Recurrence relations for toric N = 1 superconformal blocks

: General 1-point toric blocks in all sectors of N=1 superconformal ﬁeld theories are analyzed. The recurrence relations for blocks coeﬃcients are derived by calculating their residues and large ∆ asymptotics


Introduction
Correlation functions in any 2-dimensional CFT can be expressed in terms of three-point coupling constants and some universal, model independent functions called conformal blocks [1]. Such decompositions are not unique and the equivalence of various representations yields strong restrictions for coupling constants. Simplest conditions of this kindthe crossing symmetry of the 4-point function on the sphere and the modular invariance of the 1-point function on the torus -turned out to be sufficient for the consistency of all multi-point amplitudes on closed, oriented surfaces of arbitrary genus [2]. From this point of view the 4-point spheric and the 1-point toric conformal blocks are of main interest in any CFT. Both objects are defined as power series of corresponding modular parameters with coefficients expressed in terms of the inverse of the Gram matrix of a Virasoro algebra Verma module. In this form a direct calculation of higher order terms is prohibitively complicated.

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An efficient recursive technique of calculating 4-point conformal blocks on the sphere was developed long time ago by Al. Zamolodchikov [3][4][5]. This method was used for numerical tests of crossing symmetry in the Liouville field theory [6] and in the c → 1 limit of minimal models [7]. It was also applied for the numerical analysis of the classical limit of the conformal blocks and for the verification of new relations in the classical geometry of hyperbolic surfaces [8,9]. Recently in the context of the AGT relation Zamolodchikov's method has been extended to the 1-point toric blocks [10,11]. It found its application in one of the first proofs of the AGT correspondence [12]. It was also used in [11] to prove the relations between 1-point toric and 4-point spheric conformal blocks conjectured by Poghossian [10].
A recursion representation of 4-point spheric blocks in the N=1 superconformal field theories was first derived in the Neveu-Schwarz sector [14][15][16][17][18]. The extension to the Ramond sector initiated in [19] was recently completed in [20]. These results clarified the structure of the N=1 superconformal blocks and paved the way for investigations of their analytical properties [21][22][23]. They were also used for numerical verifications of crossing symmetry in the N=1 superconformal Liouville field theory [16,17,20].
In the present paper we address the problem of recursion representation of the 1-point toric conformal blocks in the N=1 SCFT. Our main motivation is the problem of modular invariance of 1-point functions on the torus in the N=1 superconformal Liouville field theory with the structure constants derived in [24,25]. The corresponding problem in the Liouville theory was solved by showing that the modular invariance of a generic 1-point function on the torus is equivalent to the crossing symmetry of a special 4-point function on the sphere [26]. An essential step of this reasoning is a relation between the modular and the fusion matrices which can be derived using Poghosian identities [10,11]. One may expect a similar, although more complicated mechanism in the N=1 superconformal case. The recurrence representations which in the bosonic case were basic tools in analyzing relations between toric and spheric N=1 blocks are first steps along this line. They are also of some interest for the recently discovered extension of the AGT relation where on the CFT side the N=1 superconformal Liouville field theory shows up [27][28][29][30][31][32].
The paper is organized as follows. Section 2 contains a detailed discussion of toric blocks in all sectors of N=1 superconformal field theories. In section 3 we calculate the residues of blocks coefficients. The method employed is a simple extension of the techniques developed for the spheric case. The main technical point is discussed in section 4 where we calculate the large ∆ asymptotics. The derivation is based on properties of the Gram matrix and the matrix elements of chiral vertex operators. Proofs of these properties are given in appendices A and B. In section 5 we derive the recursion relations for N=1 superconformal toric blocks which are the main results of the present paper. Explicit formulae for first few block's coefficients are listed in appendix C.

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In our notation λ,λ parameterize the left and the right conformal weights of φ λ,λ (z,z) and Q = b + b −1 is related to the central charge c of the NSR algebra by c = 3 2 + 3Q 2 .

NS and NS sectors
The 1-point function of φ λ,λ on a torus with the modular parameter τ can be written as where q = e 2πiτ and the sum runs over the whole spectrum of the NS theory. The matrices calculated in the standard bases in the corresponding NS Verma modules: For K, L andK,L of the same parity one has where C λ,λ ∆,∆ ,C λ,λ ∆,∆ are the three-point constants , and ρ NN , ρ * NN are 3-point conformal blocks in the NS sector. In the formulae above and in the rest of the paper we follow the notation conventions of [23].
The toric conformal blocks are defined by

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where the symbol stands for the star or the lack of it. The 1-point functions can then be represented as In NS sector one introduces the modified conformal blocks [38] F λ and the 1-point functions take the form

R and R sectors
The toric 1-point functions in the R sector read The matrices B f calculated in the standard bases in the corresponding R Verma modules W β , Wβ:

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The representation for the 1-point toric function ofφ λ,λ reads As in the case of 4-point blocks on the sphere [20] one can show cβ,e (q), cβ,e (q), and it is enough to consider the even blocks alone.

NS and NS sectors
The method to derive the recursion relations is essentially the same as in the Virasoro algebra case [11]. The blocks' coefficients (2.3) are polynomials in the external weight ∆ λ and rational functions of the intermediate weight ∆ and the central charge c. The poles in ∆ are given by the Kac determinant formula for the NS Verma modules (r + s ∈ 2N): (3.1)

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They are related to the null states in the Verma modules V ∆rs . For a generic value of the central charge the modules V ∆rs+ rs 2 are irreducible and the poles are simple [14], hence: Following the method of [14] one gets where the coefficients A rs are given by: The exact formula for A rs (c) was proposed by A. Belavin and Al. Zamolodchikov in [35].

It reads
where m + n ∈ 2Z, (m, n) = (0, 0), (r, s). The corresponding expression in the bosonic case was first conjectured by Al. Zamolodchikov [4,5] and recently proved by Yanagida [34]. In order to calculate the residues we shall use the factorization property of the 3-point blocks [14]. For |K| + |L| ∈ N one has in particular By the same token The 3-point blocks in the formula above can be expressed in terms of the fusion polynomials (3.8) and the products run over: Using the relations where s rs = 1, * s rs = (−1) rs .

R and R sectors
The blocks' coefficients (2.10), (2.11) are polynomials in the external weight ∆ λ and rational functions of β and the central charge c. The poles in β are given by the Kac determinant formula for the positive parity subspace of an R Verma module. They are located at ±β rs where and r + s ∈ 2N + 1. The poles are related to the positive parity null states For a generic value of the central charge the modules W βrs+ rs 2 are irreducible and the poles are simple [20], hence: (3.11)
In terms of the fusion polynomials P rs Since the residues at ±β rs differ by sign one simply gets

Large ∆ asymptotics
In order to complete the derivation of recurrence relations one needs the large ∆ asymptotics of conformal blocks. Their rigorous calculation turned out however to be more difficult that in the Virasoro algebra case [11]. The method presented in this section is based on properties of the Gram matrix and the matrix elements of the chiral vertex operators, collected in Propositions 1-4. Their proofs are given in appendices A and B.

NS and NS sectors
Let B f denotes the standard basis of level f subspace of the NS Verma module: It is convenient to use a simplified notation for elements of this basis Proposition 1. Let Q be a polynomial in ∆ and let deg ∆ Q denotes its degree. Then: 2. for any u i , u j ∈ B f , u i = u j : 3. the product of the diagonal terms is the only highest degree term in the determinant of the Gram matrix with respect to the base B f i.e.
By Proposition 2, for off diagonal elements

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Suppose the first inequality holds. The minor M ji of the Gram matrix can be represented as where the sum runs over permutations τ such that τ (i) = j. By Proposition 1: hence, for every permutation τ : Taking into account the first inequality of (4.2) one thus gets If the second inequality of (4.2) holds one follows the same reasoning with a different minor representation: where p NS (f ) is defined by the generating function

R and R sectors
In order to calculate the large ∆ β behavior of the Ramond toric blocks we shall chose a special basis B f of level f even subspace of the Ramond Verma module. It is defined by: where the string of generators S −K does not include S 0 . We shall also use a simplified notation for elements of the basis above: As it is shown in appendix B the subsets B ± f are composed of the same number of elements, Proposition 3. Let Q be a polynomial in β and let deg β Q denotes its degree. Then: 3. for any u ± k , u ± l ∈ B ± n , u ± k = u ± l :

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4. the product of the diagonal terms is the only highest degree term in the determinant of the Gram matrix with respect to the base B n i.e.
Let us recall that matrix elements of arbitrary chiral vertex operators V (ν λ ), V ( * ν λ ) between even states u i , u j ∈ W β can be decomposed as [23]: The decompositions above can be seen as defining the forms ρ ±± RR,e , ρ ±∓ RR,o . They are related to 3-point blocks (2.7) by There holds:

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It follows from Proposition 4 that for any u i , u j ∈ B f , u i = u j : Following the same steps as in the previous subsection we thus get for i = j : be the matrix inverse to the Gram matrix By Proposition 4, for any u ± k ∈ B ± f : and we also get lim β→∞ ρ ∓∓ RR,e (u ± k , ν λ , u ± k ))B kk ±± = 0. (4.13) Since the leading terms in the large β limit take the form:

NS and NS sectors
The large ∆ asymptotic (4.6) suggests the following definition of the elliptic blocks in NS sector:

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and in the NS sector: Coefficients H λ,f ∆ have the same analytic properties as coefficients F λ,f ∆ and formula (3.10) yields the recursive relation:

R and R sectors
The large β behavior of the blocks with R intermediate states, (4.17) and (4.18), lead to the following definition of the elliptic blocks: β,e (q), Since h * λ(+),f β vanishes, recursive relation (3.12) implies that F * λ(+) β,e is identically zero It follows that all 1-point functions ofφ λ,λ vanish in the R sector: Using recursive relation (3.12) and asymptotics (4.18) one can also show that β,e (q). There are thus only two independent elliptic blocks

A Neveu-Schwarz sector
In this appendix we shall prove the propositions of section 4.1.
Proof of proposition 1. Part 1 is a simple consequence of the NS algebra. By the same token one has If #M + #K = #N + #L, then part 2 follows from part 1 and the inequality above. Suppose #M + #K = #N + #L. In this case the inequality of part 2 is also satisfied. Indeed, calculating the scalar product by the NS algebra rules one can get the maximal degree #M + #K if, and only if In order to prove part 3 let us observe that by part 1 the product of diagonal terms Any other term in the expression for the determinant of u i |u j takes the form where σ is a nontrivial permutation. Let us assume that for all i On the other hand by Prop. 1.2 the equations above imply that for all i and therefore deg ∆ Q > deg ∆ P in contradiction with our assumption. It follows that for an arbitrary nontrivial permutation σ there exists at least one i such that

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Proof of proposition 2. Let V ( ν λ ) : V ∆ → V ∆ be an NS chiral vertex operator with a conformal weight ∆ λ . For any u i , u j ∈ B f of the same parity one has and it is enough to consider the matrix elements u i |V ( ν λ )|u j . By Proposition 1.2 Calculating matrix elements u i |V ( ν λ )|u j one can use the Ward identities to move all the NS algebra generators to the right.
The matrix elements u i |V ( ν λ )|u j can be represented as a linear combination of and terms of the form where (S −K ) † (L −M ) † denotes product (S −K ) † (L −M ) † with at least one generator removed. The coefficients of this combination are independent of ∆. Using Ward identities one easily checks that for arbitrary This in order imply

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On the other hand and by assumption

B Ramond sector
We shall first prove eq. (4.8).
Let q(k, n) be the number of partitions of n in k distinct parts. The corresponding generating function reads For y = −1 it counts the difference between the number of partitions in an even number of unequal parts and the number of partitions in an odd number of unequal parts. Hence Proof of proposition 3. Part 1 is a simple consequence of the Ramond algebra. Part 2 follows from part 1 and the observation that maximal possible degree of u + k |u − l is odd while the diagonal elements of the Gram matrix are of even degrees. The proof of part 3 parallels the proof of Proposition 1, part 2 while part 4 is proved along the same lines as Proposition 1, part 3.
Proof of proposition 4. We shall prove part 1 using the same method as in the proof of Proposition 2.