¯v ) is a (1, 0) ( (0, 1) ) form.

The SL(2) currents in eq. (65) with a littlemodification ( changing all φ’s in the expression to φ + 12σ ) still satisfy the SL(2) KacMoody algebra, after the twisting.

The Virasoro generators are written in terms of thesecurrents as in eq.(14). The twisted SL(2, C)/SU(2) model is also solvable by successivefunctional Gaussian integration in the same way as in the untwisted case.AcknowledgementsIt is a pleasure to thank J. Horne, M. Li, S. Nojiri and A. Steif for useful conversations andcomments.

This work is supported by the Nishina Memorial Foundation and NSF GrantNo. PHY86-14185.References[1] A. M. Polyakov, Mod.

Phys. Lett.

A2 (1987) 893. [2] V. Knizhnik, A. M. Polyakov and A.

B. Zamolodchikov, Mod. Phys.

Lett. A3 (1988)819.

[3] A. Alekseev and S. Shatashvili, Nucl. Phys.

B323 (1989) 719. [4] M. Bershadskii and H. Ooguri, Comm.

Math. Phys.

126 (1989) 49.17

[5] J. Balog, L. Feher, P. Forgacs, L. O’Raifeartaigh and A. Wipf, Phys. Lett.

B 227(1990) 214; bf B 244 (1990) 435; Ann. Phys.

203 (1990) 76. [6] A.M.Polyakov, Int.J.Mod.Phys.A5(1990) 833[7] H.Verlinde, Nucl.Phys.B337(1990) 652[8] L. O’Raifeartaigh, P. Ruelle and I. Tsutsui, Phys.

Lett. B258 (1991) 359.

[9] R. Dijkgraaf, H. Verlinde and E. Verlinde, preprint PUPT-1252. [10] F. David, Mod.

Phys. Lett.

A3 (1988) 1615 ; J. Distler and H. Kawai, Nucl. Phys.B321 (1989) 509.

[11] K. Gawedzki and A. Kupiainen, Nucl. Phys.

B 320 (1989) 625. [12] D. Friedan, E. Martinec and S. Shenker, Nucl.

Phys. B271 (1986) 93.

[13] A. Gupta, S.P. Trivedi and M.B.

Wise, Nucl. Phys.

B340 (1990) 475. [14] M. Goulian and M. Li, Phys.

Rev. Lett.

66 (1991) 2051. [15] P. Di Francesco and D. Kutasov, Phys.

Lett. B261 (1991) 385; Princeton preprintPUPT-1276.

[16] E. Br´ezin and V. Kazakov, Phys. Lett.

B236 (1990) 144;M. Douglas and S. Shenker, Nucl. Phys.

B355 (1990) 635;D. J. Gross and A.

A. Migdal, Phys. Rev.

Lett. 64 (1990) 127.

[17] T. Inami and K.-I. Izawa, Phys.

Lett. B255 (1991) 521.18

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