Elliptic recursion for 4-point superconformal blocks and bootstrap in N=1 SLFT

All types of 4-point spheric conformal blocks in both sectors of N=1 superconformal field theory are introduced and analyzed. The elliptic recurrence formulae are derived for all the types of blocks not previously discussed in the literature. The results are used for numerical verification of the crossing symmetry of some 4-point functions in the N=1 superconformal Liouville field theory.


Introduction
The Liouville field theory (LFT) is one of the most important examples of 2 dimensional non-rational conformal field theories. Its numerous applications range from the 2-dim quantum gravity and matrix models [1,2] through D-brane dynamics in string theory [3] to the recently discovered AGT relation [4]. The exact analytical expression for the Liouville structure constants was first proposed by Dorn and Otto [5] and independently by Zamolodchikovs [6]. The proposal was motivated by an analytic continuation of the 3-point functions perturbatively calculated within the Coulomb gas approach. Another derivation of the DOZZ formula based on functional relations for structure constants was presented by Teschner in [7].
In principle any n-point function in the Liouville theory is given by the DOZZ structure constants and conformal blocks [8]. Such representation however is not unique and the consistency of the theory requires various decompositions of the same correlator to yield the same result. In case of a CFT on closed Riemman surfaces the consistency conditions for all the correlators are satisfied if and only if the 4-point spheric functions are crossing symmetric and the 1-point toric functions are modular invariant [9]. The first numerical check of the crossing symmetry of the 4-point functions in LFT was done in [6]. It was based on Al. Zamolodchikov's effective recursive relations for 4-point blocks worked out in a series of papers [10,11,12]. An analytical proof of the crossing symmetry was derived by Ponsot and Teschner [13,14] and Teschner [15,16]. It was recently shown [17] that the modular invariance follows from the relations between the toric 1-point and the spheric 4-point blocks [18,19]. Let us note that the proof of these relations is based on the recursive representations of both types of blocks [19].
The N = 1 supersymmetric generalization of the Liouville theory is much less developed. Although the structure constants on closed surfaces have been known for a long time [20,21] and certain couplings on bordered surfaces were successfully derived [22] the computation of all 4-point functions is still an open question. The main reason is that the 4-point superconformal blocks are much more complex objects than the bosonic ones.
The first complication arises already in the Neveu-Schwarz (NS) sector where the superconformal Ward identities determine the 3-point functions up to 2 (rather than 1) structure constants [23]. This leads to eight types of NS 4-point blocks [24], [25]. The second difficulty comes from the Ward identities in the Ramond (R) sector which are considerably more involved [26]. A method of finding a proper basis for 3-point blocks and an appropriate representation of the R fields in terms of chiral vertex operators was presented in [27]. It was also shown that the correlation functions of 4 R fields decompose into eight types of corresponding 4-point blocks.
The recursive relations for the NS 4-point blocks were derived by suitable modification of Zamolodchikov's method [24], [25]. The more efficient elliptic recursion was proposed by Belavins, Neveu and Zamolodchikov [28,29] and derived in full generality in [30]. With the help of this recursion the bootstrap equations for 4-point functions in the NS sector of N = 1 SLFT were numerically verified [28,29]. An analytical check of the crossing symmetry in the NS sector based on the braiding and the fusion properties of the NS blocks was worked out in [31,32]. The recursive representations of the eight 4-point blocks corresponding to correlation functions of 4 R fields were derived in [27].
The aim of the present paper is to extend the analysis of [27] to all types of N = 1 superconformal blocks. This concerns in particular the conformal blocks corresponding to the correlation functions of 2 NS and 2 R fields. We apply the techniques developed in [27] in order to find a diagonal representation of an NS superprimary field in terms of vertex operators acting in the R sector. Such a representation suggests a convenient basis for the corresponding 3-point blocks. These new 3-point blocks together with those of [24], [27] constitute a complete set indispensable for defining all types of 4-point superconformal blocks corresponding to the correlation functions of the R primaries and the NS superprimaries.
These blocks include in particular the 4-point blocks with R intermediate states, which have not been investigated so far. In all new cases the elliptic recursive relations are derived.
Using recursive representations for the 4-point blocks we numerically check the crossing symmetry of correlators of 4 R fields and 2 R and 2 NS fields in N = 1 SLFT. The crossing symmetry of a correlator of 4 R fields can be seen as a verification of two structure constants in the R sector [20]. We preform two more checks involving all four SLFT structure constants and 4-point blocks with NS and R intermediate states. These checks not only test all the structure constants but also give a strong verification of the definitions and recursive representations of all the types of the 4-point blocks involved. This makes the considerations of the present paper a firm starting point for deriving an analytic proof of the bootstrap equations in N = 1 SLFT which was one of the motivations of the present work. Another one is a study of 1-point functions on a torus in N = 1 SLFT and their modular invariance. The recursive representation of the corresponding 1-point superconformal blocks can be found using the techniques developed in the present paper. It would be very interesting to check if there exists a relation between supersymmetric 1-point toric and 4-point spheric blocks similar to that found in the bosonic case [18,19].
The organization of the paper is as follows. In the first section we make a brief review of N = 1 SLFT and introduce our notation. Section 2 is devoted to diagonal representations of NS superprimary and R primary fields in terms of chiral vertex operators. We collect earlier results from [24], [27] 2 and derive a formula for NS superprimary field in R-R sector. In

N=Supersymmetric Liouville Theory
The N = 1 supersymmetric Liouville theory is defined by the action: where b is the dimensionless coupling constant and µ is the scale parameter. The central charge of the theory is c = 3 2 + 3Q 2 , where Q = b + 1 b is the background charge. The N = 1 superconformal symmetry is generated by the holomorphic bosonic and fermionic generators T (z), S(z) (and their antiholomorphic counterpartsT (z),S(z)) fulfilling 2 We also correct the formula for chiral decomposition of R primary fields in R-NS sector introduced in [27].
Each Liouville superprimary NS field φ a is represented by an exponential: with the conformal dimension ∆ a =∆ a = a(Q−a)

2
. We shall also use the parametrization in terms of the momentum p: The superconformal family of φ a corresponds to the tensor product V ∆a ⊗V∆ a of 1 2 Z-graded representations of the left and the right NS algebra. It contains four Virasoro primaries, φ a itself and three descendants: There are two Virasoro primary fields in the Ramond sector represented by where σ ± are the twist operators with the conformal weight 1 16 . The weights of primary fields R ± a read: The OPEs of the R primary fields with the fermionic current take the following form: where β is related to the conformal weight by We assume that the superprimary fields φ a and primaries R + a are even with respect to the common parity operator.
We define the R supermodule W ∆ as a highest weight representation of the R algebra extended by the chiral parity operator (−1) F L : The tensor product W ∆ ⊗W∆ of the left and the right R supermodules provides a representation of the direct sum R ⊕R of the left and the right extended Ramond algebras. In the Liouville theory however we need an extension of the left and the right R algebras only by the common parity operator (−1) F = (−1) F L (−1) F R . This can be achieved by reducing the where w + ∆ , w + ∆ are the even highest weight states in W ∆ ,W∆ and w − ∆ = e i π −iβS 0 w + ∆ . This is so called "small representation" [27].
Correlation functions in N = 1 SLFT are determined by the superconformal Ward identities up to 3-point structure constants of the four independent types: They were derived in [20,21] and read: dx | x=0 and Υ NS/R [33] denote: The special function Υ b (x) was introduced by Zamolodchikovs in [6]. In the strip 0 < Re(x) < Q it has the integral representation: In this paper we are interested in 4-point functions of the R fields and the bootstrap equations they should satisfy. Restricting ourselves to the R + fields and the NS superprimaries we have the following crossing symmetry conditions: (16) and

The 3-point blocks
The aim of this section is to collect the formulae for the NS superprimary and the R + primary fields written in terms of normalized chiral vertex operators. The strategy is to express 3point functions by the structure constants and suitably normalized 3-point blocks. In order to find such decompositions we will define the chiral 3-forms using the Ward identities for correlation functions of three fields with an arbitrary number of holomorphic generators. The NS and the R operators have the following block structure: with respect to the direct sum decomposition H = H NS ⊕ H R of the space of states. Since for each block there are different Ward identities, it is convenient to investigate these four cases separately.
The simplest case is that of pure NS sector [24]. The Ward identities for a 3-point function suggest the definition of the chiral 3-form (anti-linear in the left argument and linear in the central and the right ones) NN (ξ 3 , ξ 2 , ξ 1 |z) : − NN (ξ 3 , ξ 2 , L n−m ξ 1 |z) , n > −1, where |ξ i | = 0 for an even state and |ξ i | = 1 for an odd state. The 3-form is set by the definition up to two independent constants: For the highest weight state ν 2 ∈ V ∆ 2 we define the 3-point blocks by: where the indices e, o denote even and odd parity of the 3-point block respectively. The even part of the block vanishes when ξ 3 , ξ 1 are states of different parity, while the odd part vanishes for ξ 3 , ξ 1 of the same parity.
The even-even and the odd-odd products of the left and the right constants (24) yield the two structure constants (8) ).

An arbitrary 3-point function of the NS fields with a definite parity is determined by the
Ward identities up to one of the two structure constants: Thus the decomposition of the superprimary field φ N N in terms of normalized vertex opera- Due to the complicated form of the Ward identities the chiral decomposition of R primary fields is considerably more involved [27]. In contrast to NS sector any 3-point function involving R fields with definite parity depends on both R structure constants (9). The chiral Ward identities defining 3-forms 3 The vertex decomposition of the other 3 primary NS fields (4) is presented in [24] 4 The Ward identities for the Virasoro generators Ln (20)- (23) are the same in all sectors.
They determine each 3-form up to four rather than two constants: Since the R fields correspond to states from the "small representation" (7) eight even products of the left and the right NR/RN constants reduce to the two structure constants [27] C (±) where C ± 3 [2][1] are 3-point correlators of primary fields (9). The mechanism of constants' number reduction together with the properties of ρ ı NR/RN suggest the convenient basis for the 3-point blocks in each sector: Using these bases we define the chiral vertex operators: In terms of these operators the fields R + NR and R + RN have the following diagonal representation [27]: Let us stressed that R RN are expressed in terms of the vertex operators V The case of the NS superprimary field in the R-R sector has not been investigated before.
The chiral Ward identities take the form: As in the previous two cases the Ward identities determine the 3-form up to four constants: We write φ RR in terms of the non normalized chiral vertex operators in the following form: Considering the matrix elements of φ RR between primary states w ± ∆,∆ from the "small representation" (7) one can see that eight even products of the left and the right RR constants reduce to the two structure constants (28): In order to express an arbitrary 3-point correlation function in terms of these structure constants one needs several relations between the normalized 3-forms ρ ı RR . From the Ward identities it follows that the 3-forms of the same parity satisfy: for the even number of fermionic operators in both strings: #M + #N = 2N and for #M + #N = 2N + 1. The relations between the 3-forms of different parity have the following form: These identities together with the constants' number reduction (32) allow to write an arbitrary matrix element of φ RR in the diagonal form: for #I + #J = 2N and for #I + #J = 2N + 1. The basis for the 3-point blocks has the following form: Introducing the normalized chiral vertex operators the superprimary field can be written in the diagonal form similar to (31): Let us note that for the 3-point blocks (36) the relations (33)-(35) imply : The similar identities are fulfilled by the ρ Finally, we note that the blocks are functions of β i rather than the Ramond weights. It follows from (33)-(35) that the blocks with opposite signs of β i are related: and [27]: 3 The 4-point blocks

Definitions
We shall write the 4-point correlation functions (14)- (19) and four odd  A correlation function of two NS superprimaries and two R primary fields factorized on the NS states will be expressed by two even blocks: and two odd blocks where the coefficients are given by: NR,|f | (ν ∆,KM , w + β 2 , w + 1 ). (44) 6 We have corrected the definition from [27] where the blocks coefficients were written in terms of conjugated where f = e and f = o denote blocks with even and odd intermediate states, respectively. The even and odd coefficients are defined in terms of the 3-point blocks with the corresponding parity: F n,f c,β is the inverse of the Gram matrix [34,35] in the subspaces of the f parity. Each parity subspace can be decomposed into the direct sum of the subspace spanned by the basis vectors containing S 0 and the subspace spanned by all other basis vectors. Blocks of the Gram matrices with respect to these decompositions are related to each other. This in order implies some relations between blocks of inverse Gram matrices. Using these relations and the 3-point blocks property (39) one can show that the odd and the even 4-point blocks are equal: We shall need two more types of 4-point blocks corresponding to the correlators factorized on R states 7 : F n,f c,β As in the previous case, due to the properties of 3-point blocks (39) and (38), the odd coefficients are proportional to the even ones: Thus there are four independent even blocks of each type: The representations of primary fields through the vertex operators (25), (31), (37) imply the following decompositions of the 4-point correlation functions on the 4-point blocks: = dp

Classical limit of the 4-point blocks
Let us now investigate the behavior of the 4-point blocks in the classical limit b → 0, 2πµb 2 → m = const. A correlation function of two NS superprimary fields (2) and two R primaries (5) is defined by the path integral with the action (1): The twist fields have light weights and thus they do not contribute to the classical limit. If all the exponential operators have heavy weights ∆ a i : the correlator has the same asymptotic behavior as the bosonic 4-point function of primary fields: where S cl [δ 4 , δ 3 , δ 2 , δ 1 ] is the bosonic Liouville action The asymptotic behavior of the 3-point structure constants reads: where S cl [δ, δ 2 , δ 1 ] is the 3-point classical bosonic Liouville action. The b −2 coefficient in the second relation arises due to the fermionic contribution from φ 2 field (4) [30].
A similar reasoning applied to the 4-point correlator (50) projected on an even-even subspace of W ∆ ⊗W∆ yields: where the "∆-projected" classical action is given by and f δ δ 3 δ 2 δ 4 δ 1 (z) is the classical conformal block [12]. The 4-point correlator projected on an even-even (or odd-odd) subspace of V ∆ ⊗V∆ has the same behavior: Thus the equations (45)-(49) together with (51) lead to the following asymptotical behavior of the 4-point blocks: The coefficient in the last term can be derived analyzing the leading ∆ dependence of the odd 3-point blocks with an arbitrary NS state from level k [30,27]

Elliptic recurrence for the blocks with an NS intermediate weight
with ∆ rs (c) given by the Kac determinant formula for NS Verma modules: In order to calculate the residues it is convenient to choose a specific basis in the NS module [30]. Let us remind its construction. First, one introduces the states: are the coefficients of the singular vector in the standard basis of V ∆rs : The family of states S −L L −N χ ∆ rs |L|+|N |=n− rs 2 can be completed to a full basis in the NS module V ∆ at the level n > rs 2 . Working in such a basis in the NS module one obtains: where the coefficient A rs (c) is given by: The exact formula for A rs (c) is due to A. Belavin and Al. Zamolodchikov [36]: in [37].
Due to the factorization property of the 3-point blocks (A.9) the residue (54) is proportional to a coefficient of the same block R k c, rs for rs 2 ∈ N, and to a coefficient of another block R k c, rs In order to derive a closed elliptic recurrence for blocks' coefficients one has to investigate the large ∆ asymptotic. According to Zamolodchikov's reasoning [11,12] the ∆ i and c dependence of the first two terms in the 1 ∆ expansion can be read from the classical limit of the superconformal blocks. From the path-integral arguments it follows that in the classical limit the bosonic classical block occurs (53). It yields the large ∆ asymptotic in the form: for the even blocks, and: for the odd blocks. f ± (z) are functions specific for each type of block and independent of ∆ i and c.
Introducing the multiplicative factor which captures all the ∆ i and c dependence of nonsingular terms one defines the elliptic blocks: The elliptic blocks H e,o ∆ ∆ 3 ±β 2 ∆ 4 β 1 (q) can be written as sums over simple poles in ∆. The residues are given by the corresponding residues of the superconformal blocks (55),(56): Since the functions g ± are independent of ∆ i and central charge they can be read off from the special c = 3 2 blocks. The explicit expressions for the c = 3 2 blocks with the NS external weights ∆ 1 = ∆ 2 = 1 8 and the R external weights ∆ 3 = ∆ 4 = 1 16 can be calculated using the techniques of the chiral superscalar model [38]. In the present case it yields which gives The recursive representation of the 4-point blocks corresponding to a correlator of four R fields (42),(43) can be derived in the same way and it reads 8 : where g e = 1, g o = 0.

Elliptic recurrence for the blocks with R intermediate states
The where r, s ∈ N and the sum r + s must be odd. Thus the 4-point blocks can be expressed as the following sums over simple poles in β: In order to determine the residues at β rs and −β rs we need to choose a specific basis in the R module. Let us introduce the states: where we have used the relation between 3-point blocks with opposite signs of β rs (41). The is of the form [36]: where m + n ∈ 2Z + 1, (m, n) = (0, 0). The equations (63),(64) together with factorization formulae for 3-point blocks (A.11) lead to the expressions for the residues: where corresponds to the shifted weight ∆ rs + rs 2 and P rs c ± β ∆ı are the fusion polynomials (A.3).
By similar calculation one can determine all the other residues: In order to find a closed recurrence for the 4-point blocks with R intermediate states we will define the corresponding elliptic blocks. The first step is to determine large β asymptotic of the 4-point blocks. It follows from Zamolodchikov's reasoning [12], [30] that the ∆ i ,β i and c dependence of the first three terms in the large β expansion of the block is given by the classical limit. Since in the classical limit the bosonic classical block occurs (52), the linear in β terms in the large β asymptotic of the 4-point blocks have to be zero. The asymptotic takes the same form as in the case of the even block F e ∆ ∆ 3 ±β 2 ∆ 4 β 1 (z) (57). This suggests the following definition of the elliptic blocks: The elliptic blocks satisfy recursive relations with the coefficients at residues given by (65), : Comparing the formulae above with definitions (68), (69), (70) one gets:

Numerical check of bootstrap equations 4.1 Four R fields
The correlation function of 4 R primary fields expressed in terms of elliptic blocks (45), (61) take the following form: where q(z) = e iπτ and τ is defined by the complete elliptic integral of the first kind K(z): The crossing symmetry conditions for the 4-point function (14),(17) read: The first equation can be verified analytically due to the relations: which follow from the recursive formula (62) with the fusion polynomials satisfying (A.7).
The second crossing symmetry condition (73) can be checked numerically with the help of the recursive relations for the blocks (62). In order to simplify calculations one can choose two different non-zero external weights: With such a choice none of blocks vanishes which provides a non-trivial test of the general recursive formulae. The products of two SLFT structure constants (12), (28) take the form:  (74) with the P -dependent part: The form of the function above follows from the integral representation of Υ b (13). The bootstrap equation (73) reads: where q = e − iπ τ . Due to the highly oscillatory character of the integrant in the function r 1 , 2 1 (P ), the numerical calculations should be carefully performed. We present a check of 4 ) and for τ along the imaginary axis in the range [0.2i, 5i]. On Figure 1 the relative difference of the left and the right side of (74) as a function of τ is plotted. The three curves correspond to the expansions of the even and the odd elliptic blocks up to the terms q n and q n+ 1 2 for n = 8, 9, 10.

R intermediate weights
The correlation functions of 2 R primaries and 2 NS superprimary fields (48), (49) written in terms of elliptic blocks (69), (70) read: [1] (τ,τ ), Crossing symmetry of the 4-point functions (15), (18) implies: Since the elliptic blocks of the two types are related (71, 72): the first condition is satisfied straightforwardly. The second condition can be verified numerically. For the external weights: the products of the structure constants read: with the P -dependent part: sinh t 2b sinh bt Then the bootstrap equation takes the form: We present the sample calculation for p 1 = 0.3, p 2 = 0.4, c = 3, and for τ in the range [0.2i, 5i]. The relative difference of the both sides of (75) is plotted on Figure 2. The three curves correspond to the elliptic blocks expanded up to n = 8, 9, 10 power of q.

Functions factorized on R and NS states
The crossing symmetry of the 4-point functions (16) The first condition can be checked analytically using the relations: which follow from recursions (59),(60) and the properties of fusion polynomials (A.7),(A.8).

Factorization over an NS singular vector
The fusion polynomials corresponding to 3-point blocks with the singular NS state were defined in [24], [27]: