S-move matrix in the NS sector of N = 1 super Liouville ﬁeld theory

In this paper we calculate matrix of modular transformations of the one-point toric conformal blocks in the Neveu-Schwarz sector of N = 1 super Liouville ﬁeld theory. For this purpose we use explicit expression for this matrix as integral of product of certain elements of fusion matrix. This integral is computed using the chain of integral identities for supersymmetric hyperbolic gamma functions derived by the degeneration of the integrals of parafermionic elliptic gamma functions.


Introduction
One of the basic objects in two-dimensional conformal field theory is the Smatrix of modular transformation of one-point toric conformal blocks.
One can write the one-point correlation function on a torus with modular parameter τ in the following form: where q = e 2πiτ , F λ γ (q) are one-point conformal toric blocks and C λ, λ γ,γ are structure constants.
The S-matrix of modular transformation of the one-point toric blocks on a torus is defined by relation The S-matrix is one of the most important notions of 2D conformal field theories.The S-matrix, together with fusion F -matrix, braiding B-matrix and T -matrix, representing the second generator of the modular group τ → τ + 1, form the Moore-Seiberg groupoid [34].These generators satisfy a set of relations called Moore-Seiberg's equations [34].One of them will be discussed below in detail.The Moore-Seiberg groupoid gives in a sense the full number of operations on the space of conformal block.More abstractly one can consider the space of conformal blocks as a representation of Moore-Seiberg groupoid.For better understanding of the Moore-Seiberg's equations we recall that the space of conformal blocks in rational conformal field theory also arises as Hilbert space of states of 3D topological field theory [22,54].It was observed that the Moore-Seiberg's equations are equivalent to three-dimensional topological invariance [33,54].In fact, it was proved for rational theories that given a representation of the Moore-Seiberg groupoid, namely a solution to the Moore-Seiberg's equations, one can reconstruct the 3D topological field theory [19].
The Moore-Seiberg groupoid proved to be useful also for non-rational theories.
It was proposed in [53] that the space of conformal block of Liouville theory with its mapping class group representation is isomorphic to the Hilbert space of states of quantized Teichmuller space.Later this proposal was refined in [50,51] and it was shown that quantized Teichmuller space carries unitary projective representation of Moore-Seiberg grupoid equivalent to that of Liouville theory.Using these data it was built in [17,18], in full analogy with rational case, topological field theory based on conformal blocks of the Liouville theory, called Virasoro topological quantum field theory.In these papers this Virasoro topological quantum field theory was related to 3D quantum gravity with negative cosmological constant.
Here we used that S αβ 2 (β 5 ) in the limit β 5 → 0 becomes matrix S αβ 2 of the modular transformation of characters and S 0β 2 in ( 5) is the corresponding element of this matrix.The formula (5) was used in [52] to calculate S β 6 β 2 (β 5 ) in the Liouville field theory.The calculations are very involved and use sequence of integral identities for various products of hyperbolic gamma functions.Important point here is that all the necessary identities can be derived by some chains of the reductions from the elliptic beta integral [45] and symmetry relations of so called elliptic V -function [46,47].This becomes possible since as shown in [38,39] the hyperbolic gamma function is an asymptotic form of the elliptic gamma function.As a result it was obtained that the S L αβ 2 (β 5 )-matrix in Liouville theory is given by the expression [50,52]: where N (α 1 , α 2 , α 3 ) are some normalization coefficients of chiral vertex operators, is the corresponding component of the matrix of modular transformation of characters, and Here S b (x) is hyperbolic gamma function defined in appendix A. For the physical states all the entering parameters belong to the set Q/2 + iR.
Using identities (108) and ( 140) in appendices B and E correspondingly for r = 1 one can show that this function enjoys the following properties and Note that function ( 8) is nothing else as partition function of mass-deformed N = 4 SQED with two electron hypermultiplets, called T [SU (2)], on squashed three-sphere S 3 b and eq.( 9) expresses its mirror symmetry [27,28].This coincidence is not accidental but consequence of AGT dualty [3] between 4D N = 2 SU (2) supersymmetryc gauge theory on S 4 and Liouville theory in the presence of domain walls/defects [20,28].It was proposed in [20] that partition function of 3D S-duality domain wall inside 4D N = 2 * theory, which is the mentioned mass-deformed T [SU (2)] theory, should coincide with the kernel of S-move transformation of Liouville theory.The actual coincidence was established in [28].
It is possible to show that the invariance of the one-point correlation function (1) under the modular transformation requires S-matrix to satisfy the following Moore-Seiberg equation (see for example [17,52]): It is easy to check that using the relations ( 9) and (10) the eq.( 11) can be written as This property is closely related to the fact that the function S T αβ 2 (β 5 ) satisfies the following eigenvalue equation [24,35] where H is the finite-difference operator: This equation can be derived in the numerous ways.First one can obtain it evaluating the Moore-Seiberg equation (3) for degenerate primaries [35].It is possible directly to check [24] that S T αβ 2 (β 5 ) satisfies (13).And finally one can derive it by certain limiting procedure from the difference equations discussed in [4].The operator (14) is well known Hamiltonian of the relativistic Calogero-Sutherland model [40], sometimes also called Ruijsenaars-Schneider model, and was extensively studied recently is series of works [11][12][13].Now presence of the δ-function in the r.h.s. of ( 12) is easily understood from the observation that the Hamiltonian ( 14) is hermitian w.r.t. the scalar product given by the l.h.s. of (12).The integral in l.h.s. of (12) is computed in [11][12][13] and indeed coincides with r.h.s. of (12).We see that the function ( 8) is on the junction of three theories: Liouviile field theory, T [SU (2)] theory and relativistic Calogero-Sutherland model, and various its properties have different interpretation in each of them.
In this paper we use the formula (5) to calculate S-matrix of modular transformation of one-point toric conformal block in N = 1 super Liouville field theory (SLFT).For this purpose we should elaborate "supersymmetric counterpart" of the mentioned integral identities.At this point we would like to point out that if the essential part of the fusion matrix in the Liouville field theory is integral of product of hyperbolic gamma functions S b (x), defined in appendix A, then the essential part of the fusion matrix in N = 1 super Liouville field theory is sum of integrals of product of so called Neveu-Schwarz and Ramond hyperbolic gamma functions: where Q = b + 1/b.To derive the necessary integral formulas for them first we note that the functions ( 15) and ( 16) are particular case of so called parafermionic hyperbolic gamma functions [30,36,42], defined for any r ∈ N: where 0 ≤ m ≤ r.Obviously, for r = 2 one obtains the functions ( 15) and ( 16): Λ(y, 1) = S 0 (y) .
In paper [49], rarefied elliptic gamma functions are constructed and corresponding generalizations of the elliptic beta integral and elliptic V-function are studied.It was checked also in [42] that the functions (17) are asymptotic forms of the rarefied elliptic gamma functions.Therefore we can get the necessary identities using the same chains of the reductions as in the case r = 1.Some of the necessary integral identities were derived in this way in papers [4,42].Others are obtained in appendices B-E .Our main results are given by formulae (63) and (80).In fact these integrals describe also bulk-boundary coupling in super Liouville theory as we explained above.
In super Liouville theory we find as well that integrals describing matrix of modular transformation reside on the junction of different theories.
First one can see that integral (63) is zero-holonomy value of the partition function of the above mentioned mass-deformed T [SU (2)] theory on squashed lens space S 3 b /Z 2 found in [36].This is in agreement with AGT correspondence between N = 2 SU (2) supersymmetric gauge theory on S 4 /Z 2 and super Liouville theory [8-10, 15, 16].Thus we can consider our results as an evidence for generalization of the AGT correspondence between gauge theory on S 4 /Z 2 and super Liouville theory in the presence of the domain walls.It is important further to study correspondence between holonomies entering in partition function of T [SU (2)] theory on squashed lens space S 3 b /Z 2 and types of primaries entering in matrix of modular transformation.
The paper is organized as follows.In section 2, we review some basic facts on N = 1 Super Liouville field theory.In section 3, we calculate S β 1 β 2 (β 3 ) in the case when all β i , i = 1, 2, 3 are NS primaries.The corresponding result is given by formulae (63) and (64).In section 4, we calculate S β 1 β 2 (β 3 ) in the case when β i , i = 1, 2 are NS primaries and β 3 is ÑS primary.It is given by ( 80) and (81).In section 5, we comment our results and outline possible applications.In appendices A-E the necessary special functions, their properties and integral identities are reviewed.
2 Basics on 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 Lagrangian density: The energy-momentum tensor and the superconformal current are where with the central charge Here k and l take integer values for the Ramond algebra and half-integer values for the Neveu-Schwarz algebra.Since in the Neveu-Schwarz sector of N = 1 SLFT one has besides the Virasoro generators L m , also supercurrent generators G k with half-integer k, descendant fields are broken into two sets of integer and half-integer levels.Thus we will work with primary fields associated with the vertex operators N α = e αϕ , which we call NS field, and its supercurrent descendant Ñα = G −1/2 N α , which we call ÑS field [7].The N α primary field has conformal dimension The physical states have α = Q 2 + iP .The Ñα has the conformal dimension 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 future use we need matrix of modular transformation of the NS characters.The corresponding characters for generic P which have no null-states are where q = exp(2πiτ ) and Modular transformation of characters ( 30) is well-known: For the degenerate representations, the characters are given by those of the corresponding Verma modules subtracted by those of null submodules: Modular transformations of ( 33) is Given that the identity field is specified by (m, n) = (1, 1) one finds the vacuum component of the matrix of modular transformation: For brevity the N α and Ñα primary fields will be denoted by α and α correspondingly.Therefore, if for example we take in the relation (5) α, β 2 and β 5 of NS type, it takes the form As we mentioned before S αβ 2 (β 5 ) in the limit β 5 → 0 should coincide with the matrix of modular transformations of the characters, and S 0β 2 is given by (35).The elements of the fusion matrix in the Neveu-Schwarz sector of N = 1 SLFT are given by the formulae [25,26]: where normalization coefficients are described by the relations: , where the functions Γ N S (x), Γ R (x) are described in appendix A, and 6j-symbols of the quantum algebra U q (osp(1|2)) are [26]: i=1 ν i = 0 mod 2, and where The second equality in (44) was established in [4].Let us explain briefly how it arises.First recall the following periodicity and reflection properties of Λ(y, ν) [42,43]: For r = 2 the relation (46) implies The functions S ν (x) are supposed to be periodic in the discrete variable: S ν (x) = S ν+2 (x).This together with (48) implies Repeatedly using (49) one can arrive to the second equality in (44).One can check that the expressions (37)-( 40) are invariant under the reflection α → Q−α of any of six variables entering in the fusion matrix.Inserting (37)-( 40) in (36) we derive: where S αβ 2 (β 5 ) is computed by the same formula (36) where inserted only corresponding elements of 6j-symbols of the quantum algebra U q (osp(1|2)) appearing in the l.h.s of ( 37)-( 40): where 3 Elements of S-matrix for three NS primaries Now we turn to the calculation of the matrix S αβ 2 (β 5 ) in the case when all the entering primaries of the NS type.For this purpose we use the formula (51).
Start to calculate the left hand side integral (52).The first term in (52) is the following element of the 6j-symbol: 11 .Setting in ( 43)-( 44) To compute integral here we used formula (106) in appendix B for r = 2. Now we should compute the second term in (52): 11 .Setting in (43) where Writing in this way we separated in integral appearing in (54) β 4 -dependent terms and collected in F (y, ν) all β 4 -independent terms.The next element of the fusion matrix to be computed is 00 .Putting in (43) The last element of the fusion matrix which we need to compute is 11 .Now setting in (43) were the function F (y, ν) was defined in (55).
Since β 4 has the form β 4 = Q/2 + iP for some real P , one can write , Recalling also formulae for conformal dimensions ( 27) and ( 28) we obtain (59) Here we used the shorthand notation Λ(y ± x, m ± n) ≡ Λ(y + x, m + n)Λ(y − x, m − n).The integrals appearing in the third and the fourth lines of (58) are particular cases of the integral (114) in appendix B for r = 2. Hence using (114) in appendix B for r = 2 one has , where Changing sign of y and using that for ν = 0, 1 hold the relations: Λ(y, −ν) = (−1) ν Λ(y, ν) and Λ(y, 2 − ν) = Λ(y, ν) we can write the last integral in (60) in the form .
It is easy to see that this integral has the same structure as the integral on the left hand side of relation (128) in appendix D for r = 2, therefore we can write for him taking as Note that integral in the r.h.s. of (62) coincides with the integral in the r.h.s. of (124) in appendix C for r = 2. Thus we can write for the latter Collecting all we obtain: ) dy i or using (49): This expression has number of properties analogous to that of in usual Liouville field theory [52].First we obtain easily Using formula (140) in appendix E for r = 2 one can establish Combining ( 65) and (66) with definition of the normalization factors (41) and using formulae (103) in appendix A we obtain that is invariant under reflection of all three variables.Now let us check that (64) is self-consistent in the limit β 5 → 0. In this limit using formulae in appendix A one has that 1 S NS (β 5 ) ∼ πβ 5 → 0, and thus we should pick up only diverging part of integral.This diverging part originates due to pinching of integration contour in this limit between poles of S NS (y −Q/2+β 2 +β 5 /2) and S NS (−y +Q/2−β 2 +β 5 /2) and also between poles of S NS (−y − Q/2 + β 2 + β 5 /2) and S NS (y + Q/2 − β 2 + β 5 /2).Analyzing poles of S R (x) reviewed in appendix A we see that the second term in (64) has no diverging part.The diverging part caused by pinching is pole with coefficient given by the sum of residues: , where the product S NS (−Q + 2β 2 )S NS (Q − 2β 2 ) was computed using formulae (97) and (98) in appendix A. Recalling ( 35) and ( 32) we obtain that indeed lim 4 Elements of S-matrix for two NS and one ÑS primaries Here we consider the case when α, β 2 are NS primaries and β 5 is ÑS primary.In this case the relation (36) takes the form where S αβ 2 ( β5 ) is given by the relation: and Here δ should be taken to 0. The first terms in the first and second lines 11 00 are the same as before and computed in ( 53) and ( 56) respectively.
To compute the second fusion matrix element in the first line we set in ( 43) where we defined To derive the second fusion matrix element in the second line we set in ( 43): Collecting all, and repeating the same steps as in the first case and using the integral (114) in appendix B, we obtain where The integral (75) is of the form (128) in appendix D, which yields (77) Here θ(x) is Heaviside step function: θ(−1) = 0 and θ(1) = 1.As before the functions with double sign in arguments are used as shorthand for their product where the first term is taken with upper signs everywhere and the second term is taken with lower signs everywhere.For example the first term in integrand reads .
And finally we express the r.h.s. of (77) via (124) in appendix Collecting all we derive or using (49): Note the following properties of this expression.First we obtain easily Using formula (140) in appendix E one can establish Combining ( 82) and ( 83) with definition of the normalization factors (42) and using formulae (103) in apendix A we obtain that is invariant under the reflection of all three variables.

Conclusion
In this paper we calculated the matrix S αβ 2 (β 5 ) in two cases.These results allow us to conjecture that in all cases the integral part of the matrix S αβ 2 (β 5 ) will be given by the following function with the various choices of discrete variables k a , t a , a = 1, 2 and N : Applying some limiting procedures to the finite-difference equation derived in [4] we obtained that for the parameters t 1 , t 2 , k 1 , k 2 obeying the condition the function (85) satisfies the eigenvalue equation: As in the case of Liouville theory one might hope that the correspondence of the elements of the S-move matrix in N = 1 super Liouville theory with the eigenvalues of the l.h.s.operator could help us to prove the Moore-Seiberg condition of orthogonality (11).It is intriguing to study the relation of equation ( 87) with the supersymmetric Ruijsenaars-Sutherland problem [14].
There is also another application of S-move matrix in the AGT correspondence.Remember that logarithm of one-point toric block in semi-classical limit yields Seiberg-Witten prepotential of N = 2 * theory [21,23,32].Therefore asymptotic form of the S-move matrix of Liouville field theory implements electromagnetic duality of SW prepotential.For Liouville field theory this was studied in [23,31,32].We are planning to pursue these issues for S-move matrix in N = 1 SLFT using relation between the Nekrasov partition function of N = 2 * theory on S 4 /Z 2 and N = 1 SLFT one-point toric blocks found in [15,16].
We expect that the S-move matrix of super Liouville theory also should play important role in quantization of the super Teichmuller space [1,2] and building supersymmetric analogue of the Virasoro topological field theory studied in [17,18].
We need also relations of sypersymmetric hyperbolic gamma functions with the supersymmetric double gamma functions [26,37].
Start with defintion of Γ b (x) function.
The function Γ b (x) is a close relative of the double Gamma function studied in [5,44].It can be defined by means of the integral representation Important properties of Γ b (x) are The function S b (x) may be defined in terms of Γ b (x) as follows Similar role in the super Liouville theory play the functions: The sypersymmetric hyperbolic gamma functions are related to them by the formulae similar to (100) B Parafermionic integral identities I In paper [42] the rarefied hyperbolic beta integral has been evaluated ‡ : Here we used the shorthand notation Λ(y It is shown in [42], that in certain limit (104) yields In certain limit (106) leads to another useful identity: We can show that taking the following limit in (104): and using the asymptotics (91) one derives

C Parafermionic integral identities II
In paper [4] some parafermionic hypergeometric functions were defined.The first is the parafermionic generalization of the Ruijsenaars function [40]: Here t a , l a ∈ Z. Parameters γ j , β j and l j , t j satisfy the constraints: The second function is : We proved in [4] the following relation between (115) and (117): where The parameter δ = 0, 1 should be determined from the requirement that L 1 and L 2 are integer numbers.Let us now paparemeterize parameters β, γ in (118) in the following way: Obviously parametrized in this way β and γ satisfy (116).Now putting (123) in (118) and taking the limit A → ∞, denoting l 1 + t 4 = T , and at the end renaming indices 2, 3 → 1, 2 for all variables, what we can do since variables with indices 1, 4 disappear, we obtain the identity where and δ = 0, 1 is determined from the requirement that M 1 and therefore also M 2 are integer.This formula for r = 1 case can be found in [48,52].

D Parafermionic integral identities III
Now let us set in (118) Taking the limit A → ∞, we obtain, after renaming at the end the variables γ 3 , γ 4 , t 3 , t 4 → γ 1 , γ 2 , t 1 , t 2 , the identity and δ = 0, 1 is determined from the requirement that L 1 and therefore also L 2 are integer.For r = 1 case this formula can be found in [52].