Comments on fusion matrix in N=1 super Liouville field theory

We study several aspects of the $N=1$ super Liouville theory. We show that certain elements of the fusion matrix in the Neveu-Schwarz sector related to the structure constants according to the same rules which we observe in rational conformal field theory. We collect some evidences that these relations should hold also in the Ramond sector. Using them the Cardy-Lewellen equation for defects is studied, and defects are constructed.


Introduction
During the last decades we got deep understanding of the properties of rational conformal field theories having a finite number of primaries. Many important relations were obtained between basic notions of RCFT. In particular we would like to mention the Verlinde formula [1], relating matrix of modular transformation and fusion coefficients, Moore-Seiberg relations between elements of fusion matrix, braiding matrix and matrix of modular transformations [2][3][4]. We have formulas for boundary states [5], and defects [6,7] in rational conformal field theories. Situation in non-rational conformal field theories is much more complicated. The infinite and even uncountable number of primary fields is the main reason that progress in this direction is very slow. One of the well studied non-rational theories is Liouville field theory. Liouville field theory has attracted a lot of attention since Polyakov's suggestion to study strings in non-critical dimension. Three-point correlation function (DOZZ formula) [8,9] and fusing matrix [10] were found exactly. Other important examples of the non-rational CFT are N = 1 superconformal Liouville theory, conformal and superconformal Toda theories and more general para-Toda theories. It is interesting to mention that all of them play a role in the recently established AGT correspondence [11][12][13][14][15][16][17][18][19]. Many data have been collected also in N = 1 superconformal Liouville theory. In particular three-point functions [20,21] and the NS sector fusion matrices [22,23] have been found exactly. Some attempts to find the fusion matrix also in the Ramond sector can be found in [24,25]. In this paper we study the following relations, proved in rational CFT without multiplicities (fusion numbers N i jk = 0, 1), in N = 1 super Liouville field theory: where and which using (1) can be written also The first relation (1) is a consequence of the pentagon identity for fusion matrix [2][3][4]. The second relation (3) results from the bootstrap equation combined with the pentagon identity [5,[26][27][28]. These relations were examined in the Lioville field theory. The relation (1) in the Lioville field theory was tested in [29]. The relations (3) and (4) were examined in the Liouville field theory in [28,30]. In [28] (3) and (4) in the Liouville field theory were checked using the relation of the fusion matrix with boundary three-point function. In [30] (3) was checked using the following integral identity for the double Sine-functions S b (x): Recently it was found in [34] supersymmetric generalization of this formula (eq.(62 in text). We find that also for N = 1 super Liouville theory the susy version of this formula leads to the corresponding generalization of the relations (3) and (4) in N = 1 superLiouville theory. The paper is organized as follows. In section 2 we review basic facts on N = 1 superLiouville theory. In section 3 we compute the elements of an Ansatz for the fusion matrices with one of the intermediate states set to the vacuum. In section 4 we specialize the formulae obtained in section 3 to the fusion matrices of the NS sector found in [22]. In section 5 we analyze the Ramond sector for a degenerate entry. In section 6 we apply formulae obtained in section 5 to solve the Cardy-Lewellen equations for topological defects. In appendix some useful formulas are collected.

N=1 Super Liouville field theory
Let us review basic facts on the N = 1 Super Liouville field theory. N = 1 super Liouville field theory is defined on a two-dimensional surface with metric g ab by the local Lagrangian density The energy-momentum tensor and the superconformal current are The superconformal algebra is with the central charge where Here k and l take integer values for the Ramond algebra and half-integer values for the Neveu-Schwarz algebra.
NS-NS primary fields N α (z,z) in this theory, N α (z,z) = e αϕ(z,z) , have conformal dimensions The physical states have α = Q 2 + iP . Introduce also the field The R-R is defined as where σ is the spin field ‡ . ‡ Sometimes the Ramond field is defined as R ± α (z,z) = σ ± (z,z)e αϕ(z,z) , but in this paper the second field R − is not important.
The dimension of the R-R operator is The NS-NS and R-R operators with the same conformal dimensions are proportional to each other, namely we have where G N S (α) and G R (α) are so called reflection functions. They also give twopoint functions. The elegant way to write the reflection functions is to introduce NS and R generalization of the ZZ function [31] in the bosonic Liouville theory: The reflection functions can be written The functions (21) and (22) satisfy also the relations The degenerate states are given by the momenta: with even m − n in the NS sector and odd m − n in the R sector. For the super conformal theory, characters are defined for the NS sector, for the R sector and the NS sector. The corresponding characters for generic P which have no null-states are where q = exp(2πiτ ) and Modular transformation of characters (28) -(30) is well-known: For degenerate representations, the characters are given by those of the corresponding Verma modules subtracted by those of null submodules: Modular transformations of (35) -37) are χ N S m,n (τ ) = χ R P (−1/τ )2 sinh(πmP/b) sinh(πnbP )dP , m, n even , Note that the vacuum component of the matrix of modular transformation specified by (m, n) = (1, 1) in formulae (38) - (40) coincide with the right hand side of (25) and (26) The structure constants in N = 1 super Liouville field theory are computed in [20,21]: Fusion matrix in the NS sector is computed in [22,23]. Let us denote (50) To write the fusion matrix we use the following convention. The functions Υ i , Γ i , S i will be understood Υ N S , Γ N S , S N S for i = 1 mod 2, and Υ R , Γ R , S R for i = 0 mod 2. Now we can write the fusion matrix: 3 Values of the fusion matrix for the intermediate vacuum states Motivated by the form of structure constants (44)-(47) and fusing matrix (51) we define the following general expressions for the fusion matrix: with where η = (−1) (1+ i (ν i +µ i ))/2 . I denotes fusion matrices of different structures, and capital Latin letters here take values NS and R. The expressions similar to (58) were considered also in [25] in construction of the Racah-Wigner coefficients. Define also the following general expression for structure constants: Now consider the limit: In this limit using formulae from appendix and the definition (59) we get for the factor in front of integral: Let us now evaluate the integral part of (56) in the limit (60). For this purpose we will use the formula [34] ν=0,1 First note that in the limit (60) the arguments of S K 's in numerator and denominator coincide and they get canceled.
For the rest of S's in this limit we get for a i in the argument of S ν i (τ + a i ) and b i in the argument of S µ i +1 (−τ + b i ): From (65) we obtain Note that Let us impose also Assuming also that (63) is satisfied we get from (62) using formulas (66)-(71) (72) Requiring additionally that where these equalities as before understood in a sense, that odd sums identified with the NS sector, and even sums identified with the Ramond sectors, we get 3.2 α t → 0 limit Consider the same fusing matrix, but parametrized in the form with where η = (−1) (1+ i (ν i +µ i ))/2 . We change here notations for the capital Latin letters denoting different spin structures. This is done to keep parametrization for the capital Latin letters in the formula for structure constants (59). Alternatively we could keep the same parametrization in formula for fusing matrix and change the notations in formula for structure constants.
Consider the limit In this limit using formulas in appendix and (59) we have for the factor in front of integral .
Consider now the limit of the integrand (77). In the limit (78) the arguments of S K 's in numerator and denominator coincide and they get canceled.
For the rest of S's in this limit we get for a i in the argument of S ν i (τ + a i ) and b i in the argument of S µ i +1 (−τ + b i ): From (80) we easily obtain: Note that Assume that Under these conditions we get from the theorem (62) , using formulas (81)-(86) (87) Requiring additionally that where these equalities as before understood in a sense, that odd sums identified with the NS sector, and even sums identified with the Ramond sectors, we get

Fusion matrix in the Ramond sector
The fusion matrix in the Ramond sector unfortunately is not known in general. Although for some attempts see [24]. But for the degenerate primaries (27) fusion matrix can be computed via direct solutions of the corresponding differential equation for conformal blocks. In particular the necessary elements of the fusion matrix when one of the entries is the simplest degenerate field R −b/2 are computed in [32,33]. The degenerate field R −b/2 possesses the OPE: The corresponding structure constant can be computed in the Coulomb gas formalism using the screening integrals: The fusion matrices can be computed having explicit expression of the conformal blocks with degenerate entries: It is an easy exercise to check that the values of the structure constants (98)-(101) and fusion matrices (102)-(105) satisfy the relations: (109) One expects that similar relations should hold also for general expressions of the corresponding elements of fusion matrix in the RR sector. For example the fusions matrix with four RR entries should satisfy the relations One can hope that constraints like (110) and (111) may help to obtain the general expressions for the corresponding elements of the fusion matrix.

Defects in Super-Liouville theory
Two-point functions with a defect X insertion can be written as where and C ii is a two-point function. They satisfy the Cardy-Lewellen equation for defects [7,28,35,36] Denote Let us take j = R −b/2 . Using (96), (97) and (106)-(109) one can obtain: where The solution of the equations (117) and (118) is with m − n is even. Substituting (121) and (122) in (119) and (120) we obtain Dividing by two-point functions (23) and (24) we obtain . (126) To obtain the continuous family of defects we use the strategy developed in [37,38]. Namely consider D R (−b/2) as a parameter characterizing a defect. More precisely we define Denoting also we obtain The solution of (130) and (131) is given bỹ with a parameter u related to A by 2 cosh 2πbu = A .

Discussion
The methods of this paper can be useful to construct fusion matrix in the parafermionic Liouville field theory [15]. Parafermionic Liouville field theory is the simplest generalization of the supersymmetric Liouville theory. Whereas the supersymmetric Liouville theory is the Liouville field theory coupled to the Ising model, the parafermionic Liouville field theory is the Liouville field theory coupled to the parafermions. The structure constants of the parafermionic Liouville field theory at the level N can be written using the following generalization of the Υ N S and Υ R functions (139) It is easy to check that these functions can be written as where Γ (N ) (141) The functions Γ which is very similar to (148) and (149). Recall that these properties played crucial role in calculations in section 3. Therefore one can try to write fusion matrix in the paraferminonic Liouville field theory using the corresponding para version of the double Gamma and double Sine functions and matching the relations (1), (3), (4) with the parafermionic Liouville structure constants found in [15]. It is well known that in the AGT correspondence Wilson lines in the N = 2 SU(N) (SU(2)) superconformal gauge theory on S 4 correspond to topological defects in Toda (Liouville) conformal field theory [39]. On the other hand it is found that N = 2 SU(N) (SU(2)) superconformal gauge theory on S 4 /Z p correspond to parafermionic Toda (Liouville) field theories. In particular N = 2 SU(2) superconformal gauge theory on S 4 /Z 2 correspond to supersymmetric Liouville field theory. Thus having the topological defects in superLiouville theory one can test the AGT correspondence of SU(2) superconformal gauge theory on S 4 /Z 2 with supersymmetric Liouville field theory in the presence of the Wilson lines.
The Lagrangian of the N = 1 super Liouville field theory with the topological defect was introduced in [40]. In [36] the light and heavy semiclassical limits were used to match two-point correlation function with the Lagrangian approach for the bosonic Liouville theory in the presence of the defects. It is an interesting task to match, using various semiclassical techniques, the results of section 6 with the Lagrangian of [40].
The function Υ b (x) may be defined in terms of Γ b (x) as follows .
It has the following property: In the super Liouville theory are important the functions They have the properties: where W N S (α), W R (α) are defined in (21) and (22), and λ = πµγ bQ 2 b 1−b 2 . Γ NS (x) has a pole in zero: The structure constants in the super Liouville theory are defined in terms of the functions: . 23 They have the properties: where G N S (x) and G R (x) are defined in (23) and (24) .
To write fusion matrix we need also the functions: .
They have the properties: And finally we need the following properties which can be easily obtained from the definitions and properties above: where A takes values NS or R.