Large-c superconformal torus blocks

We study large-c SCFT2 on a torus specializing to one-point superblocks in the N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 Neveu-Schwarz sector. Considering different contractions of the Neveu-Schwarz superalgebra related to the large central charge limit we explicitly calculate three superblocks, osp(1|2) global, light, and heavy-light superblocks, and show that they are related to each other. We formulate the osp(1|2) superCasimir eigenvalue equations and identify their particular solutions as the global superblocks. It is shown that the resulting differential equations are the Heun equations. We study exponentiated global superblocks arising at large conformal dimensions and demonstrate that in the leading approximation the osp(1|2) superblocks are equal to the non-supersymmetric sl(2) block.

The aim of this paper is to study SCFT 2 on a torus in the large-c regime. 1 As a first step in this direction we consider the N = 1 superconformal Neveu-Schwarz (NS) algebra and 1-point superconformal blocks. We focus on global superblocks that are associated to the osp(1|2) subalgebra of the NS superalgebra. We develop the superCasimir approach where osp(1|2) global superblocks are realized as eigenfunctions of the super-Casimir differential operators.
The idea is that global blocks in CFT 2 on any topology play the central role in studying the large-c regime because all other semiclassical blocks including light and heavy-light JHEP08(2018)042 blocks are related to the global one [6,21,[26][27][28]. We show that similar relations are valid in the supercase. Also, we consider the regime of large conformal dimensions and study the exponentiated global superblocks. It is interesting that the supersymmetry turns out to be degenerate in the leading order because the NS supermultiplet conformal dimensions are indistinguishable in this regime, ∆ + 1 2 ≈ ∆. The paper is organized as follows. Using osp(1|2) representation theory described in section 2 we formulate two global superconformal torus blocks corresponding to different structure constants in section 3. The superCasimir approach for torus superblocks is elaborated in section 4. Here we derive two second order differential equations for the superblocks and analyze their solutions. These equations are in fact the Heun equations and we describe their local and global properties. In section 4.3 the global superblocks are studied in the regime of large conformal dimensions. We show that the resulting superblock block functions are exponentiated and find explicit expressions. Remarkably, they all are expressed in terms of the exponentiated non-supersymmetric block function. In section 5 we consider superconformal blocks of the NS superalgebra and show that global, light, and heavy-light superblocks can be obtained via particular contractions of the NS superalgebra when 1/c → 0. We close with some concluding remarks in section 6.

Representation theory of osp(1|2) superalgebra
In this section we shortly review the osp(1|2) superalgebra and Verma supermodules. It basically serves to set our notation and conventions. For detailed reviews see, e.g., [29,30].

JHEP08(2018)042 3 Global torus superblocks
Let us consider osp(1|2) superconformal theory on a two-dimensional torus. A one-point function of the primary superfield Φ ∆,∆ (x,x) with conformal dimensions ∆,∆ is given by where str is the supertrace on the (super)space of states, and (x,x) = (w,w, η,η) are the supercylindrical coordinates, see appendix A. The modular parameter q = e 2πiτ , where τ ∈ C is the torus modulus.
From now on we consider holomorphic sector only. Assuming that the space of states can be decomposed into supermodules of various dimensions ∆ we can project onto a particular supermodule. It is convenient to introduce the supertrace function with the supertrace evaluated on the Verma supermodule of weight ∆. Then, lower and upper superconformal blocks B 0 and B 1 can be defined by means of the decomposition 2 where C ∆∆ ∆ = ∆|φ ∆ (0)| ∆ and C ∆∆+ 1 2 ∆ = ∆|ψ ∆+ 1 2 (0)| ∆ are two independent structure constant, see relations (3.6) below. Note that B 0,1 are even functions.
The lower and upper torus superblocks read off from relations (3.2) and (3.3) are given by The superblocks (3.4) and (3.5) are decomposed according to parity of the exchanged channel that is manifested by respectively first and second sums of each expression (see figure 1). Figure 1. One-point superconformal blocks of the supermultiplet (φ ∆ , ψ ∆+ 1 2 ). The solid (dashed) lines stand for Grassmann even (odd) operators. The left and right tadpole graphs correspond to the lower block (3.4) and upper block (3.5) and solid-dashed loop represents taking the supertrace.

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Taking into account the form of the matrix elements in the even/odd sectors, we find the closed expressions for the lower superblock function, and for the upper superblock function, Indeed, substituting Φ ∆ = 1 into (3.2) yields the supertrace of the identity operator. In accordance with (2.3) it shows that there is one state on each level of the supermodule V ∆ (2.4). Note that B 0 (0, ∆|q) = B 1 (1/2, ∆|q) and B 0 (1/2, ∆|q) = B 1 (0, ∆|q).

SuperCasimir eigenvalue equations
It is known that CFT 2 global blocks can be described as solutions to the second-order differential equations interpreted as the sl (2) Casimir operator eigenvalue conditions imposed on exchanged channels [42]. The original construction for 4-point blocks on the sphere can be extended to higher-point conformal blocks on the sphere and torus [19,26]. The superCasimir equations for 4-point sphere blocks were previously discussed in [43][44][45]. In d dimensions (super)conformal Casimir equations were discussed in [43][44][45][46][47][48]. 3 In what follows we elaborate the superCasimir approach for torus superblocks. To this end, we note that acting with the superCasimir operator (2.5) inside the supertrace operation we get the eigenvalue equation for the exchanged channel where the factor on the right-hand side is the eigenvalue of the superCasimir on the irreducible Verma supermodule V ∆ . On the other hand, the external superfield Φ ∆ satisfies the other eigenvalue equation with the superCasimir operator given by where the osp(1|2) generators are realized as differential operators in the supercylindrical coordinates, see appendix A. Using the operator-state correspondence we can show that the eigenvalue superCasimir equation (4.2) is identically satisfied. Combining two eigenvalue conditions (4.1) and (4.2) we will obtain two equations for two superblocks. We will see that the supertrace function embracing two superblocks (3.3) provides a natural way to impose the superconformal invariance conditions.

Derivation of the eigenvalue equation
Using the approach of [19] we find the following identities with the osp(1|2) basis elements inserted into the supertrace JHEP08 (2018)042 where n = ±1 and k = ± 1 2 . Here, in particular, we used the commutator [L 0 , T s ] = −sT s with T s = (L 0,±1 , G ± 1 2 ) being osp(1|2) basis elements that yields the identity T s q L 0 = q L 0 +s T s . Also, it is easy to derive the polynomial homogeneity identity Loosely speaking, the identities (4.4) and (4.5) convert polynomial combinations of basis elements L n and G k acting on states of V ∆ to differential operators L n and G k acting on the superfield along with q-differential operators. It is crucial here that the superblocks are combined into the supertrace function that allows using the graded cyclic property and commuting even/odd operators via (graded) cyclic permutations.
Recalling the U(1) ×U(1) global symmetry of two-dimensional torus and relation (A.4) we find that L 0 acts trivially, i.e., Then, substituting the identities (4.4) and (4.5) along with the relation (4.6) into the superCasimir equation (4.1) we get where operators C 2 and Υ 2 are directly read off from (2.7) and (4.3). Now, we have to know how C 2 and Υ 2 act on the superfield Φ ∆ (x). To this end, we use the superCasimir equation (4.2) to express and substitute this relation into the equation (4.7) Now, we observe that the sl(2) Casimir operator acts non-diagonally on the supermodule To derive this relation we used formulas from the appendix A and the fact that even/odd components of the osp(1|2) superfield are themselves sl(2) conformal fields. Substituting

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this relation into (4.9) and using (3.3) we finally find two equations These are the second order ODEs that differ only in the coefficient of the third terms. Since each of the equations has two independent solutions we fix the asymptotics 13) and find that the corresponding solutions to the superCasimir equations (4.11) and (4.12) are given by functions (3.7) and (3.8).

Properties of the eigenvalue equation
One might wonder whether the local OPE data and the modular properties of the torus correlation functions fix the form of the differential equations (4.11) and (4.12). To clarify this issue, we change variables as x = q 1/2 , for example, in the first equation (4.11) to obtain the second order ODE 14) (the prime denotes x-derivative) with the (regular) singular points −1, 0, 1 and ∞, and the Riemann P-symbol From the entities α i,j of the P-symbol (which are two characteristic exponents, j = 1, 2 at singular points x i , where index i = −1, 0, 1, ∞ labels singular points) one can recognize the superCasimir eigenvalues in the exchanged channel (4.1). The equation (4.14) is Fuchsian that can be seen, in particular, by checking the Fuchs identity i,j α i,j = (p−2)n(n−1)/2, where p(= 4) is the number of singular points and n(= 2) is the order of the ODE. An important characteristic of the Fuchsian ODEs is the so-called rigidity index

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where m i,j are multiplicities of the characteristic exponents α i,j (4.15). In our case the spectral type ({m i,j }) = (1, 1; 1, 1; 1, 1; 1, 1). If the rigidity index I = 2, then the local data completely defines a differential equation, in particular, its explicit form and the monodromy group, etc. (for details see, e.g., [49]). In our case, however, the rigidity index I = 0 implying that the ODE is not rigid and contains one accessory parameter which form cannot be determined only from the local data (4.15). Doing the transformation B 0 (x) = (1−x) α 1,2 (1+x) α −1,1 x α 0,2 y(x) we get the differential equation of the form Hence, the lower superblock can be expressed in terms of the Heun function The superCasimir equation for the upper superblock (4.12) can be considered along the same lines. The resulting Heun's representation is given by

Exponentiated global superblocks
Let us consider large conformal dimensions. In this regime ∆ and ∆ tend to infinity uniformly ∆ = κ σ , ∆ = κ σ , (4.21) where the scale parameter κ 1, and σ and σ are finite rescaled conformal dimensions. Remarkably, the large rescaled dimensions ∆ + 1 2 and ∆ + 1 2 are also given by σ and σ. Given that σ and σ are fixed, the large-κ asymptotics of the superblocks (3.7) and (3.8) can be conveniently defined using the superCasimir equations (4.11), (4.12). Indeed, in this case the solutions are exponentiated, 4 In CFT2 with Virasoro symmetry the Heun equation arises as the c → ∞ limit of the BPZ equation for the 5-point conformal blocks with one degenerate light operator [50] (see also recent discussion in [51,52]).

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where b α|0 (σ, σ, q) are the classical global blocks, while b α|n (σ, σ, q) at n = 1, 2, . . . are 1/κ n corrections. Below we show that the leading asymptotics coincide b 0|0 (σ, σ, q) = b 1|0 (σ, σ, q) and the difference arises in higher orders. We note that exponential factors can be represented as follows where the lightness parameter δ is dimensionless. This formula means that the scale factor κ in (4.22) is dummy because κ −n+1 b α|n (σ, σ, q) = ∆ −n+1 b α|n (∆/ ∆, q). However, working with dimensionless parameters κ and δ turns out to be technically more convenient. Substituting the ansatz (4.22) into the superCasimir equations we find out that each leading asymptotic satisfies the same residual differential equation (4.24) Therefore, classical lower/upper superblocks coincide with each other and equal to the known classical sl(2) global block [27] b The integral is divergent in x = 0. However, the divergence can be bypassed if we expand in the lightness parameter (4.23) near δ = 0 meaning that the exchanged channel is much heavier than external primary operator (see [27] for more details). All higher-order corrections are iteratively expressed in terms of the leading contribution (4.25). For example, the O(k 0 ) corrections b α|1 (σ, σ, q) are defined by inhomogeneous first order differential equations given in appendix C. The solutions have the integral form where W = (1 − x) 2 + xδ 2 . The above integrals differ in the last two terms. Evaluating the integrals and further expanding in the smallness parameter δ we can obtain a first few terms b 0|1 (σ, σ, q) = − log 1 + q 1/2 + 1 2 In particular, recalling that these corrections are dimensionless (4.23) we observe that osp(1|2) character (3.9) is reproduced by the logarithmic terms in the limit δ → 0. Indeed, the limit can be achieved by σ → ∞ and therefore we can use (3.10).

JHEP08(2018)042 5 Conformal blocks of contracted Neveu-Schwarz superalgebras
In this section we consider the Inonu-Wigner contractions of the NS superalgebra with respect to the inverse central charge 1/c in the limit c → ∞. Analogously to the Virasoro algebra case [27] we compute associated torus superblocks and identify them with different types of semiclassical torus superblocks. 5

NS superalgebra and superblocks
Let us consider N = 1 NS superalgebra which generators L m and G r satisfy the graded commutation relations where m, n ∈ Z and r, s ∈ Z + 1/2. A primary superfield Φ ∆ (x) transformations on a torus are given in (A.4).
We introduce the supertrace function on the NS supermodule of the weight ∆ and define (cf. section 3) the NS superblocks as Here, summation over superindices is not graded and basis monomials of the NS supermodule denoted by M ∆ are given by

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In what follows we consider various semiclassical superblocks arising in the limit c → ∞ of the original NS superblocks (5.3), (5.4). We distinguish between heavy and light conformal dimensions ∆ = O(c 1 ) and ∆ = O(c 0 ) so that there are three types of onepoint superblocks with external light operators: global superblock, light superblock, and heavy-light superblock. It will be shown that these superblocks are associated to different contractions of the NS superalgebra and are related to each other.

Contracted NS superalgebras
Let us first consider contractions that leave a finite-dimensional subalgebra intact. Even and odd generators rescale as We consider two cases: type A contraction γ = 1 and type B contraction γ = 1/2. The rescaled transformations of the primary field Φ ∆ (x) with respect to l n and g s take the form (2.8), while higher order NS generators act trivially, Thus, Φ ∆ is an osp(1|2) superconformal quasi-primary field. In general, the resulting contracted NS superalgebras are isomorphic to semi-direct sum where F is an infinite-dimensional superalgebra with two branches F = F − ⊕ F + spanned by basis elements a m and b n with n ∈ 1 2 Z. The two branches F ± are highest weight osp(1|2) supermodules. The factor F is defined by a particular type A/B contraction.

(5.12)
Here, the first group of relations defines osp(1|2) superalgebra, the second group defines Abelian superalgebra, the third group defines the highest weight supermodule structure.

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Type B contraction. Here, the contracted superalgebra is where HC is the infinite-dimensional Heisenberg-Clifford superalgebra. The respective graded commutation relations are different from those of N S A (5.10)-(5.12) only in the part defining the infinite-dimensional factor. Namely, [a m , a n ] = m(m 2 − 1) 12 δ m+n,0 , {b r , b s } = 1 3 r 2 − 1 4 δ r+s,0 , [a n , b r ] = 0 . (5.14) Type C contraction. Also, there is a third type of contraction when all NS generators are rescaled so that the osp(1|2) subalgebra is also contracted, with non-zero graded commutation relations The superfield Φ ∆ is light and, therefore, HC-invariant,

Associated superblocks
In what follows we argue that the superblock functions associated to the three types of contracted NS superalgebra correspond to osp(1|2) global, light, and heavy-light one-point superblocks, respectively. Consider first the representation theory of the contracted superalgebras (5.8). A supermodule M ∆ is spanned by basis monomials where we split into osp(1|2) and F subalgebra generators, and denoted aR = a i 1 −m 1 . . . a i k −m k , with a level |R| = i 1 m 1 +· · ·+i k m k , and bS = b j 1 −s 1 . . . b j k −s k , with a level |S| = j 1 s 1 +· · ·+j k s k . The total level number is given by |N| = |N | + |R| + |S| ∈ 1 2 Z, where N labels osp(1|2) indices and |N | = s + k and s = 1, 2, . . . and k = 0, 1, cf. (2.3) and (5.5).
Let V ∆ be an osp(1|2) supermodule of weight ∆, and F be the Fock supermodule of the factor F in the truncated superalgebra (5.8). Then, considering M ∆ , V ∆ , and F on their own as linear spaces we conclude from (5.19) that M ∆ is the (graded) tensor JHEP08(2018)042 product of vector spaces M ∆ = F ⊗ V ∆ . On the other hand, the primary superfield Φ ∆ is F -invariant (5.7) and, therefore, can be represented as 1 F ⊗ Φ ∆ , where Φ ∆ is osp(1|2) quasi-primary superfield. Using that a supertrace trace on the tensor product is a product of supertraces one can explicitly show that the supertrace function (5.2) associated to the contracted type A/B superalgebras reads where the first factor here is the F character on the Fock module, while the second factor is the osp(1|2) supertrace function (3.3).
For the type A superalgebra (5.9) the F character is trivial χ F (q) = 1 so that the resulting supertrace function is Υ(∆, ∆, q|η) = B(∆, ∆, q|η) given in (3.2). Thus, we conclude that global superblocks correspond to the type A contracted NS superalgebra. In other words, a truncation of the NS superalgebra to osp(1|2) subalgebra is equivalent to a particular contraction.
For the type B superalgebra (5.13) the F character is a truncated Heisenberg-Clifford (graded) character. Indeed, the Heisenberg-Clifford character is known to be (see, e.g., [56]) On the other hand, basis elements a m and b r of the F factor are labeled by (half-)integers |m| 2 and |r| 3/2. It means that states generated by basis elements with m = 0, ±1 and r = ±1/2 do not contribute because they belong to the osp(1|2) module. The F character, if compared to the Heisenberg-Clifford character, does not take into account those lower label states so that the truncated character χ HC (q) is given by where the osp(1|2) character given by (3.9). Finally, the resulting block is L(∆, ∆, q|η) = χ HC (q)B(∆, ∆, q|η) , (5.23) where the prefactor is given by (5.22). The block on the left-hand side is the light NS superblock that equivalently can be obtained as the c → ∞ limit of the original NS superconformal block (5.2) at fixed conformal dimensions ∆, ∆. Finally, let us shortly consider heavy-light superblocks H 0,1 (∆, ∆, q) which have one heavy dimension ∆ = O(c 1 ) and light dimension ∆ = O(c 0 ) at c → ∞. A contraction of the NS superalgebra underlying the heavy-light superblocks is given by the type C superalgebra (5.16). Applying the definitions (5.3) and (5.4) to the type C we can explicitly calculate the associated superblocks that are given by the Heisenberg-Clifford character (5.21), i.e. The above relations between various superblocks (5.23), (5.24), and (5.25) are a supersymmetric version of the analogous relations between semiclassical Virasoro torus block and sl(2) global torus blocks [27]. 7

Concluding remarks
In this paper we have developed a framework to study large-c behavior of torus SCFT 2 superblocks. We have explicitly calculated various types of semiclassical one-point superblocks and established relations between them.
We have seen that two exponentiated osp(1|2) global superblocks are equal to the single non-supersymmetric exponentiated sl(2) global block. On the other hand, there is a lot of evidence that exponentiated global blocks in the leading approximation are related to the linearized classical conformal blocks [6,21,26,27]. We expect that in SCFT 2 these two types of superblocks are similarly related and, therefore, the linearized classical superblocks are equal to the linearized non-supersymmetric classical blocks. Indeed, here we have the same phenomenon that classical conformal dimensions of the supermultiplet It is similar to the argument of [57] that classical N = 1 superblocks on the sphere should have the large-c asymptotic given by the purely bosonic Zamolodchikov's classical block.
From the bulk perspective, both classical global and linearized classical torus blocks are realized as lengths of geodesic tadpole-type networks stretched in the thermal AdS 3 space [13,19,21]. One might expect that classical superblocks could be realized in terms of superparticles propagating on the particular background that solves 3d supergravity equations. 8 However, presently, we may conclude only that in the leading order of the large-c approximation the superblocks are realized by duplicated system of bosonic geodesic networks.
The other interesting question is to understand the bulk dual realization of higher-order corrections to the classical global (super)blocks of section 4.3. It appears that using the worldline approach they can be calculated by accounting for the backreaction of particles in the bulk (see [22] for recent discussion of the worldline formalism in the context of the semiclassical AdS 3 /CFT 2 correspondence).
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