Super-Liouville - Double Liouville correspondence

The AGT motivated relation between the tensor product of the N = 1 super-Liouville field theory with the imaginary free fermion (SL) and a certain projected tensor product of the real and the imaginary Liouville field theories (LL) is analyzed. Using conformal field theory techniques we give a complete proof of the equivalence in the NS sector. It is shown that the SL-LL correspondence is based on the equivalence of chiral objects including suitably chosen chiral structure constants of all the three Liouville theories involved.


Introduction
It was conjectured several years ago that partition functions of N = 2 superconformal SU(N ) gauge theories in four dimensions are directly related to correlation functions of the two-dimensional Liouville/Toda field theories [1,2]. This by now well established AGT relation has been extended and generalized in various directions leading to many new developments on both sides of the correspondence. One of the essential generalizations was the proposal that N = 2 SU(N ) gauge theories on R 4 /Z p should be related to certain coset conformal fields theories. This conjecture was first formulated in [3] where also the first checks in the case of N = p = 2 corresponding to the N = 1 super-Liouville theory were presented. It was soon clarified that the general case p > 2 should correspond to para-Liouville/Toda theories [4]. Some further checks of the AGT relation for N = p = 2 where done in the NS sector in [5,6,7] and in the R sector in [8,9]. It was in particular observed in [6,7] that the blow-up formula for the Nekrasov partition function suggests a precise relation between N = 1 super-Liouville and Liouville conformal blocks. An interesting explanation of this phenomenon on the CFT side was given in [10]. It was motivated by old results [11,12,13,14] relating various rational models realized as quotients, The case relevant for the present paper is the relation between the Virasoro minimal models V (m) = V (1, m) and the N = 1 superconformal models SV (m) = V (2, m): where the symbol ⊗ P denotes a projected tensor product in which only selected pairs of conformal families are present. The nonrational counterpart of this relation proposed in [10] takes the schematic form free fermion ⊗ N = 1 super-Liouville ∼ Liouville ⊗ P Liouville. (1.1) In the NS sector this relation has been made much more precise in [15] where it was used as an essential element of the proof of the AGT correspondence in the case of N = p = 2. The considerations of [15] are based on the instanton partition function computed on the resolved space [6,7]. It is interesting to note that a different instanton counting scheme for the ALE spaces, the orbifolded instanton counting, yields in general a slightly different partition function [16] and it is still an open question whether its CFT counterpart exists. The extension of (1.1) to the Ramond sector along with some nontrivial checks were presented in [17].
Although most of the ingredients and constructions were already discussed in [15] and [17] a precise formulation of (1.1) as an exact equivalence of CFT models is still an open problem. The product of the free fermion and the N = 1 super-Liouville theory (SL) is a perfectly well defined model of CFT but an exact definition of the double Liouville theory (LL) on the r.h.s. of (1.1) has not yet been clarified. There are also some essential elements of the proof missing. The aim of the present paper is to fill these gaps.
Let us briefly describe our proposal for the precise meaning of equivalence (1.1). The chiral symmetry algebra A NS of the SL theory is the product of the N = 1 super-Virasoro algebra SVir NS and the Heisenberg algebra H NS of fermion oscillators. One has therefore a 1 − 1 correspondence between SVir NS -and A NS -primaries. Any highest weight representation A ∆p of A NS with the central charge c = 3 2 + 3Q 2 , Q = b + b −1 and the highest weight ∆ p = Q 2 8 + p 2 2 carries a representation of two mutually commuting Virsoro algebras {L L n }, {L Γ n } with the corresponding central charges given by [10,11,12,13,14,15]: (1.2) For real b they are on the opposite sides of the c = 1 barrier. We assume b < 1 throughout the paper. Then c L > 25, 1 > c Γ and the parameterizations above coincide with the standard ones of the Liouville theory [18,19,20,21] and of the generalized minimal models (GMM, the time-like Liouvile theory) [22,23].
It was shown in [15] that the Vir⊕Vir representation on A ∆p decomposes into irreducible components where V ∆ L (p,j) , V ∆ Γ (p,j) are Verma modules of the corresponding Virasoro algebras with the highest weights Decomposition (1.3) implies that the spectrum of the double Liouville theory, although diagonal in the continuous parameter p, is non-diagonal in the discrete index j. This important novelty requires appropriate off diagonal extensions of the DOZZ [18,19,20] and the GMM [22,23] structure constants. A simple idea is to split the diagonal constants into chiral parts. As we shall see this receives strong support from our calculations. Indeed it turns out that the SL-LL equivalence is to large extend based on the relations between chiral structure constants.
In the standard normalization of the Liouville theory the reflection amplitude and the two-point function are equal [18,19,20]. This is suitable for the analytic continuation arguments and natural from the path integral point of view [24]. Other normalizations were discussed in the context of the GMM [22,25]. In the present discussion it is convenient to choose the symmetric normalization Φ α = Φ Q−α with no restrictions on the two-point functions. This simplifies the relation between fields on both sides of the correspondence.
1 A definition of the Barnes double gamma function along with some of its properties is given in Appendix C.
The same can be done for the GMM structure constant Decompositions (1.4) and (1.5) are based on the splitting of into the Barnes gamma factors and are to some extend arbitrary. What is important for further calculations is that one uses the same splitting of Υ b for terms with the same arguments in the DOZZ and the GMM structure constants. Once splittings are chosen one can introduce the off-diagonal extensions of the Liouville structure constants: We shall split in a similar manner the NS super-Liouville structure constants [26,27] (see also [28,29]). In the symmetric normalization the NS structure constants (which coincide with the SL structure constants) can be written as The modular properties of the toric partition function of the SL theory imply that the GSO projection on the even with respect to the common left and right parity subspace is a necessary consistency condition. This yields On the r.h.s. of (1.1) it corresponds to The spaces of states H SL , H LL and structure constants (1.6), (1.7), (1.8) unambiguously define both sides of the SL-LL correspondence in the NS sector. The main statement we are concerned with in this paper can be formulated as follows.
If the relative normalization condition is satisfied then there exists a map I from the algebra of local fields of the NS sector of the GSO projected SL model to the algebra of local fields of the GSO projected LL model preserving all correlation functions on the sphere: Let us remark that our motivation goes beyond a self contained CFT proof of the statement above. It characterizes only the simplest of the relations suggested by the AGT correspondence [30] and by the analogy with the rational cases [10]. One may expect for instance an exact relation between the para-Liouville theories [31] and projected tensor product of several copies of the Liouville theory. It seems that better understanding of all equivalences of this type would provide a new insight into underlying structures of at least certain class of nonrational CFT models.
The organization of the paper is as follows. In Section 2.1 the basic concepts of the free field realization of the NS sector are introduced mainly for the notational convenience. In Section 2.2 a special attention is paid to the description of the reflection map and the reflected modes in the NS Verma modules which play an essential role in the constructions of [15]. The properties of the states generated by solely fermionic modes are summarized in Propositions 1 and 2 of Subsection 2.2. They are proven in Appendix A.
Section 3 is devoted to the properties of basic chiral structures related to the chiral symmetry algebra A NS of the SL theory. In Subsection 3.1 we review decomposition (1.3) of the A NS Verma module into projected tensor product of Virasoro Verma modules constructed in [15]. In Subsection 3.2 we analyze 3-point A NS conformal blocks and derive the central for the whole paper formula for the blow up factor. This is the main and technically the most involved result of the present paper. It is based on Propositions 1 and 2 of Subsection 2.2 and on the formula for generalized Selberg integral derived in Appendix B. In Subsection 3.3 we investigate the A NS algebra 4-point conformal blocks. As a side result we obtain expressions for the NS super-Virasoro blocks in terms of the Virasoro ones and the blow up factors.
In Section 4 we complete our proof of the SL-LL equivalence for arbitrary correlation functions on the sphere in the GSO projected NS sector. To this end we analyze in Subsection 4.1 relations between chiral structures of all the there theories involved. It turns out that one can prove the SL-LL equivalence for the left and for the right chiral parts separately. It is also remarkable that although the left and the right structure constants are different the relations they satisfy involve exactly the same coefficients. This is a consequence of two types of identities for Barnes gamma functions which are derived in Appendix C. In Section 4.2 we analyze how the relations between chiral structures result in an equivalence of full CFT theories. We construct an exact map between local fields and show it preserves all the 3-point functions and the factorization of correlation functions on the sphere.
There are several possible continuations of the present paper. The most obvious is a completion of the proof for the whole GSO projected models. With the results of [17] it seems that an extension to the Ramond sector is straightforward. The same concerns an extension to arbitrary closed surfaces which would require an analysis of 1-point toric functions. More challenging is an extension to bordered surfaces.
The chiral structure constants played an essential role in the calculations of the present paper.They also show up in the relative normalization of the fusing matrix and the 6j-symbol of a continuous series of representations of U q (sl(2, R)) [21]. This relation and the orthogonality properties of the 6j-symbols yield a general expression for structure constants satisfying the crossing bootstrap equation on the Liouville theory spectrum ( [21], formula (252)). To what extend the chiral structure constants form fundamental building blocks of Liouville type models is by itself important problem related to the questions of classification and new model building. The double Liouville theory is the first example of nonrational models with the spectrum being diagonal in the continuous and non-diagonal in the discrete parameter. It is interesting if there are other consistent models of this type.
Another possible topic is to investigate already mentioned relations between the para-Liouville theories and projected tensor products of several copies of the Liouville theory which are well supported by the AGT correspondence [6,7,10]. For rational models another equivalence is known [12]. It relates the tensor products of the N = 2 super minimal models and the Ising model to projected tensor products of the N = 1 super minimal models and the parafermionic models. Although the AGT 4-dimensional counterpart is not expected in this case it would be very interesting to check if there is a non-rational version of this correspondence.

Chiral vertex operators
Let F b be the bosonic Fock space generated by modes of the Heisenberg algebra We denote by F NS the fermionic NS Fock space generated by modes of the algebra out of the vacuum |0 f obeying ψ r |0 f = 0 for r > 0. The super-scalar Hilbert space H NS of the NS sector can be introduced as a tensor product where H 0 ∼ = L 2 (R) is a representation space of the zero mode operators p, q, satisfying [p, q] = −i. A scalar product in H NS is defined by imposing modes' conjugation properties together with the normalization condition where |p = |p ⊗ |0 b ⊗ |0 f and p|p = p|p . The space H NS can be seen as a direct integral of Hilbert spaces: with the scalar product in H p determined by conjugation properties (2.3) and the normalization The space H NS carries two representations (±) of the NS algebra with the same central charge c = 3 2 + 3Q 2 : L n (±p) = 1 2 m =0,n c n−m c m + 1 2 and the same highest weight states Let us define the superscalar components Since the fermion field ψ(z) and the screening charges are odd with respect to the fermion parity operator (−1) F defined by the total number of screening charges appearing in (2.9) must be even, while the total number of screening charges appearing in (2.10) and in (2.11) must be odd.

Reflection map in the NS sector
For each level t ∈ 1 2 N we introduce the transition matrices S t (±p): The matrix S t (with a different normalization) was studied in [32] where the formula for its determinant was found. It takes the form where multiplicities P NS (f ) are defined by the generating function For real Q and p L n (p) † = L −n (p), and S t (±p) are simply related to the Gram matrix B t of the t-level Schapovalov form Let |∆ p be the normalized highest weight vector in the NS Verma module V ∆p of the highest weight ∆ p . The maps are by construction morphisms of representations. Determinant formula (2.13) and hermitian conjugation properties (2.14) imply that for real Q and p these maps are unitary isomorphisms. So is the composition called the reflection map [19,21]. 2 Up to a multiplicative constant it is uniquely determined by the intertwining property (2.17) One can study the relation between representations {L(+p), G(+p)} and {L(−p), G(−p)} introducing reflected modes in H p [15] (see also [33] for a similar construction in the Virasoro case): In principle they can be expressed as series of monomials in c m and ψ r . The general form of this transformation is however not known. It can be calculated term by term from intertwining property (2.17) which takes the form : The matrix of the reflection map r t (p) with respect to the basis c −M ψ −K |p is simply related to the transition matrix We shall also need expressions of the oscillation bases (2.20) In the Fock space representation of an NS Verma module we used above the modes c m , ψ k are elementary p-independent objects in contrast to their reflected counterparts c R m (p), ψ R k (p) which are complicated infinite series with coefficients being rational functions of momentum p. A symmetric picture can be achieved by lifting both sets of modes to the NS Verma module: (2.21) In this formulationc R m (p) =c m (−p),ψ R k (p) =ψ k (−p), (2.22) and the reflection map takes the form: Modes (2.21) can be used to construct two different orthogonal bases in the NS Verma module V ∆p (2.23) In a generic case (when the modules V ∆rs+ rs 2 are irreducible) the coefficients in (2.23) have simple poles at p = p rs and p = −p rs respectively. In the following we shall need a more detailed information about the structure of pure fermionic statesψ(p) −K |∆ p . This can be conveniently summarized in the form of two propositions below. The proofs are given in Appendix A.

Proposition 1 The only possible singularities of the coefficients of the decomposition of the statẽ
Let us define polynomials Proposition 1 implies that all coefficients of the decompositions are polynomials in p variable. One can estimate the degree of these polynomials.

Verma modules
Let V ∆p be the NS module with the central charge c = 3 2 + 3Q 2 , Q = b + b −1 and the highest weight We introduce another copy of the fermionic NS Fock space F NS generated by the modes The generators define on F NS a representation of the c = 1 2 Virasoro algebra which is a direct sum of two irreducible highest weight representations with the highest weights ∆ = 0 and ∆ = 1 2 , respectively. We shall introduce an indefinite scalar product on F NS by the relations With respect to this scalar product L f n † = L f −n and direct sum (3.2) is orthogonal with the induced scalar product positively definite on the first summand and negatively definite on the second.
On the A NS algebra Verma module A ∆p = V ∆p ⊗ F NS one can construct two sets of generators: They form two mutually commuting Virasoro algebras [11,12,13] and satisfy the standard conjugation relations: The corresponding central charges are given by (1.2).
The problem of decomposing the Vir ⊕ Vir representation on A ∆p into irreducible components has been analyzed in [15]. We shall briefly recall the main points of this derivation. Using construction (2.23) in the NS Verma module V ∆p one introduces a family of states: The modes can be conveniently organized into local fields By constructionχ It follows that for all j ∈ Z the states |p, j are of zero norm ( p, j |p, j = 0).

With the help of both representations (2.19) one can compute the commutation relations
Using these formulae one shows that states (3.5) are highest weight states with respect to both Virasoro algebras and one has two different Virasoro highest weight vectors on each j 2 2 level subspace of A ∆p except j = 0 where there is only one state |p, 0 . We have shown that In order to show that the spaces above are equal one can calculate characters of the representations involved. For real p and Q the modules are non-degenerate and Using relation (3.9) and the Jacobi triple identity one gets The equivalence above is a unitary isomorphism if we assume on the l.h.s. the scalar product such that where ν σ p,j denotes the highest weight state in the Virasoro Verma module V ∆ σ (p,j) (σ = L, Γ). Let us stress that the skew form of product (3.10) is the only one consistent with the complex weights ∆ L (p, k), ∆ Γ (p, k), k = 0 and the hermiticity of L L 0 , L Γ 0 . We shall discuss this point in some more details in Subsection 3.3.

Blow up factor
The three-point block ρ A NS (ξ 3 , ξ 2 , ξ 1 |z) with respect to the A NS symmetry algebra is defined as a solution to the A NS Ward identities normalized by the conditions: Our goal in this subsection is to calculate the block ρ A NS (ξ 3 , ξ 2 , ξ 1 |z) for z = 1 and for arbitrary states ξ p,j ≡ |p, j n = Ω(p, j) |p, j , j ∈ Z, Ω(p, 0) = 1.

By definition
where V p,j (z) are chiral vertex operators corresponding to the states |p, j n and * V p, is the super-Virasoro primary field corresponding to the highest weight state |∆ p . An explicit expression for (3.12) was proposed in [15] where it was called the blow up factor due to the role it plays on the four-dimensional side of the AGT correspondence. We shall compute it employing the free field techniques. At the first step we show that (3.12) is a polynomial and calculate an upper bound on its degree. This is based on the propositions of Subsection 2.2 proven in Appendix A. Then using possible free filed representations we find all zeros of this polynomial. Finally in order to fix the overall constant we calculate (3.12) in a simple special case. The last step requires formulae for generalized Selberg integrals which are derived in Appendix B.
From decompositions (2.25) it follows that (3.12) is a polynomial in the parameters α i , i = 1, 2, 3 with all α i dependence coming from the Ward identities in the super-Liouville factor. To determine its order observe that for arbitrary multi-indices: By the construction of |p, j n and Proposition 2 one gets: when j 1 + j 2 + j 3 is even and There are several free field representations of the matrix element n p 3 , j 3 | V p 2 ,j 2 (1) |p 1 , j 1 n . Suppose that all j i are positive. The state |p, j n can then be represented either as or There are also two different representations for the descendant field V p,j (1) : and are the free field representations of the fieldsχ(ξ),χ R (ξ) (3.6). In consequence there are eight distinct representations of the blow-up factor.
Let us first consider the one with no reflected fields. Suppose that The correlator in the integrand of the r.h.s. of (3.13) and of (3.14) can be rewritten as an integral of product of the bosonic and the fermionic factor From the commutation relations of modes one has This implies that (3.15) vanishes for r + s < j 1 + j 2 + j 3 and therefore the l.h.s. of (3.13) contains a factor 0 r,s r+s ∈ 2N r+s<j 1 +j 2 +j 3 where l NS (x, n) has been defined in (2.24) while the l.h.s. of (3.13) is proportional to Let us now consider the free-field representations of the blow-up factor containing at least one reflected field. Since the oscillators ψ R k (−p) are complicated functions of both the bosonic c n and the fermionic ψ k oscillators one cannot factorize the correlators appearing in the corresponding integrands into a bosonic and a fermionic parts. For the same reason the OPE of E α (z) and χ R (w) does not vanish. The free field representations ψ R (w) and ψ(z) of fields introduced in (3.6) commute with p. Their OPE cannot therefore contain any primary field of the form E α (z) with α = 0. The only admissible operators are thus even members of the conformal family of the identity field. Since both ψ R (w) and ψ(z) are primary fields with conformal dimension 1 2 , one gets Taking into account these properties of the reflected field ψ R (w) one can still conclude that the free field representation must vanish whenever the number of χ field insertions exceeds the number of reflected fields and screening charges. It follows in particular that representation with three reflected fields does not provide any direct information about possible zeros of the blow up factor. The remaining six representations involve the following quotients They imply the factors respectively, where ♯ =NS for even j 1 + j 2 + j 3 , while ♯ = R for j 1 + j 2 + j 3 being odd.
For a given set of positive j 1 , j 2 , j 3 not all the representations contribute. For instance if the inequalities hold, (3.17) (resp. (3.18)) and (3.26) -(3.28) are the only products with non-empty ranges of integers r, s. Since the number of pairs (r, s) satisfying 0 r + s < n ∈ 2N with r + s ∈ 2Z is equal to 1 4 n 2 , the total number of factors in (3.17) and (3.26) -(3.28) in the "even" case is equal to The product of terms (3.17) and (3.26) -(3.28) exhausts therefore the dependence of blow-up factor (3.12) on all α i . Hence, if inequalities (3.32) are satisfied, Similarly, since the number of pairs (r, s) satisfying 0 r + s < n ∈ 2N − 1 with r, s ∈ 2Z − 1 is equal to 1 4 (n 2 − 1), and Looking for non-empty ranges of products and counting the degree of the resulting polynomial one obtains formulae for the blow up factor for other ranges of positive integers j 1 , j 2 , j 3 : for j 2 > j 1 + j 3 (and, consequently j 1 + j 2 > j 3 and j 2 + j 3 > j 1 ), for j 3 > j 1 + j 2 .
In order to determine the normalization constant C j 3 j 2 j 1 (b) we assume inequalities (3.32) and use the representations (3.13) and (3.14) in the simplest non-vanishing case , i.e. for s = 0, r = j 1 + j 2 + j 3 and In this case the second line of (3.13) and (3.14) takes the form The integral h j 3 j 2 j 1 (t 1 , . . . , t N ) is by definition totally antisymmetric in t k variables. The integrand g j 3 j 2 j 1 can be calculated by means of the Wick theorem with a help of OPE-s (3.16). Since all the modes of the field χ anticommute with each other, the only contributions arise from the anticommutators between a mode of the field χ and a mode of the field ψ. Indeed, had we taken into account a contribution from a "pairing" between the fields ψ, we would have been left with a correlator containing only modes of the field χ, which vanishes. This shows that h j 3 j 2 j 1 (t 1 , . . . , t N ) → 0 for t k → t l , 1 k, l N (3. 38) and the only possible singularities may arise at t l = 1, 0, l = 1, . . . , N . Since The commutation relation {ψ(t l ), The only totally antisymmetric function with zeros (3.38), singularities (3.39), (3.40) and asymptotics (3.41) reads where σ j 3 j 2 j 1 does not depend on the variables t k .
Taking into account sign factors we get from the definition of h j 3 j 2 j 1 (t 1 , . . . , t N ) that, up to subleading terms Comparing this formula with the one above we get a recurrence relation uniquely determining σ j 3 j 2 j 1 . This yields We conclude from (3.42) that I num j 3 j 2 j 1 is expressed through a standard Selberg integral and thus can be calculated with the result where g = − 1 2 b Q, N = j 1 + j 2 + j 3 . Suppose that j 1 + j 2 + j 3 is even. In the special case under consideration the matrix element p 3 | E α 2 (1)Q N b |p 1 takes the form is a totally symmetric polynomial. We calculate this generalization of the Selberg integral in Appendix B. Using (B.27) one obtains The properties of the Euler gamma function allow to present the ratio of (3.43) and (3.45) in the form which completes the calculation of representation (3.13) of the l.h.s. of (3.33). The r.h.s. of (3.33) takes the form Comparing (3.46) with (3.47) we get: To check this result in the case of odd j 1 + j 2 + j 3 = 2m − 1 we observe that the matrix element can be calculated as a particular limit of integral (3.44). Let us rewrite (3.44) in the form For t 2m → 1 : where the dots stand for terms sub-leading for t 2m → 1.
Using the identity valid for a function h(t) left continuous at t = 1, we thus get Computing the limit one obtains Comparing the ratio of (3.43) and (3.50) with formula (3.34), calculated at ip 2 = Q 2 and ip 3 = ip 1 + Q + (2m − 1)b, one can verify formula (3.48) in the case when j 1 + j 2 + j 3 is odd.
Let us note that the cases when some or all j indices are negative can be obtained either by direct analysis or by changing the sign of corresponding momenta. The special cases when one or more j indices go to zero can be derived using exactly the same method. They can be seen as limiting cases of the situations considered above. For instance in the case of j 1 = j 3 , j 2 = 0 one gets the formula for the scalar product of normalized states n p, −j|p, j n = n −p, j|p, j n = 2 j l NS (2ip, 2j)l NS (Q + 2ip, 2j). (3.51) In Section 4 we shall need 3-point conformal blocks related to the 3-point functions (3.52) They can be expressed in terms of the 3-point block related to the matrix elements of the vertex operators (3.12) as follows In the case when all j are positive and inequalities (3.32) are satisfied, using (3.33) and (3.48) one gets for instance B R j 3 j 2 j 1 (α 3 , α 2 , α 1 ) for j 1 + j 2 + j 3 ∈ 2Z + 1, (3.54)

4-point conformal blocks
The states on the l.h.s. of (1.1) are organized in terms of A NS Verma modules A ∆p = V ∆p ⊗ F NS . In order to compare both sides of (1.1) it is convenient to choose a basis in A ∆p consistent with direct sum decomposition (1.3): where M, N are arbitrary ordered, integer multi-indices and j ∈ Z. Vertex operators related to the states |p, j n take the form (j > 0): For all j ∈ Z they are primary with respect to the energy momentum tensors Indeed one can check by explicit calculations that the OPE holds. By standard contour arguments (3.57) implies [L σ n , V p,j (z)] = (n + 1)z n ∆ σ (p, j)V p,j (1) + z n L σ −1 , V p,j (1) .
Relation (3.60) gives rise to relations between higher conformal blocks. As an example we consider the spheric four-point blocks. As in the case of the NS algebra [34] one has four types of the A NS algebra blocks. For each type there is one even, , and one odd, conformal block. Although in correlation functions the even and the odd blocks show up separately it is convenient for our present purposes to combine them into a single function In the formulae above ∆ i stand for the conformal weight ∆ i of the highest weight state ν i or the conformal weight * ∆ i = ∆ i + 1 2 of the state * ν p = G − 1 2 ν i . An explicit expression for coefficients depends on the basis used in the factorization of the corresponding four-point chiral correlator. For the basis all states with nonzero f excitations drop out from the factorization formula and one gets where B t (∆) −1 is the matrix inverse to Gram matrix (2.16) in the NS Verma module V ∆ and ρ NS denotes the three-point N = 1 superconformal block. 3 It follows that all four-point blocks of the algebra A NS in the tensor product exactly coincide with the N = 1 superconformal blocks in the super-Liouville factor: where on the r.h.s. the notation of [34] was used.
On the other hand one can factorize on basis (3.55). With identification ( 3 In order to simplify notation we use the same symbol νi for the state |∆i and |∆i ⊗ 0f .
The identity operator on V ∆ ⊕ V∆ can be expressed as For each conjugate pair of complex weights ∆,∆ it is then natural to introduce two four-point conformal blocks: and its∆ counterpart. With this definition one has where we use the simplified notation 0). Formulae for the other types of blocks follow from the relation For blocks with a single star the f excitations drop out and one has 1). The relations for double star blocks is slightly more complicated. First of all relation (3.64) implies Then using (3.60) one gets

Chiral structure constants
It was shown in Subsection 3.1 that the map is an isomorphism of Vir ⊕ Vir representations. It has its counterpart for the corresponding chiral operators It provides an equivalence of the theories if all three point functions of corresponding operators are equal. We have shown in the previous subsection that for L L n , L Γ n generators one can use the same Virasoro Ward identities on both sides of the correspondence. It is then enough to check the correlators of the operators V p,j . One obviously has the same situation in the right sector.
The following identities are responsible for the relations between the chiral structure constants 4 : The first one is the chiral versions of the identity for upsilon functions proposed in [15]. Using the identities above one gets where We shall now analyze the three point chiral functions of operators corresponding to arbitrary |p, j n states. For clarity of presentation we restrict ourselves to the case when all j andj are positive and inequalities (3.32) are satisfied. Let us first consider the even case j 1 + j 2 + j 3 ∈ 2N. Using formulae (C.1) and (C.2) of Appendix C and properties of Euler gamma functions one gets (3.54). Together with (4.5) and (4.6) it yields The comparison with formula (3.53) for the conformal block suggests the rescaling ζ p,j ≡ |p, j nn = 1 2 j l(2ip + Q, 2j)l(2ip, 2j) |p, j n . (4.9) This gives the normalization nn p, −j |p, j nn = 1 which is in line with the natural normalization we have chosen in Subsection 3.3 for Virasoro Verma modules over complex weights (3.63). If we assume this scalar product in direct sum (1.3) the maps For the new states one has In the odd case j 1 + j 2 + j 3 ∈ 2N + 1 one obtains where B R j 3 j 2 j 1 (α 3 , α 2 , α 1 ) is given by (3.54). The counterpart of (4.10) reads It follows from (4.10) and (4.11) that after an appropriate overall rescaling of the NS chiral structure constants the maps I,Ī provide equivalence of chiral correlators in the left and in the right chiral sector, respectively.

Correlation functions
We shall now compare the full correlation functions on both sides of the correspondence. If one chooses the standard scalar product on the left-right tensor products ξ ⊗ξ |χ ⊗χ = ξ |χ ξ |χ on both sides of the correspondence, then the map I : H SL → H LL , i ν L p,j,j ⊗ν Γ p,j,j for j,j ∈ 2Z + 1, where ν L p,j,j ≡ ν L p,j ⊗ν L p,j , ν Γ p,j,j ≡ ν Γ p,j ⊗ν Γ p,j , is a unitary isomorphism. Its counterpart for the local fields i Φ L p,j,j ⊗ Φ Γ p,j,j for j,j ∈ 2Z + 1, is expected to provide the SL-LL equivalence if all correlation functions of corresponding operators are equal. The choice of phase in (4.12) may seem strange at this stage but it is in fact indispensable as we shall see in the following.
If we restrict ourselves to the correlation functions on the sphere it is enough to show that all 3-point functions coincide and that the factorization procedures are the same. For the first part let us recall that for L L n , L Γ n generators one can use the same Virasoro Ward identities on both sides of the correspondence. It is then sufficient to check the 3-point functions of the operators Φ p,j,j .
Let us now conisder 3-point function with one excitation. From definition (1.8), normalization (4.9) and relation (3.64) one has on the SL side The calculations on the LL side can be readily done using chiral formulae (4.7), (4.8). If condition (1.11) is satisfied one gets Note that this partially explains our choice of phase in (4.12). Using essentially the same calculations one checks the formula above for the three other cases (j 2 ,j 2 ) = (1, −1), (−1, 1), (−1, −1).
We shall now consider the three point function of operators Φ p,j,j corresponding to arbitrary ζ p,j ⊗ ζ p,j states. For simplicity we restrict ourselves to the case when all j andj are positive and inequalities (3.32) are satisfied. Let us first assume that the sum of all left j indices is even 6 There are two subcases: all left j indices are even or two of them are odd and one is even. In the first subcase one has on the SL side 7 Then by chiral formulae (4.10) one obtains In the case of two odd and one even j's one has instead with the minus sign coming from the anti-commutation of the odd left and the odd right excitations.
On the other hand due to our choice of phase in (4.12) one gets on the LL side the factor i 2 so the 3-point functions on both sides coincide.
Let us now turn to the odd case j 1 + j 2 + j 3 ∈ 2N + 1.
As before it splits into two subcases: one left index is odd and two are even or all of them are odd. In the first case the splitting of the 3-point function into A-algebra conformal blocks does not develop any extra sign: Then by chiral formulae (4.11) one obtains In the subcase of all odd indices one has the extra minus sign on the SL side and the extra i 2 factor on the LL side, so again the 3-point functions coincide. All cases involving other inequalities and non-positive indices can be analyzed in the same way. The final result can be compactly written as follows Let us stress the role the phase i introduced in (4.12) plays in the formula above -it reproduces the factor i in the NS structure constantC NS b (1.8) and the extra minus signs coming from the anti-commutation of odd objects on the SL side.
The last step of our proof of the SL-LL equivalence is to compare the factorization of correlation functions. To this end we choose in H SL the basis The Virasoro generators in the formula above are all even operators and the map I is an isomorphisms of Vir ⊕ Vir ⊕Vir ⊕Vir representations one thus can safely drop them from the subsequent formulae. Let us consider for instance the factorization of the 4-point function on basis (4.13). In the simplified notation it takes the form On the LL side one has One thus gets the exact equivalence of the 4-point functions. It is also clear that the simple mechanism above works for any factorization of any n-point function on the sphere as well.
the poles arise due to the existence of a singular vector |χ rs ∈ V ∆rs on level rs. Let χ N L rs denote the coefficients of |χ rs in the basis L −N G −L |∆ rs , We normalize |χ rs such that the coefficient at (L −1 ) rs |∆ rs is equal 1. For rs < |K| consider vectors of the form so that |χ rs = lim p→prs D rs |∆ p . The set of these vectors can be always extended to a full basis in V |K| ∆ . Working in such a basis and using the properties of the Gram matrix B |K| c,∆ and its inverse [34] one gets where χ(p rs )| = p rs | D † rs (p rs ).
We shall show that due to special properties of pure fermionic states some of limits (A.3) vanish. Let us first observe that up to an overall normalization the singular vectors χ(p rs )| can be constructed by means of the screening charges [32]: for r s or for s r, where all the integration contours are closed 8 . It follows that the singular vector χ(p rs )| contains at most min{r, s} fermionic excitations (oscillators).
Suppose that r s and consider matrix elements of the form where, for notational convenience the argument p rs of generators L n and G l is suppressed. Since for all n > 0 : we can restrict ourselves to matrix elements of the form where l 1 < l 2 < . . . < l m and is thus a necessary condition for (A.6) not to vanish. Since all l i are different where the second inequality follows from (A.8). Using (A.7) we thus get It follows that all the states at the level 1 2 j 2 > rs 2 , generated from χ(p rs )| by the operators G l and L n are orthogonal to the state ψ − 2j−1 2 . . . ψ − 3 2 ψ − 1 2 |p rs unless condition (A.9) is satisfied.
Consider now a matrix element of the form is a combination of states of the form ψ −K |p rs , where again K is a subset of J. We can thus restrict ourselves to matrix elements of the form Suppose that for r + s > 2k there exists a multi-index L such that (A.11) is non-zero. As a polynomial in b and b −1 the matrix element (A.11) has the large b leading term We shall now calculate the large b leading term of the collerator where J \ K is the multi-index composed of those indices which complete K to J. To this end it is sufficient to keep only the "linear" parts of the generators This yields and therefore the correlator on the l.h.s. does not vanish in contradiction with our previous considerations. This completes the proof in the general case.
Moving subsequent oscillators d m j ′ , j ′ < j to the right yields new terms involving commutators with generators, yielding again the factor ǫ 1 ǫ 2 , and oscillators resulting from the previous steps. The commutation with the oscillators is nonzero only if d m j−1 meets its conjugated counterpart and gives the factor ǫ 1 ǫ 2 . Each step contributes therefore the factor ǫ 1 ǫ 2 . If after moving all bosonic oscillators to the right the result is nonzero it has the overall factor ǫ j 1 ǫ j 2 which cancels against the denominator of (A.17).
As the r.h.s. of (A.18) is a combination of linearly independent vectors and the l.h.s. scales as λ j 2 the scaling rule for each coefficient must be ω(q, j)s n (q) −1 N L,∅K → λ j 2 −2#N −#L ω(q, j)s n (q) −1 N L,∅K .
If ω(q, j)s n (q) −1 N L,∅K is a polynomial in variables q, ǫ 1 , ǫ 2 then the scaling implies We shall show that this is indeed the case. To this end let us first calculate the determinant of s n . Since, it follows from relation (2.13) that the whole q dependence of det s n is given by the factor In order to find the full ǫ 1 , ǫ 2 dependence we calculate the determinant of the matrix Up to a numerical factor it is given by where m n is the diagonal matrix m n M ′ K ′ ,M K = m n M K δ M ′ K ′ ,M K . This easily follows from the Kac formula for matrix B n and the relations Since b n = (s n ) † m n s n formula (A. 19) implies that up to a numerical factor The matrix elements s n (q) −1 N L,∅K are given by

B. Generalized Selberg integral
Our task is to calculate the integral for even N = 2m and where P N (t 1 , . . . , t N ) = ψ(t 1 ) . . . ψ(t N ) is a symmetric polynomial. Our derivation parallels the original Selberg method (see [35] for a pedagogical discussion of various methods of deriving the Selberg formula). Integral (B.1) has been already calculated long time ago (although in a different way) in [36]. It is also a special case of a more general formula discussed in [31]. We present a simpler derivation mainly for the completeness of the present paper.
Since for t 1 → t 2 : the polynomials P N satisfy the "clustering property": which is an important ingredient of the presented calculation.
Let g be a natural number. Then, The polynomial D N can be presented as Since the expansion coefficients c ν 1 ,...,ν N are totally symmetric with respect to permutation of their indices, we can rewrite the sum as where the indices ν k are ordered or, equivalently 2p+j l=1 ν l (2p + j) (2p + j − 1)g + p . (B.13) Taking into account the ordering of the indices ν l we thus get ν 2p+j (2p + j − 1)g + p (B.14) which for p = m − 1, j = 2 agrees with (B.10).
In the same way one shows that ratio (B.20) is also A−independent. We can thus write    To determine the value of X 2 (g) note that, since P 2 (t 1 , t 2 ) = 1, the integral I 2 (A, B; g) is just a "standard" Selberg integral with a well-known value Γ(1 + p + (2p + j)g) Γ(1 + g) C 1 (g).
Formula (B.27) has been derived for g ∈ N. One can however resort to the Carlson theorem [37] to check its validity also for complex g, for which the integral is convergent.
Similar methods can be used to calculate the "odd" integral

C. Gamma Barnes identities
For ℜ x > 0 the Barnes double gamma function Γ b (x) can be defined by the integral representation [38,19]