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SOME CORRELATION FUNCTIONS

arXiv:hep-th/9109037v2 24 Sep 1991CERN-TH.6243/91September, 1991SOME CORRELATION FUNCTIONSOF MINIMAL SUPERCONFORMALMODELS COUPLED TO SUPERGRAVITYL. Alvarez-Gaum´eTheory Division, CERNCH-1211 Geneva 23, SwitzerlandandPh.

ZauggD´epartement de Physique Th´eoriqueUniversit´e de Gen`eveCH-1211 Gen`eve 4, SwitzerlandABSTRACTWe compute general three-point functions of minimal superconformal modelscoupled to supergravity in the Neveu-Schwarz sector for spherical topology thusextending to the superconformal case the results of Goulian and Li and of Dotsenko.⋆Partially supported by the Swiss National Science Foundation.

1. Introduction.

The successes of the double scaling limit [1] and its connectionwith the KP-hierarchy [2] in the computation of correlation functions of minimalmodels coupled to 2D-gravity (see for example [3]) prompted a good deal of activityin trying to reproduce the same results directly in the continuum limit (for detailsand references see [4,5]).The original approach to the coupling of Conformal Field Theories (CFT) togravity appeared in [6,7]. Using the light-cone gauge these authors were able toexactly compute the gravitational dimensions of the gravitationally dressed pri-mary fields.These results were obtained subsequently in the conformal gauge[8,9] which also allowed the generalization of some of the results to non-sphericaltopologies.

Several methods have been suggested for the computation of correla-tion functions in Liouville theory coupled to minimal conformal models [10,11] inthe continuum limit in order to reproduce the results of matrix models. The firstproposal consisted of an analytic continuation in the value of the central chargeof the Virasoro algebra [12].

This technique was further explored in [13,14]. Thesecond proposal, closely related to the previous one, was a generalization of theCoulomg gas technique [15] to the Liouville case in order to include negative num-bers of screening charges [16].

So far these techniques have only allowed a directcomputation of generic one-, two,- and three-point functions and there is no clearprocedure known on how to extend the same ideas to higher point functions exceptin some special cases. The second technique was used in [17] to clarify some issuesconcerning the fusion rules in the presence of 2D-gravity.The basic technical problem in most of these computations is the evaluationof some integrals which also appear in the computation of the structure constantsof the minimal conformal models which was done in [15].

In the case of N = 1Superconformal Fields Theories [18,11,19] although there is available a Coulombgas formulation [20], there are only partial results with respect to the structureconstants of the operator algebra (see for example [21, 22, 23]). The generalizationof the results in [15] to the minimal N = 1 models in the Neveu-Schwarz sectorhas been carried out in [24].

As a simple application of the results in this paper we1

can calculate the one-, two- and three-point functions of N = 1 minimal modelscoupled to 2D-supergravity on surfaces of spherical topology. The derivation of thesupergravitational dressing and dimensions was carried out in the light-cone gaugein [25] and in the superconformal gauge in [26].

We follow the proposal in [16],although one could equally well extend the ideas in [12]. These computations havesome interest because there is as yet no analogue of the matrix model formulationfor superconformal theories coupled to supergravity (see nevertheless [27]).2.

Formulation of the problem. In the Coulomb gas formulation of the minimalsuperconformal models [20] the matter energy-momentum tensor is built in termsof a free massless scalar superfield X(Z)TM(Z) = −12 : DX∂X : + i2α0D∂X(1)where Z = (z, θ) represents a point on the superplane.

It is convenient to introducetwo quantities α+, α−satisfying α+ + α−= α0α+α−= −1. Then the centralcharge and the screening charges take the formˆc = 1 −2α20J± = eiα±X(Z)(2)The background charge α0 changes the dimension of a super-vertex operator eiαXfrom α2/2 to α(α −α0)/2.

The minimal superconformal models are obtained forspecial values of α0, α+, α−. Take two integers p′, p; p′ > p such that 1) p′ −p ≡0(mod 2), 2) if they are odd, then they are coprime, or 3) if both are even thenp′/2, p/2 are coprime.

In the N = 1 minimal models we haveα+ =sp′pα−= −r pp′ˆc = 1 −2(p′ −p)2pp′(3)The primary fields are represented as vertex operators in the NS sectorΨm′,m = eiαm′,mXm′ −m ≡0(mod 2)2

αm′,m = 12(1 −m′)α−+ 12(1 −m)α+(4)and in the Ramond sector byΨm′,m = σeiαm′,mXm′ −m ≡1(mod2)(5)Where σ is a spin field of the matter sector. The superconformal dimensions arehm′,m =18pp′[(mp′ −m′p)2 −(p′ −p)2] + 132(1 −(−1)m′−m)(6)We will be exclusively concerned with correlators of NS fields in this paper.

Therange of m′, m is 1 ≤m′ ≤p′ −1, 1 ≤m ≤p −1,mp′ −m′p ≥0.In the super-Liouville sector, after gauge fixing in the superconformal gauge [26]we can describe the Liouville part of the theory locally with an energy-momentumtensor expressible in terms of a real scalar superfield Φ :TL(Z) = −12DΦ∂Φ + Q2 D∂Φ(7)The superconformal dimension of a vertex operator in the NS sector eβΦ is−β(β −Q)/2.The Liouville background charge is determined by α0 and theghost contributions to beQ2 = 9 −ˆc2= 4 + α20(8)The dressing of a NS field eiαX is given byeiαXeβΦ,12α(α −α0) −12β(β −Q) = 12(9)making the dressed field into a (1/2, 1/2)-form which can be integrated over thesurface without any reference to the conformal factor of the background metric.3

The two solutions to the quadratic equation in (9) areβ = Q2 ± |α −α02 |(10)The microscopic branch (see [4] for details) is obtained by choosing the minus sign.With this sign the superconformal dimension agrees with the classical dimensionin the classical limit ˆc →−∞. For (4) we obtainβm′,m = p + p′ −|mp′ −m′p|2√pp′(11)As with the conformal case we can introduce two quantities, β± satisfyingβ+ + β−= Qβ+β−= 1(12)They are related to α+, α−byα+ = β+α−= −β−(13)It is useful to notice that in super-Liouville theory the cosmological constant cou-ples to the operators eβ−Φ.

Formally we can introduce two screening charges inthe Liouville sectorU± = eβ±Φ(14)If we are interested in general properties of n-point correlators we first note thatthe area constraint in the conformal case is replaced here by a ”length” constraintbecause we are counting volumes of N = 1 supersurfaces. For arbitrary values ofp′, p the field of lowest superconformal dimension is not necessarily the identity.LetΨmin be the field of lowest dimension, and let βmin be the correspondingdressing exponent.

The cosmological constant µ should be taken as the coefficientof the operator ΨmineβminΦ. If we concentrate for the time being on the super-Liouville contribution to the n-point function, we can obtain a useful expression4

for ⟨Qi eβiΦ(Zi)⟩by following step by step the arguments in the conformal case([12], see also [28],[5]). First we introduce a δ-function constraint fixing the lengthand integrate over lengths from 0 to ∞.

Second, we separate the constant piecefrom Φ(Zi) to explicitly solve the δ-function, Φ(Zi) = φ0 + ˜Φ(Zi). For a surface ofgenus h, the result is⟨YieβiΦ(Zi)⟩=1βminµsΓ(−s)⟨˜Ls Yieβi ˜Φ(Zi)⟩S(0)L (˜Φ)s = −1βmin(Xiβi −Q(1 −h))(15)and˜L =Zd2Z ˆEeβmin ˜ΦΨmin(16)ˆE is the reference zweibein and the integration over supermoduli parameters isnot explicitly exhibited.

The expectation value in (15) is computed in terms ofthe free super-Liouville action with a term representing the background chargeQ ˆR˜Φ ( ˆR is the curvature associated to the superframe ˆE; see [26] for details).Note parenthetically that we can read offfrom the µ-dependence in (15) the stringsusceptibility and the gravitational dimensions of the fields in the theory. If s werea positive integer we could easily evaluate (15) using free field techniques.

This isthe method proposed in [12]. One first computes the integral for s an integer andthen one analytically continues to arbitrary values.

This prescription could alsobe applied to our case. We choose however to follow [16].

The idea is to treat theLiouville and matter sectors in the same way using the Coulomb gas formulation.We should make a few remarks before we proceed. When the matter theory isunitary, the field of lowest dimension is the identity operator Ψmin = 1.

In thecomputation we will describe below for the general case, we take the cosmologicalconstant by definition to be the coefficient of the dressed identity operator. Inthe non-unitary case this corresponds to a fine tuning of the coupling of all the5

operators of negative dimension. The derivation of (15) goes through again, withthe only change that βmin is replaced by β−.

We can introduce vertex operatorssimilar to (4):Lm′,m = eβm′,mΦβm′,m = 1 −m′2β−+ 1 −m2β+(17)Since the dressing exponent of (4) isβ(αm′,m) = Q2 −12β−|m′ −ρm|ρ = β+β−we can represent β(αm′,m) in terms of βm′,m. There are two cases to distinguish:m′ > ρm,β(αm′,m) = Q2 −12β−(m′ −ρm) = βm′,−mm′ < ρm,β(αm′,m) = Q2 −12β−(−m′ + ρm) = β−m′,m(18)We can assign a chirality χ to a primary field Ψm′,m.

χ = 1 if m′ > ρm ; χ = −1if m′ < ρm. To guarantee the choice of the microscopic branch in (10) there aretwo possible dressingsA+m′,m = Ψ+m′,mLm′,−mχ = 1A−m′,m = Ψ−m′,mL−m′,mχ = −1(19)The dressed fields have dimensions (1/2, 1/2).

Ignoring for the moment chiralitylabels, we are interested in⟨ASANAM⟩=Z3Y1d2Zi ˆE⟨AS(Z3)AN(Z2)AM(Z1)⟩(20)using SL(2|1) invariance we can fix Z1 = (∞, ∞η), Z2 = (1, 0), Z3 = (0, 0). Di-viding out the SL(2|1) volume we are left with a single integration over the odd6

variable η:⟨ASANAM⟩=Zd2η⟨AS(0, 0)AN(1, 0)AM(∞, ∞η)⟩=Zd2η⟨ΨS(0, 0)ΨN(1, 0)ΨM(∞, ∞η)⟩⟨LS(0, 0)LN(1, 0)LM(∞, ∞η)⟩(21)The matter correlation function is evaluated using the physical structure constants[24]⟨ΨS(0, 0)ΨN(1, 0)ΨM(∞, ∞η)⟩= DevenSNM + DoddSNM|η|2(22)The contribution will come from D even or odd depending on the number ofscreening charges needed in the evaluation of (22) in the Coulomb gas formulation.For ⟨ASANAM⟩to be non-vanishing, the matter and Liouville parts must haveopposite Grassmann parity. This will be verified later by counting screenings inboth cases.

Using capital letters to label pairs of indices (M = (m′, m) ,etc), thephysical structure constants D are defined in terms of the symmetric structureconstants in the Coulomb gas definition of the correlatorsCSNM = ⟨ΨSΨNΨM(screenings)⟩(23)Defining the conjugate fields ΨS ≡Ψ−S, (s′, s) = (−s′, −s), the asymmetric struc-ture constants are defined according toCMSN = ⟨ΨSΨNΨM(screenings)⟩(24)In analogy with [15] it is shown in [24] for the superconformal case that the physicalstructure constants DSNM are simply related to CSNM. There are four equivalentways of writing D in terms of C:DSNM =(CMSNCSMNaN)1/2=(aNaSa−1M )1/2CMSN=(aNaMa−1S )1/2CSMN=Ω−1(aNaSaM)1/2CSNMaS =(C1SS)−1Ω= −ρ4∆(ρ −12)∆(ρ′ + 12)(25)7

Let Q = (q′, q) be the number of screenings in the matter sector. q′ (resp.

q)counts the number of J−(resp. J+) screening charges in the correlator.

The parityof q′ + q is the same for the four representations in (25). The number of screeningsin the Liouville sector depends on q′, q.

Consider first the case with all chiralitiesχ = −1, ⟨−−−⟩:⟨L−s′,s(0, 0)L−m′,m(1, 0)L−n′,n(∞, ∞η)⟩(26)The Liouville screenings areq′L =−s′ −n′ −m′ −12= −q′ −1qL =s + n + m −12= qhence the parity of the Liouville sectorq′L + qL = −q′ −1 + q ≡q′ + q −1 (mod 2)is opposite to the matter parity as required for the non-vanishing of three-pointcorrelators. For other chiralities it is easy to show that the same conclusion holds.The three-point functions can be expressed as products of ratios of Γ-functions,with the products ranging up to q−1 or q′−1.

For the matter sector these numbersare positive. In the Liouville case however one of them is negative.

The analyticcontinuation advocated in [16] is to use the definition−l′−1Yi=1f(i) =l′Yi=01f(−i)l′ > 0(27)This together with the results of [24] is all we need to write the explicit form ofthe three-point correlators. Before we write the results, we have to determine theµ-dependence of the correlators.

The coefficient of U−in the action is µ. However,the coefficient of U+ in the Liouville action is taken as in [16] to be determined byits gravitational dimension, and it is given by µρ. This guarantees that the powerof µ in front of the correlators is exactly s as defined in (15).8

3. Computations.

There are four cases depending on the chirality: ⟨−−−⟩,⟨+−−⟩, ⟨++−⟩, ⟨+++⟩. The computations are very similar in all four cases.

Al-though the matter and Liouville correlators are separately very cumbersome, whenboth of them are put together almost everything cancels leaving only a set of termssimilar to Polyakov leg factors [29]. Depending on the chirality of the matter fieldsthe cancellation is made more evident by choosing one of the representations (25).We list now each chirality case together with the number of screenings involved inthe Liouville and matter sectors.

Take k′, k to be the number of −, + screeningoperators in the Liouville sector, and l′, l in the matter sector. The four cases tobe considered arei).

⟨−−−⟩k′ =12(−s′ −n′ −m′ −1) = −l′1 −1k =12(s + n + m −1) = l1l′1 =12(s′ + n′ + m′ −1)l1 =12(s + m + n −1)(28)and in performing the cancellation it is best to take DSNM ∼CSNM.ii). ⟨+ + −⟩k′ =12(s′ + n′ −m′ −1) = l′2k =12(−s −n + m −1) = −l2 −1l′2 =12(s′ + n′ −m′ −1)l2 =12(s + m + n −1)(29)and we take DSNM ∼CMSN.iii).

⟨+ −−⟩k′ =12(s′ −n′ −m′ −1) = −l′3 −1k =12(−s + n + m −1) = l3l′3 =12(−s′ + n′ + m′ −1)l3 =12(−s + m + n −1)(30)and DSNM ∼CSNM.9

iv). ⟨+ + +⟩k′ =12(s′ + n′ + m′ −1) = l′1k =12(−s −n −m −1) = −l1 −1(31)and DSNM ∼CSNM.The matter three-point function up to irrelevant constants can be representedas a suface integrallimR→∞R4∆(αM)⟨VαM(R, Rη)VαN(1, 0)VαS(0, 0)Zl′Y1d2Z′i Vα−(Z′i, Z′i)ZlY1d2Zi Vα+(Zi, Zi)⟩= limR→∞R4∆(αM)Zl′Y1d2Z′iZlY1d2Zi |R −1|2αMαN|R|2αMαS|R −zi −Rηθi|2αMα+|R −z′i −Rηθ′i|2αMα−l′Yi=1|1 −z′i|2αNα−|z′i|2αSα−l′Yi

whereξ =|1 −αMα+Xiηθi −αMα−Xiηθ′i|2a′ =α−αs′,sa = α+αs′,sb′ =α−αn′,nb = α+αn′,nc′ =α−αm′,mc = α+αm′,mThe integral (33) is given by the structure constants CSNM = C−MSN which can befound in [24]:(const. )ρ2ll′ ρ′22M ′l′ ρ22Ml+2M ′l′+1lY1∆(−l′ + ρ + 12i −Mi)l′Y1∆(ρ′ + 12i −M′i)l−1Y0∆(1 + a + ρ −12i + Mi)∆(1 + b + ρ −12i + Mi)∆(−a −b −ρ(l −1) + l′ + ρ −12i + Mi)l′−1Y0∆(1 + a′ + ρ′ −12i + M′i)∆(1 + b′ + ρ′ −12i + M′i)∆(−a′ −b′ −ρ′(l′ −1) + l + ρ′ −12i + M′i)(34)whereM′i = [ i2]Mi = −l′ + [l′ + i2]with [x] = integer part of x and ∆(x) = Γ(x)/Γ(1 −x).

Furthermore from thescreening conditions we have−a′ −b′ −ρ′(l′ −1) + l = 1 + c′−a −b −ρ(l −1) + l′ = 1 + c11

The same techniques apply to the Liouville partlimR→∞R4∆(βM)⟨VβM(R, Rη)VβN(1, 0)VβS(0, 0)Zk′Y1d2Z′i Vβ−(Z′i, Z′i)ZkY1d2Zi Vβ+(Zi, Zi)⟩=Zk′Y1d2Z′iZkY1d2Zi ξk′Yi=1|z′i|2a′|1 −z′i|2b′ k′Yi

We finally have(const. )(−ρ)2kk′ −ρ′22M ′k′ −ρ22Nk+2M ′k′+1kY1∆(−k′ −ρ −12i −Ni)k′Y1∆(−ρ′ −12i −M′i)k−1Y0∆(1 + a −ρ + 12i + Ni)∆(1 + b −ρ + 12i + Ni)∆(−a −b + ρ(k −1) + k′ −ρ + 12i + Ni)k′−1Y0∆(1 + a′ −ρ′ + 12i + M′i)∆(1 + b′ −ρ′ + 12i + M′i)∆(−a′ −b′ + ρ′(k′ −1) + k −ρ′ + 12i + M′i)(36)12

nowM′i =[ i2]Ni = −k′ + [k′ + i2]1 + c′ = −a′ −b′ + ρ′(k′ −1) + k1 + c = −a −b + ρ(k −1) + k′Using (27) and standard properties of Γ-functions we obtain after some tediouscomputationsi). ⟨−−−⟩⟨Ψ−m′,m(∞, ∞η)Ψ−n′,n(1, 0)Ψ−s′,s(0, 0)⟩⟨L−m′,m(∞, ∞η)L−n′,n(1, 0)L−s′,s(0, 0)⟩= µsC1∆(1 −s2 + s′2 ρ′)∆(1 −n2 + n′2 ρ′)∆(1 −m2 + m′2 ρ′)(37)ii).

⟨+ + −⟩= µsC2∆(1 −s′2 + s2ρ)∆(1 −n′2 + n2ρ)∆(1 + m′2 + m2 ρ)(38)iii). ⟨+ −−⟩= µsC3∆(1 + s2 −s′2 ρ′)∆(1 −n2 + n′2 ρ′)∆(1 −m2 + m′2 ρ′)(39)iv).

⟨+ + +⟩= µsC4∆(1 −s′2 + s2ρ)∆(1 −n′2 + n2ρ)∆(1 −m′2 + m2 ρ)(40)There is a common infinite factor in Ci which can be absorbed in the normalizationof the correlation function.Recall also that s is given by (15) but with βminreplaced by β−. As a particular application, and to compare with the results inthe conformal case [12, 16] we consider the unitary case p′ = p + 2 and diagonal13

fields Am,m. It is convenient to evaluate a combination of correlators where thedependence on µ cancels⟨Am1,m1Am2,m2Am3,m3⟩2Z⟨Am1,m1Am1,m1⟩⟨Am2,m2Am2,m2⟩⟨Am3,m3Am3,m3⟩=m1m2m3(1 + ρ)ρ(ρ −1)(41)In the case when p′ = p + 2 = 2(n + 1) these correlators cannot be distinguishedfrom the same diagonal correlators in the (n + 1, n) conformal theory.In thiscase not only the string susceptibility and the gravitational dimensions coincide,but also the special combination (41) of three-point functions.The argumentsof the Γ-functions in (37)-(40) can be written in a simpler form if we combinethe quantities α, β of a dressed vertex operator eiαXeβΦ into a Minkowskian two-dimensional vector p = (β, α).

Defining the background two-vector b = (Q, α0),the dressing (on-shell) condition takes the form (p −b/2)2 = 0, where for any two-vector p2 = β2 −α2. Then the argument 1 −m/2 + m′ρ/2 appearing for examplein (37) can be written as 12 + p22 .

Finally the singularities of the factors in (37)-(40)appear in the boundary of the Kac table for values of (m′, m) = (0, m), with meven in analogy with the conformal case [29]. These states should very likely havean interpretation as boundary operators as suggested for the conformal case in [30].We find it rather intriguing that in the case when p′ = p + 2 is even (the casewhen the theory has a supersymmetric ground state before coupling to gravity),the zero-, one-, two- and three-point functions in the NS sector of the theory cannotbe distinguished from the conformal case (1 + p2, p2).While this paper was being typed we received three papers where similar issuesare addressed [31,32,33].ACKNOWLEDGEMENTS.

We would like to thank M. Ruiz-Altaba for manydiscussions during the course of this work.14

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