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STRUCTURE CONSTANTS IN THE

arXiv:hep-th/9109050v1 25 Sep 1991CERN-TH.6242/91September, 1991STRUCTURE CONSTANTS IN THEN=1 SUPEROPERATOR ALGEBRAL. Alvarez-Gaum´eTheory Division, CERNCH-1211 Geneva 23, SwitzerlandPh.

Zaugg∗D´epartement de Physique Th´eoriqueUniversit´e de Gen`eveCH-1211 Gen`eve 4, SwitzerlandABSTRACTUsing the Coulomb gas formulation of N=1 Superconformal Field Theories,we extend the arguments of Dotsenko and Fateev for the bosonic case to evaluatethe structure constants of N=1 minimal Superconformal Algebras in the Neveu-Schwarz sector.⋆Partially supported by the Swiss National Science Foundation.

1. INTRODUCTIONIn this paper we compute the structure constants of the operator algebra forsome Superconformal Field Theories in the Neveu-Schwarz sector (NS).

For theminimal Conformally Invariant Theories [1] and for the unitary subseries, [2] thestructure constants of the operator algebra were computed in general using theCoulomb gas representation for the models in a set of classic papers by Dotsenkoand Fateev [3,4,5]. The N = 1 Superconformal Field Theories [6,2,7] were foundsoon after the discovery of the minimal models [1], and some of their propertieswere analyzed in [8,9,10].

In particular, in [9] the Coulomb gas description in [3]was extended to the superconformal case. This construction was used to determinethe fusion rules and some of the general properties of four-point correlators [11].However, the full determination of the structure constants of the operator algebrafor the minimal superconformal theories in analogy with the analysis of [3,4,5] isnot available in the literature.We extend the contour manipulation techniques of the work of Dotsenko andFateev to the case of supercontours and supercontour integral representations ofsuperconformal blocks.

We follow their methodology closely, although the super-conformal case presents some peculiarities of its own. There are several ways to testthe accuracy of our results.

The first one consists of verifying that the structureconstants have zeroes exactly where indicated by the fusion rules. We can alsocheck that our results agree with those of the tricritical Ising model, where thestructure constants can be read offdirectly from [4,5] or [10].

The third and lesstrivial check of the consistency of our method is presented in the appendix. It willbe argued in the text that in the computation of the structure constants we need toknow two things.

First we need to know a set of monodromy matrices expressingthe behavior of the conformal blocks under braiding and second we need to knowexplicitly a number of normalization superintegrals. There are two independentways of computing the monodromy matrices.

The first one, as we will discuss indetail, is to take them from [4] after some appropriate changes are made, and the1

second is to evaluate them directly in terms of the normalization integrals. Thesetwo methods are independent and give the same result thus providing a good veri-fication of our results and methods.

This paper is very technical and for the readerinterested only in the results, we have summarized the main results at the end ofeach section.The structure of this paper is as follows. In section two we collect a number ofuseful formulae in the theory of superconformally invariant field theories.

We havefollowed the presentations in [12,13] and in [9] for the Coulomb gas formulation ofthe Superconformal minimal models. We write the Coulomb gas representation ofthe chiral blocks which need to be computed and at the end of the section we havea short discussion on the issue of open versus closed supercontours in the represen-tation of superconformal blocks and in the solution of super-differential equations.Here, and perhaps unexpectedly, one finds a phenomenon familiar in Superstringperturbation theory with regard to integration ambiguities in supermanifolds withboundaries (see for details and references) although here the problem is not nearlyas severe.

We also collect a number of useful formula about SL(2|1) and its invari-ants which are necessary in writing three- and four-point correlators.In section three we extend to the superconformal case the general analysis ofconformal blocks that was carried out for the conformal case in [1] and [4]. Thegeneral structure of the operator product expansion (OPE) of two superfield ispresented, we define the monodromy matrices and the monodromy invariants usedin building physical correlation functions.We write explicitly the holomorphicsuperconformal blocks to be computed in later sections, we determine carefully thegeneralization of the conformal block normalization conditions in [1] and itemizeall the possible case and how to determine the general form of the monodromyinvariant metric allowing us to put together the holomorphic and anti-holomorphicblocks.We do this for the thermal subalgebras and the general algebra of NSsuperfields.

We determine here the type of integrals we need to compute in orderto explicitly evaluate the structure constants of the superoperator algebra. Thissection therefore outlines the computation to follow.2

In section four we compute the normalization integrals for the thermal series.We have to separate two substantially different cases depending on whether thenumber of screening charges is even or odd. The odd case is more difficult than theeven case because two types of integrals are necessary.

We find the generalization ofthe recursion relations and functional relations for them which generalize the workof [4], relate the even to the odd integrals by some specific limiting procedures andwe are able finally to write the explicit form of the integrals in terms of rather longproducts of Γ-functions. The most difficult part of the computation appears in thedetermination of a set of integers appearing in the arguments of the Γ-functions.In section five we compute the normalization constants in the general case.Again the most difficult part is the determination of some integers in the argu-ments of the Γ-functions, however, here we can use the results of the thermal seriesto determine them.

In section six we compute the non-symmetric structure con-stant of the superoperator algebra. These are the ones we can read offdirectly fromthe formulae in section three.

At the end of this section we collect a number ofuseful integrals analogous to those appearing in appendix B of [4]. In section sevenwe extend the arguments in [5] to compute the symmetric (physical) structureconstants.

Finally, in the appendix we provide a way to compute the monodromymatrices different from the one used in the text. We believe that the methods pre-sented in this paper can be extended to compute the structure constants involvingtwo Ramond fields although we have not yet tried to do so.Note added.

When this work was completed we discovered the paper by Ki-tazawa et al. [15] where the N = 1 superconformal structure constants werecomputed.

Our results agree with those of ref. [15], and provide a non-trivial ver-ification of the whole computation.

Our method is a manifestly superconformallyinvariant generalization of the work of Dotsenko and Fateev [3, 4, 5]. The resultsare complicated enough that an independent computation of these structure con-stants is not unreasonable.

We also feel that some of the technical problems wetackled in dealing with supercontour integrals are interesting in their own right,and they can provide a basis for extension to other cases like for example N = 23

models.4

2. GENERAL PROPERTIES OFSUPERCONFORMAL THEORIESIn this section we collect some of the general properties of SuperconformalField Theories (SCFT) [12].Since there is abundant literature on the subjectwe mostly establish our notation and review the basic formulae of the Coulombgas formulation of N = 1 SCFT [9].

We also present some useful properties ofSL(2|1)-transformations. Further details can be found in the literature.2.1.

SUPERCONFORMAL TRANSFORMATIONS, SL(2|1)We follow mainly D. Friedan’s lectures in [12]. A superpoint in the complexsuperplane C1|1 will be denoted by Z = (z, θ).

The superderivative D is given byD = ∂∂θ + θ ∂∂z(2.1)A function f(Z, Z) is superanalytic if it satisfies Df = 0. In components f =φ+θψ+θλ+θθχ and the analyticity condition implies λ = χ = 0.

Furthermore, φ, ψare holomorphic functions of z: f = φ(z)+θψ(z). A superconformal transformationis a change of coordinatesZ →eZ(Z) = (ez(Z), eθ(Z))(2.2)under which D transforms covariantlyD = Deθ eD(2.3)This impliesDez = eθDeθ(2.4)A super-Riemann surface is built by gluing C1|1 patches with superconformal trans-formations.

In analogy with Conformal Field Theory (CFT) [1] we can introduce5

tensor fields. If dZ denotes the superdifferential transforming according tod eZ = Deθ dZ(2.5)under superconformal transformations, a superconformal tensor of rank h is a func-tion φ(Z) such thatφ(Z)dZ2h(2.6)is invariant under superconformal transformations.

As usual, we omit the anti-holomorphic dependence whenever possible. In componentsφ(Z) = φ0(z) + θφ1(z)The standard conformal dimensions of φ0, φ1 are respectively h and h + 1/2.

Theanalogues of abelian differentials are tensors of type 1/2. For these we can definesupercontour integrals.

RecallingIdθ θ = 1Idθ 1 = 0(2.7)one definesICdZ ω(Z) =ICdzIdθω(Z) =ICdzω1(z)(2.8)It is also possible to define line integralsf(Z1, Z2) =Z1ZZ2dZ ω(Z)(2.9)according tof(Z2, Z2) = 0D1f(Z1, Z2) = ω(Z1)(2.10)6

For exampleθ12 ≡θ1 −θ2 =Z1ZZ2dZZ12 ≡z1 −z2 −θ1θ2 =Z1ZZ2dZZZZ2dZ′(2.11)A superanalytic function can be expanded in power series:f(Z1) =∞Xn=01n!Z12n∂n2 (1 + θ12D2)f(Z2)= f(Z2) + θ12D2f(Z2) + z12∂2f(Z2) + . .

. (2.12)The fundamental formulae of superconformal calculus are12πiIC2dZ1 Z12−n−1 = 012πiIC2dZ1 θ12Z12−n−1 = δn,0(2.13)yielding the generalization of Cauchy’s formula12πiIC2dZ1 f(Z1)θ12Z12−n−1 = 1n!∂n2 f(Z2)12πiIC2dZ1 f(Z1)Z12−n−1 = 1n!∂n2 D2f(Z2)(2.14)A special type of transformations is the fractional linear transformations.

WritingeZ = g(Z)ez(Z) = az + b + αθcz + d + βθeθ(Z) = ¯αz + ¯β + ¯Aθcz + d + βθ(2.15)one easily solves (2.4) to obtain¯α = aβ −cα√ad −bc¯β = bβ −dα√ad −bc¯A =pad −bc −3αβ(2.16)7

Define the superdeterminant of g assdet g ≡ad −bc −αβ(2.17)One easily verifies the following properties:Deθ =√sdet gcz + d + βθeZ12 = sdet gZ12(cz1 + d + βθ1)(cz2 + d + βθ2)(2.18)The supergroup SL(2|1) has dimension 3|2. Therefore given any four points Zi, i =1, 2, 3, 4 we can fix for example z1 = 0, z2 = 1, z3 = ∞; θ1 = θ2 = 0.

In general ann-point function of Neveu-Schwarz (NS) fields will depend on n−3|n−2 parameters.The harmonic ratioZ12Z34Z13Z24(2.19)is SL(2|1)-invariant. Given any three points Z1, Z2, Z3 we can construct an oddSL(2|1) invariant quantityη = (Z12Z13Z23)1/2(θ1Z23 + θ2Z31 + θ3Z12 + θ1θ2θ3)(2.20)It is convenient to perform the transformation from Z1, .

. .

, Z4 to (0, 0), (1, 0),(∞, θ), (z4, θ4) in two steps. First we apply the transformationez(Z) = Z·1Z23Z·3Z21eθ(Z) = −1Z·3rZ23Z12Z31(θ1Z·3 + θ3Z1· −θZ13 + θ1θθ3)(2.21)with Z.j = z −zj −θθj.

Then we apply:ˆz( eZ) = ez(1 + eθ2eθ)ˆθ( eZ) = eθ −eθ2ez2(2.22)Choosing Z1 = (z1, θ1), Z2 = (z2, θ2), Zǫ3 = (z3+ǫ, θ3), Z4 = (z4, θ4), the applicationof the previous two transformations has the desired result as ǫ →0.8

Since the NS vacuum is SL(2|1)-invariant, we can write the general form ofthe n-point function for NS-fields:⟨nYi=1Φi(Zi)⟩=Yi

. , n −3α = 1, .

. .

, n −2Xj̸=iγji = 2hiγij = γjiγii = 0(2.23)In particular, the three-point function takes the form:⟨Φ1(Z1)Φ2(Z2)Φ3(Z3)⟩= Z12−γ12Z13−γ13Z23−γ23(a + bη)(2.24)with η given in (2.20). Both (2.23), (2.24) are direct consequences of (2.15), (2.18).The coefficients a, b are the structure constants of the superconformal operatoralgebra, and their computation is the main object of this paper.2.2.

FREE SUPERFIELDS AND BACKGROUND CHARGEThe generator of superconformal transformations is the super-energy-momentumtensorT(Z) = TF (z) + θTB(z)(2.25)δvΦ(Z2) =IC2dZ1v(Z1)T(Z1)Φ(Z2)(2.26)For primary superfields the operator product expansion (OPE) of T(Z) and φ(Z)isT(Z1)Φ(Z2) = θ12Z122hΦ(Z2) + 1/2Z12D2Φ(Z2) + θ12Z12∂2Φ(Z2) + . .

. (2.27)The OPE defining the super-Virasoro algebra isT(Z1)T(Z2) =ˆc4Z123 + 3θ122Z122T(Z2) + 1/2Z12D2T(Z2) + θ12Z12∂2T(Z2) + .

. .

(2.28)9

The mode expansions defining Ln, Gn areTB(z) =XnLnz−n−2TF (z) =Xn12Gnz−n−3/2(2.29)where for TF , n ∈Z + 1/2 in the NS sector and n ∈Z in the Ramond (R) sector.Using Cauchy’s formula (2.14)[Ln, Φ(Z)] = znz ∂∂z + (n + 1)(h + 12θ ∂∂θ)Φ(Z)[ǫGn+1/2, Φ(Z)] = ǫznz( ∂∂θ −θ ∂∂z) −2h(n + 1)θΦ(Z)(2.30)A double check on (2.23) can be obtained by writing the SL(2|1) generatorsG±1/2, L±1, L0 as super-differential operators according to (2.30) and then showingthat (2.23) is annihilated by them.The standard value of the central charge of the Virasoro algebra is c = 3ˆc/2.The simplest realization of the superconformal algebra is provided by a free mass-less scalar superfield.T(Z) = −12 : Dφ∂φ : (Z)φ(Z1)φ(Z2) ∼log Z12(2.31)One easily verifies that ˆc = 1 and that the conformal dimension of a vertex operatoris given byh(eiαφ) = 12α2(2.32)The n-point correlators of vertex operators vanish unless the charge neutralitycondition Pi αi = 0 is satisfied. The central charge ˆc can be changed to any valueby adding a background charge [9] in analogy with the Virasoro case [3,4,5].

Define10

a new energy-momentum tensorT(Z) = −12 : Dφ∂φ : (Z) + i2α0D∂φ(Z)(2.33)nowˆc = 1 −2α02c = 32 −3α02(2.34)The conformal dimension of a vertex operator also changesh(eiαφ) = 12α(α −α0)(2.35)and notice that both α and α = α0 −α give fields with the same conformaldimension. As in the standard Coulomb gas construction [3] there are two screeningfields of dimension 1/2 leading after contour integration to the screening chargesQ±:Q± =IdZeiα±φ(2.36)with α± satisfying12α±(α± −α0) = 12(2.37)orα+ + α−= α0α+α−= −1(2.38)α± =12√2(√1 −ˆc ±√9 −ˆc)(2.39)The charge neutrality condition for the correlation function of n vertex operatorsis changed to Pi αi = α0.

In the NS sector the bosonic (φ0) and fermionic (φ1)fields combine into a superfield φ(Z). They are both periodic around z = 0.

In theRamond sector one has instead G(e2πiz) = −G(z); φ1 is antiperiodic, φ0 is periodicand they do not combine to form a superfield. Furthermore we have to take into11

account the spin fields σ±(z) associated to φ1. The vertex operators in the R sectortake the general form σ±(z)eiαφ0(z) with conformal dimension 116+ 12α(α−α0).

Thisformula is again invariant under α 7→α0 −α.The singular representations of the super-Virasoro algebra are labelled by twopositive integers m, m′ ≥1 with highest weightshm′,m = ˆc −116+ 18(m′α−+ mα+)2 + 132(1 −(−1)m′−m)(2.40)When m′ −m is even we have NS field, and for m′ −m odd we have R fields. In theCoulomb gas picture the singular modules are generated by the vertex operators.In the NS sectorVm′,m(Z) = eiαm′,mφ(Z)αm′,m = 1 −m′2α−+ 1 −m2α+m′ −m ≡0( mod 2)(2.41)and in the R sectorVm′,m(z) = σ(z)eiαm′,mφ0(z)αm′,m = 1 −m′2α−+ 1 −m2α+m′ −m ≡1( mod 2)(2.42)The charge screening condition for a n-point correlator of vertex operators is asusual Pi αi = α0 independently of whether we have NS or R fields.

It is useful tonote that¯αm′,m = α0 −αm′,m = α−m′,−m(2.43)The minimal superconformal theories are those satisfyingp′α−+ pα+ = 0(2.44)where p′, p are positive integers. They are both supposed to be even or odd, andtheir greatest common divisor is either 2 or 1.

The parity condition m′ −m ≡12

0 (mod 2) follows from the fact that (2.43),(2.44) taken together imply αm′,m =αp′−m′,p−m and if αm′,m is in the NS sector, αp′−m′,p−m should also be a NS field.In the rational case (2.44):ˆc = 1 −2(p′ −p)2pp′α+ =pp′/pα−= −pp/p′(2.45)and we can choose p′ > p. The unitary series occurs when [7] p′ = p + 2,hm′,m =18pp′[(mp′ −m′p)2 −(p′ −p)2] + 132(1 −(−1)m′−m)(2.46)The primary fields of the minimal theories can be further restricted to lie in thefundamental region 1 ≤m′ ≤p′ −1, 1 ≤m ≤p −1, mp′ −m′p ≤0. The chargeassignments (2.41) can also be derived by requiring the non-vanishing in the four-point function of the lowest component of Vα: ⟨VαVαVαVα⟩.

The charge can onlybe screened by the insertion of QN++ QN−−for the given values of α.A simpleconsequence of the Coulomb gas representation is the computation of the fusionrules. We only need to write the three possible ways of screening the three-pointfunction⟨Φm′1,m1(Z1)Φm′2,m2(Z2)Φm′3,m3(Z3)⟩(2.47)We can conjugate any of the fields in (2.47) .

For the rational case (2.45) if wecompute the fusion rules by counting screenings of both ⟨Vα1Vα2Vα3⟩, ⟨Vα1Vα2Vα3⟩,and requiring compatibility we obtain in the NS sector[m′1, m1] × [m′2, m2] =min2p′−m′1−m′2−1m′1+m′2−1X|m′1−m′2|+1′min2p−m1−m2−1m1+m2−1X|m1−m2|+1′[m′, m](2.48)Where the prime in the sum means that m′, m jump in steps of two units.13

2.3. CORRELATORS IN THE COULOMB GAS REPRESENTATIONWe write down in this section the contour integral representation of chiralcorrelators for minimal N = 1 theories in the NS sector.Using (2.21,22) andthe covariance properties of conformal superfields, we can transform the four-pointfunction ⟨Φ4(Z4)Φ3(Z3)Φ2(Z2)Φ1(Z1)⟩into the formlimR→∞R2h4⟨Φ4(R, Rη)Φ3(1, 0)Φ2(z, θ)Φ1(0, 0)⟩(2.49)Representing the superfields as Coulomb gas vertex operators eiαiφ(Zi), the genericchiral four-point function takes the formlimR→∞R2h4⟨eiα4φ(R,Rη)eiα3φ(1,0)eiα2φ(z,θ)eiα1φ(0)ImYa=1dZam′Ya′=1dZa′eiα+φ(Za)eiα−φ(Za′)⟩= limR→∞R2h4Rα1α4(R −1)α3α4(R −z −Rηθ)α2α4ImY1dZam′Y1dZa′mYa=1uα1α+a(1 −ua)α3α+(z −ua −θθa)α2α+mYa

Using (2.21,22)14

one can add to (2.50) the appropriate prefactors giving the four-point function forarbitrary points Zi. In the derivation of (2.50) the charge screening condition wascrucial.

Collecting all powers of R we obtain a prefactorR2h4+α4(α1+α2+α3+mα++m′α−)(2.51)where m, m′ are the number of +, −screening charges respectively. Using 2h4 =α4(α4 −α0) and the screening condition α1 + α2 + α3 + α4 + mα+ + m′α−= α0the exponent in (2.51) vanishes and there is no R-dependence in the R →∞limit.In the next chapter we analyze how to put together chiral conformal blocks toconstruct physical conformal blocks.2.4.

OPEN AND CLOSED CONTOURSWe come now to a rather delicate issue in the contour definition of supercorre-lators and normalization factors. The typical example in the bosonic case is givenby the decoupling for a level two null-vector [1].

This yields a hypergeometric dif-ferential equation whose solutions can be represented in terms of contour integralsaround the singular points according to standard results in the theory of ordinarydifferential equations [16]. The regular singular points of the hypergeometric equa-tion can be chosen at (0, 1, ∞) for convenience.

The solutions F1(z), F2(z) can beexpressed as contour integrals around (0, z) and (1, ∞). When the hypergeometricequation is applied to the computation of conformal blocks, the normalization ofits solutions is determined by the monodromy invariance conditions.

It is thereforeconvenient to write F1(z), F2(z) as open line integrals along the cuts joining thesingular points. In analogy with (2.50) F1,2 are represented up to constants byIC1tα(1 −t)β(z −t)γIC2tα(t −1)β(t −z)γ(2.52)15

opening the contours we obtainzZ0tα(1 −t)β(z −t)γ∞Z1tα(t −1)β(t −z)γ(2.53)The line integral representation (2.53) is simpler to use in the computation ofmonodromy matrices but it has disadvantages in determining the fusion rules andthe internal channels in a given conformal block. For pratical purposes we will usealmost exclusively the representation (2.53).

Notice that taking the parameters(α, β, γ) in the range where the integral converges, the integrands vanish at theend points of the line integrals or they have integrable divergences.The procedure briefly outlined in the previous paragraphs can be extendedwithout difficulty to multi-contour integrals in the bosonic case. For super-contourswe have to be more careful and potential ambiguities may show up in going fromclosed to open contours.

The possible existence of integration ambiguities in super-integrals is a phenomenon encountered in Superstring Perturbation Theory (for aclear exposition of the problem with references to the relevant literature, see [14]).The case at hand is a milder version of this problem. From the definitions (2.8,9)we can compare open and closed contour integrals.

We explicitly compute (2.9)satisfying (2.10):F(Z) = F0(z) + θF1(z)f(Z1, Z2) =Z1ZZ2dZF(Z) =z1Zz2dzF1(z) + θ1F0(z1) −θ2F0(z2)(2.54)The limits of integration on the right-hand side of (2.54) are determined by theeven parts of the points Z1,2. When we open a closed supercontour we should use(2.54).

It is often unavoidable to generate a nilpotent contribution at the endpoints16

z1, z2. The projected line integral is defined by ignoring the terms linear in θ1,2 in(2.54).

In other words it is defined as a closed contour in θ but as open in z. Thisprescription is inconsistent in general. However, in our case, the integrals of thetype (2.50) and similar integrals to be considered in later sections are such thatθF0(z) vanishes (possibly in the sense of analytic continuation) at the end pointsof the line integrals.

Effectively this produces a projected integration prescriptionwhich is preserved under split superconformal transformationsez = f(z)eθ = θr∂f∂z(2.55)One easily checks that under (2.55)eZ1ZeZ2d eZF( eZ) =z1Zz2dz∂f∂z F1(f(z)) + eθ1F0(ez1) −eθ2F0(ez2)(2.56)Hence, if θF0(z) vanishes at the end points of the integration interval, the pro-jected prescription will be maintained by split transformations. Notice that, strictlyspeaking, only their difference has to vanish in order to preserve this prescription.Under non-split transformations this prescription has to be modified.

In later sec-tions we will only need to use split transformations and the previous argumentsjustify the use of the projected prescription. We will remind the reader of theseapparently obscure considerations when the case arises.17

3. SUPERCONFORMAL BLOCKSIn this section we detail the structure of super-OPE, superconformal blocks,and their integral representation.

Then we discuss the necessary steps to computethe structure constants of the operator algebra. For simplicity we start with thethermal subalgebra of fields (1, m) or (m′, 1).

The presentation is tailored closelyafter the papers [1],[3, 4] and we only emphasize the intrinsic features of the su-persymmetric case.We learned in the previous chapter that the three-point function depends ontwo arbitrary constants. Through the relation between three-point functions andOPE we derive that the super-OPE involves two independent sets of structure con-stants.

More precisely, SL(2|1)-invariance constrains the OPE of two NS primarysuperfields to be of the formΦm(Z1)Φn(Z2) =XpZ12−γmnpApmn[Φp(Z2)]even + Z12−γmnp−1/2Bpmn[Φp(Z2)]odd(3.1)where γmnp = hm + hn −hp, Apmn and Bpmn are the structure constants of theoperator algebra, and [Φp] denotes a superconformal family with all its descendantfields. The first representatives are[Φp(Z2)]even = Φp(Z2) + hm −hn + hp2hp(θ12D2Φp(Z2) + Z12∂2Φp(Z2)) + .

. .

[Φp(Z2)]odd = θ12Φp(Z2) +12hpZ12D2Φp(Z2) + hm −hn + hp + 1/22hpZ12θ12∂2Φp(Z2) + . .

. (3.2)The subindex refers to the Grassmann parity of the expansion.

These formulaeare derived by requiring the OPE to be compatible with SL(2|1)-invariance. Thefull OPE is obtained by combining the holomorphic and anti-holomorphic contri-butions.18

3.1. THE STRUCTURE OF THERMAL FOUR-POINT FUNCTIONSIn the Coulomb gas representation the number of screening charges determinesthe Grassmann parity of the correlator.

This is a consequence of the use of vertexoperators to represent primary fields and of the integration over the fermionicvariable implied in every screening charge super-contour. Using (2.50) we see thatto an even (resp.

odd) number of screening charges corresponds an even (resp.odd) function of the SL(2|1)-invariant variables.A similar argument holds forthe three-point function. We shall use these remarks in the decomposition of thefour-point function into superconformal blocks.For convenience we denote the thermal primary superfields in the NS sectorby Φm(Z) = Φ(1,2m+1)(Z) and its conjugate field by Φm(Z).

The out-state is asusual⟨Φm(∞)| = limR→∞R2hm⟨0|Φm(R, Rη)The fusion rule (2.48) readsΦm × Φn =min(p−m−n−2,m+n)Xq=|m−n|Φqand the three-point function is⟨Φ¯k(∞)|Φl(1, 0)Φq(0, 0)⟩= A¯klq + B¯klqη(3.3)where the number of screenings is N = l + q −k.The four-point function can be expanded in terms of three-point functionswhen we use the OPE:G = ⟨Φ¯k(∞)|Φl(1, 0)Φm(z, θ)Φn(0, 0)⟩=Xqz−γmnqAqmn⟨Φ¯k(∞)|Φl(1, 0)[Φq(0, 0)]even⟩+ z−γmnq−1/2Bqmn⟨Φ¯k(∞)|Φl(1, 0)[Φq(0, 0)]odd⟩=XqGq(3.4)19

where the number of screening charges is M = l + m + n −k. Trading the index qfor r = q −|m −n| we haveN = l + |m −n| + r −k ≡M + r(mod 2)Using the previous observation on the Grassmann parity of the three- and four-point functions, and the expansion (3.2), we see that depending on the values ofN, M(mod 2) four types of conformal blocks Gq appear.M even :Gq is an even function of z, η, θ.N even : Gq ∼A¯klqAqmn z−γ (1 + ηθ(..) + z(..) + .

. .

)N odd : Gq ∼B¯klqBqmn z−γ−1/2 (ηθ + z(..) + zηθ(..) + . .

. )(3.5)M odd :Gq is an odd function of z, η, θ.N even : Gq ∼A¯klqBqmn z−γ−1/2 (θ + zη(..) + zθ(..) + .

. .

)N odd : Gq ∼B¯klqAqmn z−γ (η + θ(..) + zη(..) + . .

. )the ellipsis denotes constants.

The knowledge of this structure is essential in orderto normalize correctly the superconformal blocks. In deriving (3.5) we have chosenΦk as the conjugate field.

One readily sees that taking the conjugate to be anyother field leads to the same values for N, M(mod2) which is all the results dependon.3.2. THERMAL SUPERCONFORMAL BLOCKS AND MONODROMYINVARIANTSAccording to (2.50) the integral representation of a chiral superconformal block20

in the thermal subalgebra isJ(m)k(a, b, c, ρ; Z) = T∞Z1dV1 . .

.∞Z1dVm−kZZ0dVm−k+1 . .

.ZZ0dVm−1(1+a1ηθ + a2ηm−1X1θi)m−1Y1vaim−kY1(vi −1)b(vi −z −θiθ)cm−1Ym−k+1(1 −vi)b(z −vi −θθi)cm−1Yi

3.1. It is often more convenient to work with the ordered integralI(m)k(a, b, c, ρ; Z) =∞Z1dV1V1Z1dV2 .

. .Vm−k−1Z1dVm−kZZ0dVm−k+1 .

. .Vm−2Z0dVm−1{as in (3.6)}(3.8)which is simply related to J(m)kby (see sec.

4.1 for details)J(m)k(a, b, c, ρ; Z) = λk−1(ρ)λm−k(ρ)I(m)k(a, b, c, ρ; Z)(3.9)A physical correlation function is a combination of holomorphic and anti-holomorphic21

superconformal blocksG(Z, ¯Z) =mXk,l=1XklJ(m)k(Z)J(m)l(Z)(3.10)with the requirement that it should be monodromy invariant when we analyti-cally continue the variable Z (and obviously Z) along the curves shown in fig.3.2. The constants Xkl are determined using the monodromy properties of J(m)karound the points (0, 0), (1, 0).The integrals J(m)kcan be thought of as solu-tions to the superdifferential equations obtained by decoupling null-vectors in thesuperconformal modules.

The equations have regular singularities at the points(0, 0), (1, 0), (∞, η∞). We can compute the monodromy when we analytically con-tinue J(m)kalong C0 or C1 in fig.

3.2:Ci :J(m)k(Z) →(gi)klJ(m)l(Z)(3.11)where gi are (m −1) × (m −1) matrices. With the choice of contours made, thematrix g0 is diagonal and unitary.

Monodromy invariance under g0 requiresG(Z, ¯Z) =mXk=1XkJ(m)k(Z)J(m)k(Z)(3.12)The constants Xk are fixed if we require invariance under g1 as well.For thispurpose it is convenient to use another basis eJ(m)k(Z) with diagonal monodromyat (1, 0) and expand J(m)kin this new basis. Explicitly eJ(m)k(Z) are given by theintegralseJ(m)k(a, b, c, ρ; Z) = T−∞Z0dV1 .

. .−∞Z0dVm−kZZ1dVm−k+1 .

. .ZZ1dVm−1(1+a1ηθ + a2ηm−1X1θi)m−1Y1(1 −vi)bm−kY1(−vi)a(z −vi −θθi)cm−1Ym−k+1vai (vi −z −θiθ)cm−1Yi>jV ρij(3.13)with the contours ordered as in fig.3.3.

Denoting by βkl the coefficients in these22

linear expansions we get:J(m)k(a, b, c, ρ; Z) = βkl(a, b, c, ρ) eJ(m)l(a, b, c, ρ; Z)eJ(m)k(a, b, c, ρ; Z) = eβkl(a, b, c, ρ)J(m)l(a, b, c, ρ; Z)(3.14)Rewriting G(Z, ¯Z) in terms of eJ(m)k(Z) and requiring invariance under g1 yieldsthe conditionsmXk=1Xkβklβkn = 0l ̸= n(3.15)As in [4] the solution to these constraints can be chosen asXk = βmmeβmkeβmmβkmXm(3.16)This expression is invariant under simultaneous rescaling of the coefficients Xk.This amounts to changing the normalization of the four-point function. This free-dom will be used to simplify the formulae for the structure constants of the operatoralgebra.

Before doing this however we need to compute the matrices βkl and toproperly normalize the chiral blocks J(m)k.The computation of the matrix β simplifies considerably once we realize thatit relies essentially on the monodromy properties of the integrand in (3.6) andtherefore we can read offβ from the results in [4] with some minor modifications.The main difference between the ordinary chiral conformal blocks in [4] and herelies in the terms (vi−vj−θiθj)ρ replacing the terms (vi−vj)2ρ in the integrands andthe fact that the supercontour integrals are Grassmann odd. Thus, analyticallycontinuing Vi over Vj yields a phase e±iπ(ρ−1) in the superconformal case, to becompared with the phase factor e±iπ2ρ in the conformal case (this will be explainedin more detail in section four).

Hence to compute the matrices β we only need toreplace ρ by (ρ −1)/2 in the matrices α of Dotsenko and Fateev [4]. As a non-trivial check of our computations we evaluate independently in Appendix A thematrix elements βmk.

This will indeed be an important check of the normalization23

integrals computed in section four. The matrix eβ can be derived from β using therelationeJ(m)k(a, b, c, ρ; (z, θ); η) = ǫmJ(m)k(b, a, c, ρ; (1 −z, iθ); −iη)(3.17)where ǫm = 1 (ǫm = −i) when m −1 is even (m −1 is odd).

(3.17) is obtainedby performing the change of variables (vj, θj) →(1 −vj, √−1θj) in (3.13). Conse-quently we findeβkl(a, b, c, ρ) = βkl(b, a, c, ρ)(3.18)This is not surprising if we recall the relation with the N = 0 case in [4].We next turn to the normalization of J(m)k. These integrals possess a singularityas z →0.

Thus we first extract the singularity and then evaluate the resultingintegral which yields an analytic function of z near z = 0. If in the 0 →Z integralsin (3.8) we make the change of variables Vm−k+i = (vm−k+i, θm−k+i) →Si =(si, ωi) = (vm−k+i/z, θm−k+i/z1/2) we obtainI(m)k(a, b, c, ρ; Z) = z(k−1)(1/2+a+c+(k−2)ρ/2)∞Z1m−kY1dVi1Z0k−1Y1dSi(1 + a1ηθ + a2ηm−kX1θi + a2z1/2ηk−1X1ωi)m−kY1vai (vi −1)b(vi −z −θθi)c Yi

We deal with this last singularity by expanding all the terms containing θ24

or z1/2 in the integrand. This yieldsI(m)k(a, b, c, ρ; Z) = z∆k∞Z1dVi1Z0dSi(1 + a1ηθ + a2ηm−kX1θi + a2z1/2ηk−1X1ωi)(1 −cX θθivi −z)(1 −cz1/2Xθωi1 −si)(1 −ρz1/2 Xθiωjvi −zsj)m−kY1vai (vi−1)b(vi −z)c Yi

Depending on the values of m, k we findby inspection the following behaviours:(m −1) even :I(m)kis an even function of z, η, θ.i) (k −1) even : I(m)k∼z∆k (N(m)k+ ηθ(..) + . .

. )ii) (k −1) odd : I(m)k∼z∆k−1/2 (ηθN(m)k+ z(..) + .

. .

)(3.21)(m −1) odd :I(m)kis an odd function of z, η, θ.iii) (k −1) even : I(m)k∼z∆k (ηN(m)k+ θ(..) + . .

. )iv) (k −1) odd : I(m)k∼z∆k−1/2 (θN(m)k+ zη(..) + .

. .

)where (..) denote some constants. As expected we recognize here the same expan-sions as in (3.5).

The analysis of (3.5) was useful to indicate the leading term tobe normalized in the expansions of the superconformal blocks. Once the last z1/2has been taken into account by (3.21) we can set z = 0 in the integrand of (3.20)and the 1 →∞; 0 →1 integrals decouple.

We get for N(m)k(a, b, c, ρ):i) N(m)k=∞Z1m−kY1dViva+c+ρ(k−1)i(vi −1)b Yi

ii) N(m)kηθ = −a2c∞Z1m−kY1dVi(Xjηθj)va+c+ρ(k−1)i(vi −1)b Yi

We only need at this point the following properties(see (4.72))I2n(α, β, ρ) = ˆI2n(α, β, ρ)αeIΣ2n+1(α, β, ρ; η) = ηˆI2n+1(α, β, ρ) = ηˆI2n+1(β, α, ρ)(3.23)These relations are essential in simplifying (3.22). We use (3.23) in (3.22ii,iv) to26

absorb the constant c. For a2 we make use of the charge screening condition for afour-point function with m −1 screenings:4X1αi + (m −1)α+ = α0 = α+ + α−(3.24)thena2 = −α4α+ = −1 −a −b −c −ρ(m −2)(3.25)Hence (3.22) reduces to a nice form:N(m)k(a, b, c, ρ) = (−1)m−1 ˆIm−k(−1−a−b−c−ρ(m−2), b, ρ) ˆIk−1(a, c, ρ) (3.26)valid for all values of m and k.Summarizing, we find that the ordered integral representation of the thermalsuperconformal blocks can be written asI(m)k(a, b, c, ρ; Z) = N(m)k(a, b, c, ρ)F(m)k(a, b, c, ρ; Z)(3.27)Where F(m)k(Z) are the normalized thermal superconformal blocksF(m)k(a, b, c, ρ; Z) = z∆(m)k f(m)k(a, b, c, ρ; Z)(3.28)with the regular functions f(m)k(Z) having the expansions:i)f(m)k= 1 + ηθ(· · ·) + . .

.ii)f(m)k= ηθ + z(· · ·) + . .

.iii)f(m)k= η + θ(· · ·) + . .

.iv)f(m)k= θ + zη(· · ·) + . .

. (3.29)and ∆(m)kcan be read offfrom (3.21).

Finally the correlation function of four27

thermal vertex operators (2.50) can be written asG(Z, ¯Z) ∼mX1S(m)k(a, b, c, ρ)F(m)k(a, b, c, ρ; Z)2(3.30)where the quantitiesS(m)k(a, b, c, ρ) = Xk(a, b, c, ρ)N(m)k(a, b, c, ρ)2(3.31)contain all the information about the structure constants of the operator alge-bra. These will be evaluated in section six after the normalization integrals arecomputed.3.3.

GENERAL SUPERCONFORMAL BLOCKS AND MONODROMYINVARIANTSWe now extend the analysis of four-point functions to the full algebra of (m′, m)fields. Both +, −screening charges will be involved although the same programcan be carried out to the end without major differences.

The structure of the four-point function is identical to the thermal case (3.5) where M (resp. N) denotes thetotal number of + and −screenings in the four-point (resp.

three-point) function.The integral representation for a chiral superconformal block isJ(nm)lk(a, b, c, ρ; Z) = T∞Z1n−lY1dUiZZ0n−1Yn−l+1dUi∞Z1m−kY1dViZZ0m−1Ym−k+1dVi(1 + a1ηθ + a2ηn−1X1θi + a3ηm−1X1ωi)n−1,m−1Yi,j(ui −vj −θiωj)−1n−1Y1ua′in−lY1(ui −1)b′(ui −z −θiθ)c′n−1Yn−l+1(1 −ui)b′(z −ui −θθi)c′ n−1Yi

where Ui = (ui, θi), Vi = (vi, ωi) anda = α1α+b = α3α+c = α2α+ρ = α+2a′ = α1α−= −ρ′ab′ = α3α−= −ρ′bc′ = α2α−= −ρ′cρ′ = α−2 = ρ−1a1 = −α4α2a2 = −α4α−= −ρ′a3a3 = −α4α+(3.33)The ordering prescription T is explained in sections 4.1, 5.1 and the contours in(3.32) are ordered according to fig. 3.4.

One can also define the superconformalblocks eJ(nm)lk(a, b, c, ρ; Z) by a straightforward extension of (3.13). The relationwith the ordered integralsI(nm)lk(a, b, c, ρ; Z) =∞Z1dU1U1Z1dU2 .

. .Un−l−1Z1dUn−lZZ0dUn−l+1 .

. .Un−2Z0dUn−1∞Z1dV1 .

. .Vm−k−1Z1dVm−kZZ0dVm−k+1 .

. .Vm−2Z0dVm−1{as in (3.32)}(3.34)isJ(nm)lk(a, b, c, ρ; Z) = λl−1(ρ′)λn−l(ρ′)λk−1(ρ)λm−k(ρ)I(nm)lk(a, b, c, ρ; Z)(3.35)Notice that the form of the coupling terms (ui−vj−θiωj)−1 and the relations (3.33)imply that we can permute the C and S contours without affecting the value of theintegral.

This also implies that the monodromy properties of J(nm)lk(a, b, c, ρ; Z) canbe derived from those of the thermal case, and indeed, we have for the β matrices:β(lk)(rs)(a, b, c, ρ) = βlr(a′, b′, c′, ρ′)βks(a, b, c, ρ)(3.36)This in turn implies that the coefficients Xks in the decomposition of the four-point29

function:G(Z, ¯Z) =n,mXk,s=1XksIks(Z)Iks(Z)(3.37)also split in product of the thermal results (3.16)Xks = Xk(a′, b′, c′, ρ′)Xs(a, b, c, ρ)(3.38)As in the conformal case, the full algebra results are not just products of thermalquantities because of the normalization factors appearing in the superconformalblocks (3.34). The procedure for computing the factors N(nm)lk(a, b, c, ρ) is identicalto the thermal case although a bit more cumbersome.

We only state some of thenecessary steps. After a simple change of variables for the 0 →Z integrals andexpanding the terms containing either θ or z1/2 we end up with a cumbersomeexpression to evaluate:I(nm)lk(Z) = z∆lk∞Z1n−lY1dUi1Z0l−1Y1dTi∞Z1m−kY1dVi1Z0k−1Y1dSi(1 + a1ηθ + a2ηn−lX1θi + a2z1/2ηl−1X1ζi + a3ηm−kX1ωi + a3z1/2ηk−1X1νi)(1 −c′ X θiθui −z)(1 −c′z1/2Xθζi1 −ti)(1 −ρ′z1/2 Xθiζjui −ztj)(1 + z1/2 Xθiνjui −zsj)(1 −cX ωiθvi −z)(1 −cz1/2Xθνi1 −si)(1 −ρz1/2 Xωiνjvi −zsj)(1 + z1/2 Xζiωjzti −vj)n−lY1ua′i (ui −1)b′(ui −z)c′ YUρ′ijl−1Y1ta′i (1 −zti)b′(1 −ti)c′ YT ρ′ijm−kY1vai (vi −1)b(vi −z)c YV ρijk−1Y1sai (1 −zsi)b(1 −si)c YSρijY(ui−ztj)ρ′ Y(vi −zsj)ρ Y(UV )−1ijY(TS)−1ijY(ui −zsj)−1 Y(zti −vj)−1(3.39)30

with Ti = (ti, ζi), Si = (si, νi) and∆lk = (l−1)(12+a′+c′+ρ′2 (l−2))+(k−1)(12+a+c+ρ2(k−2))−(l−1)(k−1) (3.40)Depending on the values of n, m, l, k we may have to compute integrals of the typesgiven in sections 5.1, 5.2. The normalization constants are(n + m) even :i) (l + k) even : N(nm)lk= In−l,m−k(−a −b −c −ρ(m −2) + n −2, b, ρ)Il−1,k−1(a, c, ρ)ii) (l + k) odd : N(nm)lkηθ = −a2c′eIΣn−l,m−k(−a −b −c −ρ(m −2) + n −2, b, ρ; η)eIΣl−1,k−1(c, a, ρ; θ)(3.41)(n + m) odd :iii) (l + k) even : N(nm)lkη = −a2eIΣn−l,m−k(−a −b −c −ρ(m −2) + n −2, b, ρ; η)Il−1,k−1(a, c, ρ)iv) (l + k) odd : N(nm)lkθ = c′In−l,m−k(−a −b −c −ρ(m −2) + n −2, b, ρ)eIΣl−1,k−1(c, a, ρ; θ)Using (3.33) the charge screening condition for a four-point function with (n −1)−screenings and (m −1) + screenings and the definition (5.39) of ˆI(α, β, ρ) wefinally get a unified expression for the normalization constantsN(nm)lk(a, b, c, ρ) = ˆIn−l,m−k(−a−b−c−ρ(m−2)+n−2, b, ρ) ˆIl−1,k−1(a, c, ρ) (3.42)Thus we can define the normalized superconformal blocks F(nm)lk(Z) asI(nm)lk(a, b, c, ρ; Z) = N(nm)lk(a, b, c, ρ)F(nm)lk(a, b, c, ρ; Z)(3.43)whereF(nm)lk(a, b, c, ρ; Z) = z∆(nm)lkf(nm)lk(a, b, c, ρ; Z)(3.44)f(nm)lkhas an expansion similar to (3.29) and ∆(nm)lkis defined by (3.40) with a−1/2 correction depending on the values of l, k, n, m.31

Finally the general correlation function of four vertex operators can be writtenasG(Z, ¯Z) ∼Xk,lS(nm)lk(a, b, c, ρ)F(nm)lk(a, b, c, ρ; Z)2(3.45)whereS(nm)lk(a, b, c, ρ) = Xl(a′, b′, c′, ρ′)Xk(a, b, c, ρ)N(nm)lk(a, b, c, ρ)2(3.46)encompasses the full information on the structure constants of the operator algebra.32

4. NORMALIZATION INTEGRALSIN THE THERMAL SERIES4.1.

CONTOUR ORDERINGFrom the analysis of conformal blocks in the previous section we are left withthe problem of evaluating a set of normalization integrals before we can explicitlywrite down the structure constants of the operator algebra. Since the evaluation ofthese integrals is involved we present the method first in detail for the correlators inthe thermal series.

The extension of the method to the general case presents somesubtleties which require the explicit result for the thermal series. This extensionwill be the subject of the next section.As previously mentioned, the type ofanalysis we pursue here is tailored after the treatment of the conformal case in[3,4,5].

There are specific complications to the superconformal case as the readerfamiliar with these references will see in this and the next section.Two types of integrals can be distinguished depending on the number of super-contours. We may have an even or an odd number of contours.

In the even casethe integrals involved are of the formJ(p, q) = T0∞Z1qY1dTitαi (ti −1)β Yi>j(ti −tj −θiθj)ρ1Z0pY1dUauαa(1 −ua)β Ya

For odd integrals33

there are two subcases to considerJk(n −p, p) = T0∞Z1nYn−p+1dTitαi (ti −1)β1Z0n−pY1dTitαi (1 −ti)β ηθkYi>j(ti −tj −θiθj)ρ(4.2)andeJk(n −p, p) = T0∞Z1nYn−p+1dTitαi (ti −1)β1Z0n−pY1dTitαi 1 −(ti)β ηθktkYi>j(ti −tj −θiθj)ρ(4.3)The normalization integrals are onlyJΣ(n −p, p) =nXk=1Jk(n −p, p)eJΣ(n −p, p) =nXk=1eJk(n −p, p)(4.4)The contours in (4.2,3) are ordered as in fig. 4.2.

The evaluation of (4.2,3) isharder than (4.4). Fortunately only the latter is needed in our computations.

Theordering prescription is best illustrated with fig. 4.3 and the integralJm = T0XZ0mY1dTiYi

T0 orders the dθi’s as well as the integrandT0n mY1dθimYi

. dθmmYi t2 > .

. .

> tm= −dθ2dθ1dθ3 . .

. dθm(t2 −t1 −θ2θ1)ρe−iπρ .

. .for t2 > t1 > t3 > .

. .

> tm(4.6)According to this definition no residual phase is encountered in the first case in(4.6) and when we braid (analytically continue) ti around tj for i > j along thecurves in fig. 4.3 from the region ti < tj to the region ti > tj we pick up a phasee−iπ(ρ−1).

When there are two sets of contours (ti, θi), i = 1, . .

. , m; (ua, ωa), a =1, .

. .

, n with ua > ti the odd differentials are ordered as dω1 . .

. dωndθ1 .

. .

dθm foru1 > u2 > . .

. > un > t1 > .

. .

> tm. The Grassmann variable with the greatestreal variable comes first on the left.

With this choice we can define an orderedintegral Im:Im =XZ0dT1T1Z0dT2 . .

.Tm−1Z0dTmYi

Integrals with this ordering are related to Im asin (4.8) but without the ǫm’s.35

4.2. EVEN INTEGRALSWe begin by deriving a set of relations satisfied by (4.1).

In the region of(α, β, ρ) making the integral converge we pull the top contour labelled by p from0 →1 through the upper half plane. This leads to a relation between J(p, q) andJ(p −1, q + 1) (the part of the deformed contour from 1 to ∞) and an integralJ(0)(p −1, q) with one contour running from 0 →−∞.

Similarly we can pull thebottom contour through the lower half plane to obtain another relation betweenthese three integrals. Being careful with phases and ordering prescriptions, the topcontour yieldsJ(p, q) = eiπαJ(0)(p −1, q) −e−iπ(ρ−1)(p−1)−iπβJ(p −1, q + 1)(4.9a)while pulling the bottom contour yieldsJ(p, q) = e−iπ(ρ−1)(p−1)−iπαJ(0)(p −1, q) −eiπβ+iπ(ρ−1)qJ(p −1, q + 1)(4.9b)Eliminating J(0)(p −1, q) impliesJ(p, q) = −eiπ ρ−12 (q−p+1) s(α + β + (ρ −1)(p −1 + q/2))s(α + ρ−12 (p −1))J(p −1, q + 1)(4.10)Iterating this relation we can pull all contours from the region 0 →1 into 1 →∞contours:J(n, 0) = (−1)nn−1Yj=0eiπ ρ−12 (2j−n+1) s(α + β + (ρ −1)(n −1 −j/2))s(α + ρ−12 (n −1 −j))J(0, n)=n−1Yj=0s(α + β + (ρ −1)(n −1 −j/2))s(α + ρ−12 j)J(0, n)n even(4.11)Since Pn−10(2j −(n −1)) = 0, the phase in (4.11) reduces to (−1)n = 1 becausen is even.

We can now change from J(0, n) to J(n, 0) by a split superconformal36

change of variablesUi = (ui, ωi)ui = f(ti)ωi = θip∂ifDiωi =p∂if(4.12)withui = 1tiωi =√−1tiθiDiωi =√−1ti(4.13)J(0, n)(α, β, ρ) = T0∞Z1nY1dTitαi (ti −1)β Yi>j(ti −tj −θiθj)ρ=∞Z1dTndTn−1 . .

. dT1(.

. .

)= J(n, 0)(−1 −α −β −ρ(n −1), β, ρ)(4.14)leading toJ(n, 0)(α, β, ρ) =n−1Yj=0s(α + β + (ρ −1)(n −1 −j/2))s(α + ρ−12 (n −j −1))J(n, 0)(−1 −α −β −ρ(n −1), β, ρ)(4.15)Furthermore, using the change of variables in (4.1)ui = 1 −tiωi =√−1θi(4.16)we obtain a useful symmetryJ(n, 0)(α, β, ρ) = J(n, 0)(β, α, ρ)(4.17)In deriving these formulas we have to be careful in keeping track of the signsoriginating in the exchange of the dθ’s necessary to bring them to the correctordering. We should also recall the remarks at the end of section two about openversus closed contour integrals.

As in [4] (4.10) and (4.11) can be thought of as37

analytic continuations of the function J(n, 0)(α, β) to the complex α-plane. Sincethe integrals we are dealing with have milder singularities than those treated in [4]we need not repeat their arguments here.

The next step consists in determining thebehaviour of J(n, 0)(α, β) as α →∞. This is achieved using a split superconformaltransformation in (4.1) withui = e−ti/αDiθi =√−αe−ti/2α(4.18)Keeping track of the phases carefully we arrive atJ(n, 0)(α, β, ρ) ∼α−n/2−nβ−ρn(n−1)/2 (C0 + C1α−1 + .

. .

)n even(4.19)With a little extra work we find the asymptotic behaviour of odd integrals likeJk(n, 0) = T01Z0nY1dTitαi (1 −ti)β θkYi>j(ti −tj −θiθj)ρn odd(4.20)and ˜Jk(n, 0) (replace in (4.20) θk →θk/tk). The answer isJk(n, 0) ∼eJk(n, 0) ∼α−(n+1)/2−nβ−ρn(n−1)/2 (C′0+C′1α−1+.

. .

)n odd (4.21)Using (4.15),(4.17),(4.19) we can write an Ansatz for Jn(α, β, ρ) = J(n, 0)(α, β, ρ):Jn(α, β, ρ) =n−1Yj=0Γ(1 + α + ρ−12 j + Mj) Γ(1 + β + ρ−12 j + Mj)Γ(1 + α + β + ρ−12 (n −1 + j) + Nj)µn(α, β, ρ)(4.22)From the reflection condition (4.14) we obtain a relation between the integers Mand N:Nn−1−j + Mj = n −1(4.23)The function µ(α, β, ρ) is symmetric under the exchange of α and β and it satisfiesµn(α, β, ρ) = µn(−1 −α −β −ρ(n −1), β, ρ). We can obtain more constraints by38

matching the large α behaviour (4.19) :n−1Xp=0Mp = n(n −2)4n even(4.24)Since µ(α, β, ρ) is analytic in α and behaves as a constant in the large α limit, weconclude that µ is only a function of ρ. We can obtain more information aboutthe Mj’s if we require them not to depend on n. Since the case n = 2 can becomputed explicitly, M0 = M1 = 0, if we subtract (4.24) for n and n+ 2 we obtainthe relationM2k + M2k+1 = 2k(4.25)If [x] stands for the integer part of x, one easily checks that Mj = [j/2] solves(4.24) and (4.25).

As we shall see later this turns out to be the correct answer. Tosummarize this subsection we have learned that for even nJn(α, β, ρ) =n−1Yj=0Γ(1 + α + ρ−12 j + Mj) Γ(1 + β + ρ−12 j + Mj)Γ(1 + α + β + ρ−12 (n −1 + j) + n −1 −Mj)µn(ρ)n−1Xp=0Mp = n(n −2)4(4.26)Next we turn to the odd case.4.3.

ODD NUMBER OF CONTOURSIn deriving relations out of pulling contours in the odd case it will be convenientto write one more entry in the arguments of (4.2,3). We writeJk(0, n −p, p) = Jk(n −p, p)eJk(0, n −p, p) = eJk(n −p, p)(4.27)The first entry keeps track of contours running from 0 →−∞.

The ordering ofcontours is shown in fig. 4.2.

In this case we have to extend the work of [4] to39

reach definite answers. The distinguished contour k in (4.27) (fig.

4.2) may be anyof the n contours. In pulling the top and bottom contours we have to be carefulwith the position of k. We can distinguish five cases:1) 1 < k < n −p1a) Open the top contour between 0 →1Jk(−)(0, n −p, p) = eiπαJk(−)(1, n −p −1, p)(4.28a)−e−iπ(ρ−1)(n−p−1)−iπβJk(−)(0, n −p −1, p + 1)1b) Open the bottom contour between 0 →1Jk(−)(0, n −p, p) = e−iπα−iπ(ρ−1)(n−p−1)Jk−1(−)(1, n −p −1, p)(4.28b)−eiπβ+iπ(ρ−1)pJk−1(−)(0, n −p −1, p + 1)These formulae look cumbersome, but the meaning is simple.

By Jk(−) we meanthat the distinguished variable θk corresponds to the odd integration variable ofthe k-th contour from 0 →1 in fig. 4.2 counting from the bottom.

This is why onthe right-hand side of (4.28b ) the label of J has become k −1(−). By pulling thebottom contour we have changed the numbering.

The arguments (1, n −p −1, p)mean that we have one contour running from 0 →−∞, n −p −1 from 0 →1 andp from 1 →∞. Similarly, if θk is related to a contour from 1 →∞we label thecorresponding integral by Jk(+).Notice that the phases in the contour deformations do not depend on k. Thesame happens in all subsequent cases.2) n −p + 1 < k ≤n2a) Open the top contour between 0 →1Jk(+)(0, n −p, p) = eiπαJk−1(+)(1, n −p −1, p)(4.29a)−e−iπ(ρ−1)(n−p−1)−iπβJk(+)(0, n −p −1, p + 1)40

2b) Open the bottom contour between 0 →1Jk(+)(0, n −p, p) = e−iπα−iπ(ρ−1)(n−p−1)Jk−1(+)(1, n −p −1, p)(4.29b)−eiπβ+iπ(ρ−1)pJk−1(+)(0, n −p −1, p + 1)3) k = 13a) Open the top contourJ1(−)(0, n −p, p) = eiπαJ1(−)(1, n −p −1, p)(4.30a)−e−iπ(ρ−1)(n−p−1)−iπβJ1(−)(0, n −p −1, p + 1)3b) Open the bottom contourJk(−)(0, n −p, p) = e−iπα−iπ(ρ−1)(n−p−1)J(−1)(1, n −p −1, p)(4.30b)−eiπβ+iπ(ρ−1)pJn(+)(0, n −p −1, p + 1)4) k = n −p4a) Open the top contourJn−p(−)(0, n −p, p) = eiπαJ(−1)(1, n −p −1, p)(4.31a)−e−iπ(ρ−1)(n−p−1)−iπβJn−p(+)(0, n −p −1, p + 1)4b) Open the bottom contourJn−p(−)(0, n −p, p) = e−iπα−iπ(ρ−1)(n−p−1)Jn−p−1(−)(1, n −p −1, p)(4.31b)−eiπβ+iπ(ρ−1)pJn−p−1(−)(0, n −p −1, p + 1)5) k = n −p + 15a) Open the top contourJn−p+1(+)(0, n −p, p) = eiπαJn−p(+)(1, n −p −1, p)(4.32a)−e−iπ(ρ−1)(n−p−1)−iπβJn−p−1(+)(0, n −p −1, p + 1)5b) Open the bottom contourJn−p−1(+)(0, n −p, p) = e−iπα−iπ(ρ−1)(n−p−1)Jn−p(+)(1, n −p −1, p)(4.32b)−eiπβ+iπ(ρ−1)pJn−p(+)(0, n −p −1, p + 1)41

Multiplying all the bottom contours relations by eiπα+iπ(ρ−1)(n−p−1) and all the topcontour by e−iπα and then adding the top and subtracting the bottom contourswe arrive atJΣ(0, n −p, p) = −e−iπ ρ−12 (n−1−2p) s(α + β + (ρ −1)(n −1 −p/2))s(α + ρ−12 (n −1 −p))JΣ(0, n −p −1, p + 1)(4.33)I terating (4.33) we obtainJΣ(0, n, 0) = (−1)n−1Yj=0s(α + β + (ρ −1)(n −1 −j/2))s(α + ρ−12 j)JΣ(0, 0, n)(4.34)The same argument can be carried out with eJΣ. The only differences are signchanges in the eJ(−1) term in pulling the top contour in 4a) and in pulling thebottom contour in 3b).

However, in the sum the same cancellation takes place andwe end up witheJΣ(0, n, 0) = (−1)n−1Yj=0s(α + β + (ρ −1)(n −1 −j/2))s(α + ρ−12 j)eJΣ(0, 0, n)(4.35)If we next attempt to relate JΣ(0, 0, n) with JΣ(0, n, 0) by performing the super-conformal change (4.13) we find a surprise:JΣ(0, 0, n)(α, β, ρ) = eJΣ(0, n, 0)(−1 −α −β −ρ(n −1), β, ρ)eJΣ(0, 0, n)(α, β, ρ) = JΣ(0, n, 0)(−1 −α −β −ρ(n −1), β, ρ)(4.36)Hence we need to do some extra work before we can write an Ansatz for the oddintegrals.42

A second important difference arises in the symmetry with respect to the ex-change of α and β. For JΣ the change of variables (4.16) producesJΣ(0, n, 0)(α, β, ρ) = JΣ(0, n, 0)(β, α, ρ)(4.37)while for eJΣ the correct symmetry isα eJΣ(0, n, 0)(α, β, ρ) = β eJΣ(0, n, 0)(β, α, ρ)(4.38)To prove the last equation, writeeJk(α, β) = T01Z0YidUiuαi (1 −ui)β ηωkukYi

The boundary terms vanish at0 and 1. To see that (4.41) vanishes note that Qk = θk ∂∂uk −∂∂θk is the super-symmetry generator acting on (uk, θk).

After summing over k, P Qk annihilatesQi

(4.41) reduces to an integral of the form−T01Z0YidUiXk∂∂θkYiuαi (1 −ui)βηYi

4.4. Startwith Jk(p, 0, n −p).

Pulling the top and bottom contours on the right and elimi-nating the term J(p, 1, n −p −1) we obtainJΣ(p, 0, n −p) = −e−iπ ρ−12 (n−1−2p)s(α + ρ−12 p)s(β + ρ−12 (n −1 −p))JΣ(p + 1, 0, n −p −1)(4.43)IteratingJΣ(0, 0, n) = (−1)n−1Yj=0s(α + ρ−12 j)s(β + ρ−12 j)JΣ(n, 0, 0)(4.44)Now by the SL(2|1) transformation ti →1 −ti, θi →√−1θi we can transform oneintegral into the other:Jk(n, 0, 0)(α, β) = −Jk(0, 0, n)(β, α)(4.45)which together with (4.44) yieldsJΣ(0, 0, n)(α, β) =n−1Yj=0s(α + ρ−12 j)s(β + ρ−12 j)JΣ(0, 0, n)(β, α)(4.46)Using (4.36):eJΣ(0, n, 0)(−1 −α −β −ρ(n −1), β) =n−1Yj=0s(α + ρ−12 j)s(β + ρ−12 j)eJΣ(0, n, 0)(−1 −α −β −ρ(n −1), α)(4.47)44

Definingγ = −1 −α −β −ρ(n −1)(4.48)we obtaineJΣ(0, n, 0)(γ, β, ρ) =n−1Yj=0s(γ + β + (ρ −1)(n −1 −j/2))s(β + ρ−12 j)eJΣ(0, n, 0)(γ, −1 −γ −β −ρ(n −1), ρ)(4.49)This relation can be solved as in the even caseeJΣn (α, β, ρ) = ηfn(α)n−1Yj=0Γ(1 + β + ρ−12 j + Nj)Γ(1 + α + β + ρ−12 (n −1 + j) + n −1 −Nj)µn(ρ)(4.50)Since eJ1 is explicitly calculable, we know that f1(α) = Γ(α). Imposing the sym-metry relation α eJn(α, β) = β eJn(β, α) leads tofn(α) =n−1Yj=0Γ(1 + α + ρ −12j + eNj)(4.51)witheN0 = −1,N0 = 0,eNp = Np ,p > 0(4.52)Summarizing:eJΣn (α, β, ρ) = ηn−1Yj=0Γ(1 + α + ρ−12 j + eNj) Γ(1 + β + ρ−12 j + Nj)Γ(1 + α + β + ρ−12 (n −1 + j) + n −1 −Nj)µn(ρ)(4.53)The large α behaviour yieldsn−1Xp=0Np = (n −1)24n odd(4.54)45

Assuming the Np’s to be independent of n and using N0 = 0 would implyN2k+1 + N2k+2 = 2k + 1(4.55)Using (4.36) we finally obtainJΣn (α, β, ρ) = ηn−1Yj=0Γ(1 + α + ρ−12 j + Nj) Γ(1 + β + ρ−12 j + Nj)Γ(1 + α + β + ρ−12 (n −1 + j) + n −1 −eNj)µn(ρ)eJΣn (α, β, ρ) = ηn−1Yj=0Γ(1 + α + ρ−12 j + eNj) Γ(1 + β + ρ−12 j + Nj)Γ(1 + α + β + ρ−12 (n −1 + j) + n −1 −Nj)µn(ρ)N0 = 0eN0 = −1n−1Xp=0Np = (n −1)24n odd(4.56)To complete the computation we need to determine the integers Mp, Np and thefunctions µn(ρ) for both even and odd n. This we do by relating J2m to JΣ2m−1and JΣ2m+1 to J2m.4.4. FROM J2m TO JΣ2m−1Start with J(0, 2m) and pull one contour into the (0, 1) region as shown in fig.4.5.

Pulling the top and bottom contours yieldsJ(0, 2m) = −e−iπ ρ−12 (2m−1)s(α)s(α + β + ρ−12 (2m −1))J(1, 2m −1)(4.57)In the limit α = −1 + ǫ as ǫ →0, s(α) develops a zero. Since the left-hand side46

does not vanish, the integral on the right-hand side should develop a poleJ(1, 2m −1) = T0∞Z12mY2dTitαi (ti −1)β1Z0dt1dθ1tα1 (1 −t1)β2mY2(ti −t1 −θiθ1)ρ2mYi>j=2T ρij= T0∞Z12mY2dTitαi (ti −1)β1Z0dt1tα1 (1 −t1)β2mY2(ti −t1)ρ ρ2mXk=2θktk −t1!2mYi>j=2T ρij(4.58)The leading divergence is obtained by expanding the integrand in powers of t1/ti.Multiplying by η and taking the limit as ǫ →0 leads toηJ(0, 2m)(−1, β, ρ) = πρe−iπ ρ−12 (2m−1)s(β + ρ−12 (2m −1))eJΣ(0, 2m −1)(−1 + ρ, β, ρ) (4.59)From the reflection formulae (4.14) and (4.36) we arrive atηJ(2m, 0)(−β −ρ(2m −1), β, ρ) = πρe−iπ ρ−12 (2m−1)s(β + ρ−12 (2m −1))JΣ(2m −1, 0)(−β −ρ(2m −2), β, ρ)(4.60)Substituting the product representations in (4.60), collecting all β dependence onone side, and dropping the η factor:Q2m−10Γ(1 −β −ρ(2m −1) + ρ−12 p + Mp)Γ(1 + β + ρ−12 p + Mp)Q2m−20Γ(1 −β −ρ(2m −1) + ρ−12 p + Np)Γ(1 + β + ρ−12 p + Np)× s(β + ρ−12 (2m −1))π= ρe−iπ ρ−12 (2m−1)µ2m−1µ2m×Q2m−10Γ(1 −ρ(2m −1) + (ρ −1)(2m −1 −p/2) + 2m −1 −Mp)Q2m−20Γ(1 −ρ(2m −1) + (ρ −1)(2m −2 −p/2) + 2m −2 −eNp)(4.61)To cancel s(β + ρ−12 (2m −1)) on the left-hand side we take the (2m −1)-th term47

in the numerator and find that only for M2m−1 = m −1 is there a cancellation.Notice that the right-hand side of (4.61) is independent of β, and therefore allβ-dependence should disappear. By inspection one finds thatM2m−1 = m −1,Mp = Np(4.62)is the only way to make (4.61) β-independent.

Once the β-dependence is cancelled,we obtain a recursion relation between µ2m and µ2m−1:µ2m(ρ) = (−1)me−iπ ρ−12 (2m−1)ρ2Q2m−10Γ(1 −ρ−12 p −Mp)Q2m−20Γ(1 −ρ −ρ−12 p −eNp)µ2m−1(ρ)(4.63)4.5. RELATING J2m+1 TO J2mAs in the previous section we start with eJΣ(0, 2m + 1)(α, β, ρ) and pull onecontour into the 0 →1 region.

We obtaineJΣ(0, 2m + 1) = −e−iπ(ρ−1)ms(α)s(α + β + (ρ −1)m)eJΣ(1, 2m)(4.64)Looking at the eJk component in eJΣ we easily learn that the leading α →0 singu-larity in eJΣ comes from eJ1. Taking α = ǫ and letting ǫ →0 we arrive ateJΣ(0, 2m + 1)(0, β, ρ) = πηe−iπ(ρ−1)ms(β + (ρ −1)m) J(0, 2m)(ρ, β, ρ)(4.65)This yields a relation between eJΣ(2m + 1, 0) and J(2m, 0).

Following the steps ofthe previous section leads to M2m = m. Now we can determine Mp to beMp =hp2i(4.66)48

and the recursion relations becomeµ2m(ρ) = (−1)me−iπ ρ−12 (2m−1)ρ2Γ(1−ρ2 )Γ(1 −mρ) µ2m−1(ρ)µ2m+1(ρ) = (−1)m+1e−iπ(ρ−1)mΓ(1−ρ2 )Γ(12 −ρ2m+12) µ2m(ρ)(4.67)Together, they completely determine µn(ρ)µ2m(ρ) = e−iπ(ρ−1)m(m−1/2) ρ2mΓ(1−ρ2 )2mQm1 Γ(1 −ρp)Γ(12 −ρ(p −12)) (4.68a)µ2m+1(ρ) = −e−iπ(ρ+1)m(m+1/2) ρ2mΓ(1−ρ2 )2mQm1 Γ(1 −ρp)Γ(12 −ρ(p + 12))(4.68b)These two formulae can be combined into a single oneµn(ρ) = (−1)neiπMn/2e−iπρ n(n−1)4ρ2MnΓ(1−ρ2 )nQn1 Γ(1 −ρ+12 p + Mp)(4.69)Finally we collect the formulae derived in this section for easy referenceJn(α, β, ρ) =n−1Yp=0Γ(1 + α + ρ−12 p + Mp) Γ(1 + β + ρ−12 p + Mp)Γ(1 + α + β + ρ−12 (n −1 + p) + n −1 −Mp)µn(ρ)JΣn (α, β, ρ) = ηn−1Yp=0Γ(1 + α + ρ−12 p + Mp) Γ(1 + β + ρ−12 p + Mp)Γ(1 + α + β + ρ−12 (n −1 + p) + n −1 −fMp)µn(ρ)eJΣn (α, β, ρ) = ηn−1Yp=0Γ(1 + α + ρ−12 p + fMp) Γ(1 + β + ρ−12 p + Mp)Γ(1 + α + β + ρ−12 (n −1 + p) + n −1 −Mp)µn(ρ)(4.70)µn(ρ) = (−1)neiπMn/2e−iπ ρ−12n(n−1)4ρ2MnΓ(1−ρ2 )nQn1 Γ(1 −ρ+12 p + Mp)fM0 = −1,Mp =hp2i,fMp>0 = Mp>049

It is useful to introduce a new function ˆJn defined byˆJn(α, β, ρ) =n−1Yp=0Γ(1 + α + ρ−12 p + Mp) Γ(1 + β + ρ−12 p + Mp)Γ(1 + α + β + ρ−12 (n −1 + p) + n −1 −Mp)µn(ρ)(4.71)ThenJn(α, β, ρ) = ˆJn(α, β, ρ)n evenα eJΣn (α, β, ρ) = η ˆJn(α, β, ρ)n odd(4.72)If one works instead with path ordered integrals, the relation between I- and J-integrals is given in (4.8). We may as well introduce the functions ˆIn(α, β, ρ) byˆJn(α, β, ρ) = λn(ρ)ǫn(ρ)−1 ˆIn(α, β, ρ)λn(ρ) =nY1s(iρ−12 )s(ρ−12 )ǫn(ρ) =n−1Y0eiπ ρ−12 k(4.73)50

5. NORMALIZATION INTEGRALS: THE GENERAL CASEWe now extend the arguments of the previous section to the case when wehave both +, −screening charges.

There are some important differences in thedetermination of the integers M, N appearing in the arguments of the Γ-functions,but many of the arguments can be translated directly from the thermal case. Wetherefore present less details than in the previous section.

We begin once againwith the case of even integrals.5.1. EVEN INTEGRALSWe want to evaluateJnm(α, β, ρ) =T0ZCinYi=1dTiZSjmYj=1dSjnY1tα′i (1 −ti)β′nYi

5.1.With the notation of section two,ρ′ = 1ρ = α−2 =1α+2α′ = −ρ′αβ′ = −ρ′β(5.2)The ordering prescription is as in the previous chapter, and we should notice thatthe coupling terms (ti −sj −θiωj)−1 do not contribute to the monodromy if weinclude the signs coming from the exchange of dTi and dSj. One can check as in [4]that exchanging the C and S contours does not change the answer.

This impliesthat the monodromies of the conformal blocks will be given as a product of themonodromy matrices for the thermal integrals obtained by ignoring the couplingterms. In all the contour pulling manipulations the Ci and Sj contours do not feeleach other.

To compute (5.1) we define first the integrals J p′q′pq(α, β, ρ) asshown in fig. 5.2.51

By opening the top Cp′ and the bottom C1 contours we can decrease p′ by oneunit and increase q′ by one unit. In this way we can move all the 0 →1 contoursp′ to 1 →∞contours.

After we are finished with the C-type contours we applythe same procedure to the S-type contours. The final result isJ n0m0(α, β, ρ) =(−1)n+mn−1Y0s(α′ + β′ + (ρ′ −1)(n −1 −i/2))s(α′ + ρ′−12 i)m−1Y0s(α + β + (ρ −1)(m −1 −i/2))s(α + ρ−12 i)J 0n0m(α, β, ρ)(5.3)The matrix label ⊃a′b′ab means that there are a′ (resp.

a) Q−(resp. Q+) contoursfrom 0 →1 and b′ (resp.

b) Q−(resp. Q+) contours from 1 →∞.

Next we usethe split superconformal transformation ti →1/ti, si →1/si, and keeping in mindthe remarks in section 2.4 we obtainJnm(α, β, ρ) =n−1Y0s(α′ + β′ + (ρ′ −1)(n −1 −i/2))s(α′ + ρ′−12 i)m−1Y0s(α + β + (ρ −1)(m −1 −i/2))s(α + ρ−12 i)Jnm(−1 −α −β −ρ(m −1) + n, β, ρ)(5.4)This reflection property suggests the AnsatzJnm(α, β, ρ) =n−1Y0Γ(1 + α′ + ρ′−12 j + M′j) Γ(1 + β′ + ρ′−12 j + M′j)Γ(1 + α′ + β′ + ρ′(n −1) −m −ρ′−12 j −M′j)m−1Y0Γ(1 + α + ρ−12 j + Mj) Γ(1 + β + ρ−12 j + Mj)Γ(1 + α + β + ρ(m −1) −n −ρ−12 j −Mj)µnm(ρ)(5.5)With foresight we write µnm(ρ) without any dependence on α, β. The Ansatz (5.5)is symmetrical under the exchange of α and β because the original integral had thissymmetry.

Matching the large α behaviour of (5.1) and (5.5) leads to a constraint52

on the integers M′p, Mp2m−1Xp=0Mp + 2n−1Xp=0M′p = m(m −2)2+ n(n −2)2−nm(5.6)Before analyzing the case (n + m) odd, we can relate the even to the odd case aswe did in the thermal case. Starting with J 0n0mand pulling one S-contourinto the (0, 1) region we obtainJ 0n0m= −e−iπ ρ−12 (m−1)s(α)s(α + β + ρ−12 (m −1))J 0n1m−1(5.7)ExplicitlyJ 0n1m−1= T0∞Z1nY1dTitα′i (ti −1)β′ Yi

Since we are interested in the limit α = −1 + ǫ as ǫ →0, wecan expand in powers of s1 and pick up the pole term in ǫ which cancels the zerofrom s(α) in (5.7). Multiplying by the odd η variable we obtain after some simplemanipulationsηJ 0n0m(−1, β, ρ) = πe−iπ ρ−12 (m−1)s(β + ρ−12 (m −1))eJΣ 0n0m−1(−1 + ρ, β, ρ)(5.9)(for the definition of the odd integral eJΣ see below).

The same argument worksfor the C-contours. Now we set α′ = −1 + ǫ, take the small ǫ limit and obtainηJ 0n0m(ρ, β, ρ) = −πρ′e−iπ ρ′−12(n−1)s(β′ + ρ′−12 (n −1))eJΣ 0n−10m(−1 + ρ, β, ρ) (5.10)Using (5.3) and a similar formula for eJΣ to be derived in the next subsection we53

obtainηJnm(−β−ρ(m−1)+n, β, ρ) = πe−iπ ρ−12 (m−1)s(β + ρ−12 (m −1))eJΣn,m−1(−β−ρ(m−1)+n, β, ρ)(5.11)ηJnm(−1−β−ρm+n, β, ρ) = −πρ′e−iπ ρ′−12(n−1)s(β′ + ρ′−12 (n −1))eJΣn−1,m(−1−β−ρm+n, β, ρ)(5.12)Equations (5.11,12) will allow us to obtain recursion relations for µn,m(ρ).5.2. ODD INTEGRALSWe consider next the (n + m) odd case.

We define two types of integrals JΣnm,eJΣnm.JΣnm(α, β, ρ) =T0ZCinYi=1dTiZSjmYj=1dSj ρηmX1ωk −ηnX1θk! nY1tα′i (1 −ti)β′nYi

The contours Ci, Si are as shown in fig. 5.1.

Similarly wedefineeJΣnm(α, β, ρ) = T0ZCinYi=1dTiZSjmYj=1dSj ρηmX1ωksk−ηnX1θktk! {same as in (5.13)}(5.14)For contour manipulations it is convenient to define Jk′ 0n−pp0m−qqandJk 0n−pp0m−qqas in fig.

5.3. The superindex k′ indicates that the factor θk(θk/tk) belongs to the C-contours, and k that it belongs to the S contours.

The54

matrix of labels a′b′c′abccounts contours. The first column indicates the con-tours from 0 →−∞.

The second column counts contours from 0 →1 and thelast column from 1 →∞. The first row refers to Q−-contours and the second toQ+-contours.

The arguments leading from (4.28) to (4.35) can be repeated herefor both JΣ and ˜JΣ. Using the definitions of fig.

5.3. we can write JΣnm asJΣnm = ρmXk=1Jk 0n00m0−nXk′=1Jk′ 0n00m0(5.15)Repeating (4.28)−(4.35) in the present context is more cumbersome and leads toJΣ 0n00m0(α, β, ρ) = (−1)n+mn−1Yj=0s(α′ + β′ + (ρ′ −1)(n −1 −j/2))s(α′ + ρ′−12 j)m−1Yj=0s(α + β + (ρ −1)(m −1 −j/2))s(α + ρ−12 j)JΣ 00n00m(α, β, ρ)(5.16)The same relation holds for eJΣ. The change ti →1/ti; si →1/si mixes JΣ andeJΣ:JΣ 00n00m(α, β, ρ) = eJΣ 0n00m0(−1 −α −β −ρ(m −1) + n, β, ρ)eJΣ 00n00m(α, β, ρ) = JΣ 0n00m0(−1 −α −β −ρ(m −1) + n, β, ρ)(5.17)As in the thermal case we can pull the 1 →∞to 0 →−∞contoursJΣ 00n00m(α, β, ρ) = (−1)n−1Yj=0s(α′ + ρ′−12 j)s(β′ + ρ′−12 j)m−1Yj=0s(α + ρ−12 j)s(β + ρ−12 j)JΣ n00m00(α, β, ρ)(5.18)Now changing the variables ti →1 −ti; si →1 −si yieldsJΣ n00m00(α, β, ρ) = −JΣ 00n00m(β, α, ρ)(5.19)55

This identity together with (5.17,5.18) implieseJΣnm(γ, −1 −α −γ −ρ(m −1) + n,ρ) =n−1Yj=0s(α′ + ρ′−12 j)s(α′ + γ′ + ρ′(n −1) −m −ρ′−12 j)m−1Yj=0s(α + ρ−12 j)s(α + γ + ρ(m −1) −n −ρ−12 j)eJΣnm(γ, α, ρ)(5.20)As in the thermal case we can show thatα eJΣnm(α, β, ρ) = β eJΣnm(β, α, ρ)(5.21)This together with (5.20) allows us to write an Ansatz for eJΣnm and JΣnm. The casen + m = 1 can be computed explicitly.

Introducing the integers Ni, N′i, eNi, eN′i weobtain the Ans¨atzeeJΣnm(α, β, ρ) = ηn−1Y0Γ(1 + α′ + ρ′−12 j + eN′j) Γ(1 + β′ + ρ′−12 j + N′j)Γ(1 + α′ + β′ + ρ′(n −1) −m −ρ′−12 j −N′j)m−1Y0Γ(1 + α + ρ−12 j + eNj) Γ(1 + β + ρ−12 j + Nj)Γ(1 + α + β + ρ(m −1) −n −ρ−12 j −Nj)µnm(ρ)(5.22)andJΣnm(α, β, ρ) = ηn−1Y0Γ(1 + α′ + ρ′−12 j + N′j) Γ(1 + β′ + ρ′−12 j + N′j)Γ(1 + α′ + β′ + ρ′(n −1) −m −ρ′−12 j −eN′j)m−1Y0Γ(1 + α + ρ−12 j + Nj) Γ(1 + β + ρ−12 j + Nj)Γ(1 + α + β + ρ(m −1) −n −ρ−12 j −eNj)µnm(ρ)(5.23)Matching the large α behaviour we obtain a relation for the integers Np, N′p, eNp, eN′p:n−1X0eN′p + N′p +m−1X0eNp + Np = −12 + n(n −2)2+ m(m −2)2−nm(5.24)Finally we reduce to an even number of contours by taking α = ǫ or α′ = ǫ, ǫ →056

as in the previous subsection. Omitting the details, the results areeJΣ 0n0m(0, β, ρ) = ηπρe−iπ ρ−12 (m−1)s(β + ρ−12 (m −1))J 0n0m−1(ρ, β, ρ)eJΣ 0n0m(0, β, ρ) = −ηπe−iπ ρ′−12(n−1)s(β′ + ρ′−12 (n −1))J 0n−10m(−1, β, ρ)(5.25)Equivalently, using (5.17),JΣnm(−1 −β −ρ(m −1) + n, β, ρ) =ηπρe−iπ ρ−12 (m−1)s(β + ρ−12 (m −1))Jn,m−1(−1 −β −ρ(m −1) + n, β, ρ)JΣnm(−1 −β −ρ(m −1) + n, β, ρ) = −ηπe−iπ ρ′−12(n−1)s(β′ + ρ′−12 (n −1))Jn−1,m(−1 −β −ρ(m −1) + n, β, ρ)(5.26)5.3.

COMPUTATION OF µn,m(ρ)To complete the computation we have to determine the integers Mp, Np, etc.This can be done by using the recursion relations established in the two previoussubsections. There is, however, a simpler method of obtaining the same answer.Consider first the even caseJnm =n−1Y0Γ(1 + α′ + ρ′−12 p + M′p) Γ(1 + β′ + ρ′−12 p + M′p)Γ(1 + α′ + β′ + ρ′(n −1) −m −ρ′−12 p −M′p)m−1Y0Γ(1 + α + ρ−12 p + Mp) Γ(1 + β + ρ−12 p + Mp)Γ(1 + α + β + ρ(m −1) −n −ρ−12 p −Mp)µnm(ρ)The integers Np, Mp, N′p, M′p are independent of ρ.Hence if we take the limitρ →−1 (still within the domain of definition of the integrals if α, β > 0), the57

original integral (5.1) becomesJnm = T01Z0n+mY1dTiuαi (1 −ui)β Yi

. , n −1Mp = Cn+p −np = 0, 1, .

. .

, m −1(5.29)There is a certain arbitrariness in this choice. We could have taken instead M′p =Cm+p −m, Mp = Cp.

This however does not affect the final result. An argumentsimilar to the one employed in the thermal case leads to the same answer.

Noticethe dependence of M on n. The same analysis can be carried out for JΣnm and itleads toN′p = CpNp = −n + Cn+peN′p = eCpeNp = −n + eCn+p(5.30)Then, when n ̸= 0, eN′0 = −1 and when n = 0 (i.e. there is no Qn−10) eN0 = −1.With this choice the symmetries α eJΣ(α, β) = β eJΣ(β, α) and JΣ(α, β) = JΣ(β, α)are automatically satisfied.

In the reduction from Jn,m to Jn−1,m or Jn,m−1 wehave to be careful in taking into account the n-dependence in M, N. The recursionrelations obtained for µnm areµnm(ρ) = (−1)1−Mme−iπ ρ−12 (m−1)ρnρ2Γ(1−ρ2 )Γ(1 + n −ρ+12 m + Mm)µn,m−1(ρ)(n+m) even(5.31)58

andµnm(ρ) = (−1)1−Mme−iπ ρ−12 (m−1)ρnΓ(1−ρ2 )Γ(1 + n −ρ+12 m + Mm)µn,m−1(ρ)(n+m) odd(5.32)The only difference between these two expressions is the factor of ρ/2. Iterating therecursion relations we end up in the thermal case which has already been solved.After some algebraic manipulations we arrive at (up to some irrelevant sign), forn ≥1µnm(ρ) = ρnmρ′2M ′n ρ2Mm+M ′n+1 e−iπ ρ′−12n(n−1)4e−iπ ρ′−12m(m−1)4Γ(1−ρ′2 )nΓ(1−ρ2 )mQn1 Γ(1 −ρ′+12 p + M′p) Qm1 Γ(1 + n −ρ+12 p + Mp)(5.33)For n = 0 we have the thermal resultJΣ0,m(α, β, ρ) = ρJΣm(α, β, ρ)µ0,m(ρ) =12M ′mρMm+1 e−iπ ρ−12m(m−1)4Γ(1−ρ2 )mQn1 Γ(1 −ρ+12 p + M′p)(5.34)FinallyJnm(α, β, ρ) =n−1Y0Γ(1 + α′ + ρ′−12 p + M′p) Γ(1 + β′ + ρ′−12 p + M′p)Γ(1 + α′ + β′ + ρ′(n −1) −m −ρ′−12 p −M′p)m−1Y0Γ(1 + α + ρ−12 p + Mp) Γ(1 + β + ρ−12 p + Mp)Γ(1 + α + β + ρ(m −1) −n −ρ−12 p −Mp)µnm(ρ)(n + m) even(5.35a)59

JΣnm(α, β, ρ) = ηn−1Y0Γ(1 + α′ + ρ′−12 p + M′p) Γ(1 + β′ + ρ′−12 p + M′p)Γ(1 + α′ + β′ + ρ′(n −1) −m −ρ′−12 p −fM′p)m−1Y0Γ(1 + α + ρ−12 p + Mp) Γ(1 + β + ρ−12 p + Mp)Γ(1 + α + β + ρ(m −1) −n −ρ−12 p −fMp)µnm(ρ)(n + m) odd(5.35b)whereM′p =hp2iMp = −n +n + p2fM′p = eCpfMp = −n + eCn+pwitheC0 = −1, eCp>0 =hp2i(5.36)These results can be unified in a way useful for the computation of N(nm)lk, thenormalization constants in the conformal blocks. DefineˆJnm(α, β, ρ) = {same as in (5.35a ), (n + m) even or odd}(5.37)Up to an irrelevant sign one can show thatJnm(α, β, ρ) = ˆJnm(α, β, ρ)(n + m) evenα′ eJΣnm(α, β, ρ) = η ˆJnm(α, β, ρ)(n + m) odd(5.38)In terms of ordered integrals,ˆJnm(α, β, ρ) = λn(ρ′)ǫn(ρ′)−1λm(ρ)ǫm(ρ)−1 ˆInm(α, β, ρ)λm(ρ) =mY1s(iρ−12 )s(ρ−12 )ǫm(ρ) =m−1Y0eiπ ρ−12 k(5.39)In all our previous results we have systematically ignored some signs because inthe quantities of interest only the square of the normalization constants is used.60

6. STRUCTURE CONSTANTS OFTHE OPERATOR ALGEBRAWe have now established all the necessary formulae needed for the computationof the quantities S(m)kout of which we shall extract the structure constants.

Wefind it convenient to deal first with the thermal case.6.1. THERMAL STRUCTURE CONSTANTSRecall that we are considering the NS thermal four-point functions, representedwith the help of vertex operators as⟨Φ1,t(Z4)Φ1,q(Z3)Φ1,n(Z2)Φ1,s(Z1)⟩= ⟨V¯α4(Z4)Vα3(Z3)Vα2(Z2)Vα1(Z1)Qm−1+⟩(6.1)It was shown in previous sections that the four-point correlator takes the form⟨V¯α4Vα3Vα2Vα1Qm−1+⟩∼mX1S(m)kF(m)k(a, b, c, ρ; Z)2(6.2)The quantities S(m)kwere given in (3.31):S(m)k(a, b, c, ρ) = Xk(a, b, c, ρ)N(m)k(a, b, c, ρ)2(6.3)withN(m)k(a, b, c, ρ) = (−1)m−1 ˆIm−k(−1 −a −b −c −ρ(m −2), b, ρ) ˆIk−1(a, c, ρ) (6.4)By writing X(m)kinstead of Xk in (6.3) we explicitly indicate that we make aconvenient rescaling of Xk (in other words we choose a particular value for Xm in(3.16)).

To compute X(m)kwe need the matrices βkl(a, b, c, ρ) which can be derived61

from the Dotsenko and Fateev results [4] through the substitution ρ →(ρ −1)/2.This yieldsβmk(a, b, c, ρ) =m−k−1Y0s(1 + a + b + c + (ρ −1)(m −2) −ρ−12 i)s(b + c + (ρ −1)(m −2) −ρ−12 (m −k −1 + i))k−2Y0s(1 + b + ρ−12 i)s(b + c + ρ−12 (k −2 + i))βkm(a, b, c, ρ) =Qm−11s(iρ−12 )Qk−11s(iρ−12 ) Qm−k1s(iρ−12 )m−k−1Y0s(1 + c + ρ−12 (k −1 + i))s(b + c + ρ−12 (m + k −3 + i))k−2Y0s(1 + b + ρ−12 (m −k + i))s(b + c + ρ−12 (m −2 + i))(6.5)and after some algebra . .

.X(m)k= βmm(a, b)βmk(b, a)βmm(b, a)βkm(a, b)C(m)m−2Y0s(1 + a + ρ−12 i)s(1 + c + ρ−12 i)s(a + c + ρ−12 (m −2 + i))m−1Y1s(iρ −12)= C(m)k−1Y1s(iρ −12)k−2Y0s(a + ρ−12 i)s(1 + c + ρ−12 i)s(a + c + ρ−12 (k −2 + i))m−kY1s(iρ −12)m−k−1Y0s(1 + b + ρ−12 i)s(a + b + c + (ρ −1)(m −2) −ρ−12 i)s(a + c + (ρ −1)(m −2) −ρ−12 (m −k −1 + i))(6.6)We have introduced the constants C(m) = (−1)m−1π2−2mΓ(1+ρ2 )2m−2 for laterconvenience.Repeatedly using the identity s(x)Γ(x) = π/Γ(1 −x) and doingsome appropriate shifts in the arguments of the sine functions in (6.6) we obtain areasonably nice expression for S(m)k:62

S(m)k=ρ22Mk−1 k−1Y1∆(ρ + 12i −Mi)k−2Y0∆(−a −c −ρ(k −2) + ρ −12i + Mi)∆(1 + a + ρ −12i + Mi)∆(1 + c + ρ −12i + Mi)ρ22Mm−k m−kY1∆(ρ + 12i −Mi)m−k−1Y0∆(1 + a + c + ρ(k −1) + ρ −12i + Mi)∆(1 + b + ρ −12i + Mi)∆(−a −b −c −ρ(m −2) + ρ −12i + Mi)(6.7)where we have defined ∆(x) = Γ(x)/Γ(1 −x) and Mn = [n2].This equation is more symmetrical that it appears at first sight. Since we restrictourselves to the thermal subalgebra, the only way to meet the charge screeningrequirement is by taking a single conjugate vertex operator, which we take to beVα4.

Defining d = α+α4 and using the charge screening condition we obtain¯d = −1 −a −b −c −ρ(m −2)(6.8)Then the second product of S(m)kin (6.7) becomesρ22Mm−k m−kY1∆(ρ + 12i −Mi)m−k−1Y0∆(−b −¯d −ρ(m −k −1) + ρ −12i + Mi)∆(1 + b + ρ −12i + Mi)∆(1 + ¯d + ρ −12i + Mi)(6.9)with the same structure as the first product in (6.7). Introducing explicitly theKac labels for the vertex operators, we get the different parameters :a = α1,sα+ = 1 −s2ρb = α1,qα+ = 1 −q2ρc = α1,nα+ = 1 −n2ρ¯d = α1,tα+ = −1 + 1 + ¯t2ρ(6.10)where s, n, q, ¯t are positive odd integers (NS sector) related to the number of screen-63

ing charges m −1 throughm = 12(s + n + q −¯t)(6.11)Then, S(m)kbecomesSk(snq¯t) =ρ22Mk−1 k−1Y1∆(ρ + 12i −Mi)k−2Y0∆((s + n2−k + 1)ρ + ρ −12i + Mi)∆(1 + 1 −s2ρ + ρ −12i + Mi)∆(1 + 1 −n2ρ + ρ −12i + Mi)ρ22Mm−k m−kY1∆(ρ + 12i −Mi)m−k−1Y0∆(1 + (q −¯t2−m + k)ρ + ρ −12i + Mi)∆(1 + 1 −q2ρ + ρ −12i + Mi)∆(1 + ¯t2ρ + ρ −12i + Mi)(6.12)From this formula we can read offthe asymmetric structure constants. The indexk labels the intermediate channels contributing to the four-point function.

We canestablish also the connection between the number k and the conformal dimensionof the field exchanged in the corresponding internal channel. This is achieved bychoosing a configuration of superpoints Zi in (6.1) such that|Z12| ∼|Z34| ∼r ≪R ∼|Z13| ∼|Z24|(6.13)Evaluating the four-point function using the OPE of the fields at Z1, Z3 we canwrite⟨Φ1,t(Z4)Φ1,q(Z3)Φ1,n(Z2)Φ1,s(Z1)⟩∼∼Xpr−2(hs+hn+hq+h¯t−2hp)C ¯p¯tqCpns⟨[Φ1,p(Z3)][Φ1,p(Z1)]⟩(6.14)For brevity we collectively denote by Cpns the two structure constants Apns andBpns and by [Φ1,p(Z1)] the superconformal tower of descendant fields including init the factors Z−1/221when necessary.

This fine structure of the OPE and of the64

superconformal blocks is not necessary in the present discussion. For the choice(6.13) we obtainF(m)k(a, b, c, ρ; Z) = z(k−1)( 12+a+c+ ρ2(k−2)) (1 + .

. .

)(6.15)Here again we discard the occasional factors z−1/2. Including the factors relating(6.1) to (6.2) and after some algebra we arrive atp = s + n + 1 −2k−¯p = −¯t + q + 1 −2(m −k + 1)(6.16)Furthermore we can identify the constantsSk(snq¯t) = C ¯p¯tqCpns(6.17)Finally we can read offthe asymmetric structure constantsCpns =ρ22Mk−1 k−1Y1∆(ρ + 12i −Mi)k−2Y0∆(1 + 1 −s2ρ + ρ −12i + Mi)∆(1 + 1 −n2ρ + ρ −12i + Mi)∆(1 + p2ρ + ρ −12i + Mi)C ¯p¯tq =ρ22Ml−1 l−1Y1∆(ρ + 12i −Mi)l−2Y0∆(1 + 1 −q2ρ + ρ −12i + Mi)∆(1 + ¯t2ρ + ρ −12i + Mi)∆(1 + 1 −¯p2ρ + ρ −12i + Mi)(6.18)where the integersk = 12(s + n −p + 1)l = 12(q + ¯p −¯t + 1)(6.19)are now chosen to be functions of the quantum numbers s, n, p and q, p, t. It isvery interesting to notice that although we had to distinguish in our analysis insection 3.1 between even and odd structure constants, we end up here with a65

common expression for both. This will be the same when we compute the physicalstructure constants in the next section.

Using the analyticity properties of the Γfunctions, it is easy to see that the structure constants we have found reproducethe correct fusion rules mentioned in section two.6.2. GENERAL STRUCTURE CONSTANTSThis subsection follows the steps of the previous one except for the fact thatthe computations are more tedious.

We will give few details. Once again we areinterested in⟨Φt′,t(Z4)Φq′,q(Z3)Φn′,n(Z2)Φs′,s(Z1)⟩= ⟨V¯α4(Z4)Vα3(Z3)Vα2(Z2)Vα1(Z1)Qn−1−Qm−1+⟩∼Xk,lS(nm)lkF(nm)lk(a, b, c, ρ; Z)2(6.20)The quantity S(nm)lkis given in (3.46)S(nm)lk(a, b, c, ρ) = Xl(a′, b′, c′, ρ′)Xk(a, b, c, ρ)N(nm)lk(a, b, c, ρ)2(6.21)For X(n)l, X(m)kwe take the same normalization as in (6.6).

From (3.42) we obtainN(nm)lk(a, b, c, ρ) = ˆIn−l,m−k(−a−b−c−ρ(m−2)+n−2, b, ρ) ˆIl−1,k−1(a, c, ρ) (6.22)Combining X(n)lwith the product of Γ-functions containing the primed quanti-ties and similarly for X(m)kfor the unprimed quantities, we end up after a long66

calculation withS(nm)lk= ρ2(l−1)(k−1)ρ′22M ′l−1 ρ22Mk−1+2M ′lk−1Y1∆(1 −l + ρ + 12i −Mi)l−1Y1∆(ρ′ + 12i −M′i)k−2Y0∆(1 + a + ρ −12i + Mi)∆(1 + c + ρ −12i + Mi)∆(−a −c −ρ(k −2) + l −1 + ρ −12i + Mi)l−2Y0∆(1 + a′ + ρ′ −12i + M′i)∆(1 + c′ + ρ′ −12i + M′i)∆(−a′ −c′ −ρ′(l −2) + k −1 + ρ′ −12i + M′i)×ρ2(n−l)(m−k)ρ′22M ′n−l ρ22Mm−k+2M ′n−l+1m−kY1∆(l −n + ρ + 12i −Ni)n−lY1∆(ρ′ + 12i −M′i)m−k−1Y0∆(1 + b+ρ −12i + Ni)∆(a + c + ρ(k −1) −l + 2 + ρ −12i + Ni)∆(1 −a −b −c −ρ(m −2) + n −2 + ρ −12i + Ni)n−l−1Y0∆(1 + b′+ρ′ −12i + M′i)∆(a′ + c′ + ρ′(l −1) −k + 2 + ρ′ −12i + M′i)∆(1 −a′ −b′ −c′ −ρ′(n −2) + m −2 + ρ′ −12i + M′i)(6.23)whereMi = 1 −l +l −1 + i2Ni = l −n +n −l + i2M′i = i2(6.24)Using the charge screening condition arising from (6.20) we define67

¯d = ¯α4α+ = −a −b −c −ρ(m −2) + n −2¯d′ = ¯α4α−= −a′ −b′ −c′ −ρ′(n −2) + m −2(6.25)Which helps make the second set of products in (6.23) more similar to the firstone.Repeating the arguments leading to (6.16) we find the Kac labels of theintermediate channelsp = s + n + 1 −2kp′ = s′ + n′ + 1 −2l(6.26)With the simplifying notation CPNS = C(p′,p)(n′,n),(s′,s) we findS(nm)lk= eC¯P¯TQCPNS(6.27)Finally, introducing the Kac labels for the parameters a, b, c . .

. we obtain theasymmetric structure constants:CPNS = ρ2(l−1)(l′−1)ρ′22M ′l′−1 ρ22Ml−1+2M ′l′l−1Y1∆(1 −l′ + ρ + 12i −Mi)l′−1Y1∆(ρ′ + 12i −M′i)l−2Y0∆(1 + s′2+ 1 −s2ρ + ρ −12i + Mi)∆(1 + n′2+ 1 −n2ρ + ρ −12i + Mi)∆(1 −p′2+ 1 + p2ρ + ρ −12i + Mi)l′−2Y0∆(1 + s2+ 1 −s′2ρ′ + ρ′ −12i + M′i)∆(1 + n2+ 1 −n′2ρ′ + ρ′ −12i + M′i)∆(1 −p2+ 1 + p′2ρ′ + ρ′ −12i + M′i)(6.28)withl = 12(s + n −p + 1)l′ = 12(s′ + n′ −p′ + 1)Mi = 1 −l′ +l′ −1 + i2M′i = i268

We do not write explicitly the other structure constants eCPT Q since they are simplyrelated to (6.28) byeC¯P¯TQ = C¯T¯PQ(6.29)One readily checks that these structure constants ( non-symmetrical ) do not re-produce the correct fusion rules due to some cancellations between zeroes and polesof various ∆factors. The physical structure constants do agree however with thecorrect fusion rules, and they are the subject of the next section.A direct application of this result is the evaluation of some surface integralscorresponding to correlators where the screening charges are integrated over thewhole plane instead of contours.

Consider the three-point functionJ(2D)l′l(a, b, ρ) = limR→∞R4hM⟨VαM(R, Rη)VαN(1, 0)VαS(0, 0)Zl′Y1d2Z′i Vα−(Z′i, Z′i)ZlY1d2Zi Vα+(Zi, Zi)⟩=Zl′Y1d2Z′iZlY1d2Zi ξl′Yi=1|z′i|2a′|1 −z′i|2b′l′Yi

Then with the help of (6.28) we can express it asJ(2D)l′l(a, b, ρ) = (−1)M ′l′+l πl′+ll′!l!∆(1 −ρ′2)l′∆(1 −ρ2)lρ2l′lρ′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)(6.31)with Mi = −l′ +hl′+i2i, M′i = i2. The proportionality factor in (6.31) is deter-mined by taking the limit ρ →0 for the thermal case, in which limit the surfaceintegrals decouple and are easily evaluated.

In deriving these results we choosedthe convention thatRd2θ|θ|2 = 1 and we omitted to write the factor |η|2 thatappears in the right-hand side when (l′ + l) is odd. Equation (6.31) is indeed thegeneralization to the super case of equation (B.10) in [4].70

7. PHYSICAL STRUCTURE CONSTANTSAll the material necessary for the computation of the physical structure con-stants has already been collected in previous sections.We follow closely themethodology of [5].

By physical structure constants we mean the constants DPSNentering the OPE of two NS primary fieldsΦS(Z1, ¯Z1)ΦN(Z2, ¯Z2) =XPDPSN |Z12|−2(hS+hN−hP ) [ΦP (Z2, ¯Z2)] oddeven(7.1)For convenience S = (s′, s), . .

. and no distinction is made between the odd andeven parts of this expansion (this can easily be done by counting screening chargesin the three-point function).

We shall impose the normalization conditionD1SS = 1(7.2)corresponding to a diagonal two-point function⟨ΦS(Z1, ¯Z1)ΦN(Z2, ¯Z2)⟩= δS,N |Z12|−4hS(7.3)and to totally symmetric structure constants. Their determination is made withthe help of the quantities SP (TQSN) defined in section 6.

Consider a four-pointfunction with the choice of arguments as in (6.13),⟨ΦS(Z4)ΦN(Z3)ΦS(Z2)ΦN(Z1)⟩=XPDPSNDPSNr4(hS+hN−hP )⟨[ΦP (Z3)][ΦP (Z1)]⟩⟨ΦS(Z4)ΦS(Z3)ΦN(Z2)ΦN(Z1)⟩=XPDPSSDPNNr4(hS+hN−hP )⟨[ΦP (Z3)][ΦP (Z1)]⟩(7.4)also written in section 6 as⟨SNSN⟩∼XPSP(SNSN)FP(Z)2⟨SSNN⟩∼XPSP(SSNN)FP(Z)2(7.5)With the present normalization the coefficient at the main singularity correspond-ing to the identity intermediate channel is equal to unity whereas S1(SSNN)71

generally is not equal to 1. Hence the appropriate definition for the square of thephysical structure constants isDPSN2= SP(SNSN)S1(SSNN)(7.6)Using the asymmetric structure constants computed in section 6 we arrive atDPSN2=eCPSNCPSNeC1SSC1NN= CSPNCPSNC1NN(7.7)We shall shortly prove the relationCSNP ≡C−PSN = −ρ4∆(ρ −12)∆(ρ′ + 12)C1PPCPSN(7.8)showing that C1PP plays the role of a metric for raising or lowering indices.

Sub-stituting in (7.7) we obtainDPSN = −4ρ′∆(3 −ρ2)∆(1 −ρ′2)C1SSC1NNC1PP−1/2 CSNP(7.9)As expected we see the symmetry of DPSN under interchange of pairs of indices.The proof of (7.8) is tedious and will be roughly sketched here. One uses the∆-functions properties listed below∆(1 −x) = ∆(x)−1∆(x + n) = (−1)nn−1Y0(i + x)2∆(x)∆(x −n) = (−1)nn−1Y0(−1 −i + x)−2∆(x)(7.10)In the definition of CPSN the bounds of the products arel′ = 12(s′ + n′ −p′ + 1)l = 12(s + n −p + 1)(7.11)72

whereas in CSNP = C−PSN they areel′ = 12(s′ + n′ + p′ + 1)el = 12(s + n + p + 1)(7.12)We first show that in the ratio CSNP /CPSN the (s′, s) and (n′, n) dependence cancelout (up to some power of ρ). This is achieved using the simple relations for theproductsl−2Y0f(i) =l−2Y0f(l −2 −i)el−2Y0f(i) =l−2Y0f(i)p−1Y0f(i + l −1)(7.13)Then the (p′, p) dependent part of CSNP cancels against µl′l (and similarly for the(p′, p) dependence in CPSN against eµ˜l′˜l) up to some factors that turn out to be pro-portional to C1PP.

The NS condition is frequently used during these simplifications.The determination of the power of ρ in (7.8) is a rather delicate issue. Actually,CSNP contains poles and zeroes that cancel each other, and the exponent of ρ is adirect consequence of the regularization procedure.

We confirmed the consistencyof this procedure by comparing the result obtained for DPSN using either (7.7) or(7.9). More on this ?73

APPENDIXThis appendix is devoted to the computation of the matrix elements βmk en-tering in the linear expansionI(m)m (a, b, c, ρ; Z) =mXk=1βmkeI(m)k(a, b, c, ρ; Z)(A.1)From the normalization procedure (3.19) for I(m)kand the relation between I(m)kand eI(m)kwe know that when z →1, eI(m)khas the singular behavioreI(m)k(Z) ∼(1 −z)(k−1)(1/2+b+c+(k−2)ρ/2) ( integral )(A.2)As in (3.19) the integral is not necessarily a regular function as z →1. Dependingon the values of k, m it may exhibit a (1 −z)−1/2 singular behaviour.

Neverthelessthe power in front of the integral in (A.2) characterizes the block sufficiently.The procedure to evaluate βmk is to find in I(m)kthe same singular behaviouras z →1 as in (A.2).The coefficient in front of this divergence will only beproportional to βmk because we are not taking into account the normalization ofthe blocks eI(m)k(Z). We start with the integral representation of I(m)mI(m)m= z∆01Z0dSi(1 + a1ηθ + a2z1/2ηXωi)m−1Y1sai (1 −zsi)b(1 −si −θωiz1/2 )c Yi

Performing on the first k −1 variablessi the change of variablesti =1 −z1 −si + 1 −zθi =ti(1 −z)1/2 ωi(A.4)and letting:ǫ = ti(si = 0) = 1 −z2 −z →0when z →11ǫ = sk−1(tk−1) = 1 −(1 −z) 1 −tk−1tk−1→1when z →1(A.5)74

we obtain after relabelling the Si and expanding the terms containing either θ or(1 −z)1/2:I(m)k=z∆0(1 −z)(k−1)(1/2+b+c+(k−2)ρ/2)1ZǫdT1Tk−2ZǫdTk−11ǫZ0dS1Sm−k−1Z0dSm−k(1 + a1ηθ + a2z1/2(1 −z)1/2ηk−1X1θiti+ a2ηm−kX1ωi)(1 −c(1 −z)1/2Xθθi1 −ti)(1 −cXθωi1 −si)(1 −ρ(1 −z)1/2 Xθiωj(1 −(1 −z)1−titi−sj)ti)k−1Y1t−1−b−c−ρ(k−2)i(1 −(1 −z)1 −titi)b(1 −ti)c Yi

ii) β′mkηθ = −a2c1Z0k−1Y1dTi(Xθθk1 −tk)t−1−b−c−ρ(k−2)i(1 −ti)c Yi

with ǫ1 = 1 except for case iv) where ǫ1 = −1.In order to obtain the coefficients βmk we only need to divide β′mk by thenormalization factor of the superconformal block eI(m)k,βmk(a, b, c, ρ) = β′mk(a, b, c, ρ)eN(m)k(a, b, c, ρ)(A.11)The normalization factor eN(m)kis derived from N(m)kby using (3.17). A carefulanalysis of the four possible cases leads toeN(m)k(a, b, c, ρ) = ǫ2N(m)k(b, a, c, ρ)(A.12)with ǫ2 = 1 except for case iii) where ǫ2 = −1.

The last relation together with(3.26) enables us to writeβmk =ˆIm−k(a, b + c + ρ(k −1), ρ) ˆIk−1(c, −1 −b −c −ρ(k −2), ρ)ˆIm−k(−1 −a −b −c −ρ(m −2), a, ρ) ˆIk−1(b, c, ρ)(A.13)(ǫ1 and ǫ2 combine and cancel the sign (−1)m−1 entering N(m)k). The µ factorsappearing in the ˆI integrals cancel out and we are left with a ratio of products ofΓ-functions.

Using Γ(x)Γ(1 −x) = π/s(x) repeatedly we finally obtainβmk = (−1)m−1m−k−1Y0s(1 + a + b + c + (ρ −1)(m −2) −ρ−12 i)s(b + c + (ρ −1)(m −2) −ρ−12 (m −k −1 + i))k−2Y0s(1 + b + ρ−12 i)s(b + c + ρ−12 (k −2 + i))(A.14)This result is exactly the same as the result obtained by Dotsenko and Fateev [4](3.16) provided we implement in their formulae the substitution ρ →(ρ −1)/2.The sign difference (−1)m−1 arises from the difference between our definition ofconformal blocks and that of [4]. This result was expected, as explained in thetext, but it provides a rather non-trivial test of our evaluation of the normalizationintegrals, the main computational difficulty in this paper.77

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Figure captionsFig.3.1. Contour ordering for J(m)k(Z).Fig.3.2.

Contours for the analytic continuation of J(m)k.Fig.3.3. Contour ordering for eJ(m)k(Z).Fig.3.4.

Contour ordering for J(nm)lk(Z).Fig.4.1. Ordering of contours chosen in (4.1).Fig.4.2.

Contour ordering in (4.2,3).Fig.4.3. Explicit definition of contour ordering.Fig.4.4.

Contours used in the definition of JΣ(p, 0, n −p).Fig.4.5. Pulling one contour in J(0, 2m).Fig.5.1.

Integration contours in (5.1).Fig.5.2. Contours used for the evaluation of the even integral Jp′+q′,p+q.Fig.5.3.

Contours used for the evaluation of the odd integral Jk′nm.79

m-k+1m-11m-k0z101xzFig 3.1.Fig 3.2.0z1m-k+1m-11m-kFig 3.3.0z1CCCC1n-ln-1n-l+1SSSS1m-km-k+1m-1Fig 3.4.

0111pqFig 4.1.011kn-pnn-p-1Fig 4.2.1m-1m01Fig 4.3.nn-p+1p101Fig 4.4.

011232mFig 4.5.n11m01Fig 5.1.Jp' q'p q= T01q'1qp'1p101n-pk'1m-q1n-p+1nm-q+1m0 n-p p0 m-q qJk'=01Fig 5.3.Fig 5.2.

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