Wilson lines construction of $\mathfrak{osp}(1|2)$ conformal blocks

We study N=1 superconformal theory in the context of AdS/CFT correspondence in the large central charge limit using Chern-Simons formulation of $3d$ gravity. In this limit conformal dimensions of a subclass of so-called light primary superfields remain finite and are governed by $\mathfrak{osp}(1|2)$ subalgebra of N=1 super-Virasoro algebra. We describe the construction of $\mathfrak{osp}(1|2)$ conformal blocks in terms of Wilson lines of the Chern-Simons $3d$ gravity. We consider examples of two and three-point blocks on the sphere and one-point torus blocks of light superprimary fields, which belong to finite-dimensional representations of $\mathfrak{osp}(1|2)$. We study the correlation function for lower and upper components of the primary $\mathfrak{osp}(1|2)$ doublets and show that the associated conformal blocks are obtained via Wilson line construction in Chern-Simons theory.

In the Virasoro case the four-point correlation function on the sphere of four primary fields ϕ i with conformal dimensions h i ,h i is where C 12h , C h34 are the structure constants, (z i ,z i ) are coordinates on the complex plane and w(h s , h i , z i ) is the four-point holomorphic conformal block 1 . Similarly, the correlation functions are defined on the torus, for example, the one-point correlation function of a primary field ϕ 1 on the torus is 2 ϕ 1 (z 1 ,z 1 ) = Tr h q L 0qL0 ϕ 1 (z 1 ,z 1 1 For the holographic description of higher point conformal blocks see [33][34][35][36][37][38][39]. 2 Throughout this paper we omit factor (qq) − c 24 which can be easily restored.
where T r h is the trace taken over descendent states associated with the intermediate primary field ϕ h , q is the elliptic parameter of the torus q = e 2πiτ and L 0 is the Virasoro generator, L 0 |h = h|h . Here F(h, h 1 , q) is the one-point holomorphic torus conformal block (for details see [40][41][42][43]) 3 .
In [20,26,48,49] the dual description of sl (2) and sl(3) conformal blocks (which are obtained from the Virasoro and W 3 conformal blocks respectively in the large central charge limit for light primary fields 4 ) on the sphere in the context of CS theory was found. It was shown that sl(2) 2-point, 3- The relation between conformal blocks and the expression (1.3) involves two basic ingredients: the singlet state |s representing a specific channel of OPE and the boundary state |hw i associated with the boundary primary field ϕ i |hw i → ϕ i (z i ). (1.4) In [45] (see also [58]) it was proposed that sl(2) one-point torus conformal block F(−j, −j 1 , q), where −j and −j 1 are respectively the intermediate and external conformal dimensions 5 , can be computed by a prescription similar to (1.3) by introducing the following modifications: Wilson line operators at the point z b are joined using the intertwining operator and two boundary points are identified modulo 2πτ , and the trace is taken over corresponding representation. Thus the proposal reads (1.5) Here T r j is taken over the representation with spin j, z 1 is the point on the boundary of the solid torus which is a geometric representation of the thermal AdS 3 , z b is an arbitrary point in the bulk of AdS 3 , |lw 1 is the lowest weight state of the representation with spin j 1 and I j;j,j 1 is the intertwining operator associated with the representations j and j 1 . The factors 3 For AdS/CFT correspondence in thermal AdS, relevant for the torus topology, see [12,[44][45][46][47]. 4 The conformal dimension h of a light primary field (resp. heavy primary field ) ϕ h by definition behaves For the holographic description of conformal blocks in the context of Virasoro heavy-light fields see [50][51][52][53][54][55][56][57]. 5 In the dual description j and j1 are the spins of finite-dimensional representations of sl(2) CS gauge group.
W a [x, y] (for a = j, j 1 ; x = z b and y = z b + 2πτ, z 1 ) denote the Wilson line operators where Ω is the flat connection where J 1 and J −1 are the lowering and raising operators of sl (2) in the representation a.
The purpose of this work is to investigate the generalization of this construction on the osp(1|2) algebra. As explained in A, this algebra is relevant in the large c limit of the N = 1 super CFT. We are interested in studying the correlation functions of superprimary fields 6 in the Neveu-Schwarz (NS) sector where (z,z) are holomorphic and antiholomorphic coordinates, and (θ,θ) are Grassmann variables, ϕ i is a primary field with conformal dimensions We are interested in the correlation functions of light superfields in the spherical and toroidal topologies. The conformal dimensions of degenerate superfields are given by the Kac formula [60], in the large c limit these conformal dimensions become thus light degenerate NS superfields in the limit c → ∞ have conformal dimensions −j = 1−m 4 , where m is odd, therefore j can take integer or half-integer values. Below, j will be identified with the superspin of a finite-dimensional representation of osp(1|2).
On the sphere, we are interested in studying the Wilson lines formulation for the correlation functions of light superfields. By using the expansion (1.8) we can express the correlation function Φ 1 (z 1 , θ 1 ,z 1 ,θ 1 )...Φ n (z n , θ n ,z n ,θ n ) (1.10) in terms of components, thus (1.10) contains 4 n terms, we call each of these terms components of the correlation function. We will concentrate on the components to which ϕ i and ψ i contribute, and we will investigate only their holomorphic dependence (due to the factorization of the symmetry into the holomorphic and antiholomorphic sectors, the consideration of the antiholomorphic dependence and of the componentsψ i , ϕ i is identical, see [41]). To this end it is convenient to formally setθ i = 0, then we get Φ 1 ...Φ n = ϕ 1 ...ϕ n + ... + θ 1 ...θ n ψ 1 ...ψ n . (1.11) The correlation functions on the torus can be also expressed in terms of the components of the rhs of (1.8). We restrict ourselves to considering the contributions of ϕ i and ψ i in the one-point correlation function. By settingθ 1 = 0 we obtain where L 0 is the generator of Virasoro subalgebra of Neveu-Schwarz algebra. In the large c limit, for light fields the trace T r h in (1.12) is reduced to the osp(1|2) subalgebra. We will see that this trace splits into the even and odd parts  (1.3) with the following prescription: |s is a singlet which belongs to the tensor product of superspin-j i representations of osp(1|2); J (i) 1 has to be replaced by L (i) 1 -the lowering generator of the CS gauge algebra osp(1|2) acting in the representation j i ; We will have two sorts of states, the highest weight states |hw i with weights j i and states with weights j i −1/2 which we denote by |hw − 1/2 i . For a component that contains the fields (ϕ i , ψ i ), each field ϕ i in this component will correspond to a state |hw i and each field ψ i will correspond to a state |hw − 1/2 i in (1.3), thus schematically we have (1.14) In the case of the one-point correlation function on the torus, we show that the conformal block of each component of the correlation function (1.12) can be computed by means of the lhs of equation (1.5) using following modifications: T r j is the trace taken over the states of a osp(1|2) finite-dimensional representation with superspin j = −h; j 1 is the superspin of a osp(1|2) finite-dimensional representation (the conformal dimension h 1 = −j 1 ); I j;j,j 1 is the osp(1|2) intertwining operator (this operator can be expressed in terms of the osp(1|2) Clebsch-Gordan coefficients); W a [z b , z] (for a = j, j 1 ; z = z b + 2πτ, z 1 ) are the Wilson line operators only this time (J 1 , J −1 ) have to be replaced by (L 1 , L −1 )-the generators of the osp(1|2) gauge algebra; Similarly to the sphere, on the lhs of (1.5) we will have two sorts of states, the lowest weight states |lw 1 with weight −j i and the states with weight −j 1 + 1/2 which we denote by |lw + 1/2 1 . For a component that contains the field ϕ 1 , the field ϕ 1 in this component will correspond to the state |lw 1 and for a component that contains the field ψ 1 , the field ψ 1 will correspond to the state |lw + 1/2 1 on the lhs of (1.5), thus schematically we have The outline of the paper is the following: In section 2 we give a brief framework of the osp(1|2) CFT and osp(1|2) finite-dimensional representation theory. In section 3 we present the Wilson lines formulation of osp(1|2) conformal blocks in the spherical and toroidal topology. In section 4 we give our conclusions and comment on further developments of the Wilson lines formulation of osp(1|2) conformal blocks. Appendixes A and B describe the notations of the Neveu-Schwarz algebra and the explicit form of sl(2) one-point torus block respectively.

osp(1|2) conformal field theory
The Neveu-Schwarz algebra (also termed N = 1 NS super-Virasoro algebra) contains the osp(1|2) subalgebra. We are interested in studying the limit c → ∞ of N = 1 NS super-Virasoro algebra, in this limit, see appendix A, we need to keep only the generators of the osp(1|2) in order to have finite inner products of states (similar to the Virasoro and W 3 algebras in the large c limit, which reduce to sl(2) and sl(3) algebras respectively). In this sense, in the large c limit, the N = 1 NS super-Virasoro algebra reduces to the osp(1|2) algebra 7 . In the N = 1 super CFT we have conformal superfields Φ i (1.8) and in the sequel we restrict ourselves to studying the (ϕ i , ψ i ) components and omit the antiholomorphic contribution, thus for simplicity, we will write superfields as follows (2.1) The osp(1|2) algebra has three even generators L ±1,0 and two odd generators G ± 1 2 , we have the following commutation and anticommutation relations where [, ], {, } are the commutator and anticommutator respectively. m, n = 0, ±1. r, s = ± 1 2 . It is assumed the conjugation rules The states in the representation V h i are obtained by applying the generators (G − 1 2 , L −1 ) to the hwv |h i associated with the field Φ i , thus any state can be written as follows where N 0 is the set of nonnegative integers. Generators (L 1 , G + 1 2 ) annihilate the hwv and 7 For a review of osp(1|2) CFT see [63,67,68].
The supermodule V h i can be written as a direct sum of two subspaces

Spherical and toroidal conformal blocks
In section 3.1 we will show how the conformal blocks (CB) of the components of the two-point and three-point correlation functions are obtained in CS theory. These two examples clarify how the idea proposed in [20,26,48,49] generalizes to the osp(1|2) algebra and provide evidence that any superconformal block could be obtained in CS theory with osp(1|2) gauge symmetry. The CBs under consideration are presented.
The two-point correlation function of superfields where The three-point correlation function of superfields (Φ 1 , Φ 2 , Φ 3 ) is given by (2.10) Similarly, using (2.1) and expanding the rhs of (2.9), we have , The one-point correlation function on the torus is given by equation (1.12). Each term of the rhs of (1.12) has been found in [69] and look as follows (2.14) For our purposes it is convenient to keep structure constants C hh 1 h and C hh 1 + 1 2 h in the definition of CBs. B 0 (h, h 1 , q) and B 1 (h, h 1 , q) are the lower and upper torus superblocks, given by

Finite-dimensional representations
In this section we recall some facts of osp(1|2) finite-dimensional representation theory which are relevant for our discussion in the subsequent sections.
A finite-dimensional representation of osp(1|2) is labelled by a nonnegative integer or halfinteger superspin j and parity λ (λ = 0 if j is integer, or λ = 1 if j is half-integer), we call these representations superspin-j representations of osp(1|2), each representation decomposes into two normal subspaces as follows where V j stands for the state space of a spin-j representation of sl(2) which is 2j + 1 dimensional, thus the direct sum (2.17) is 4j + 1 dimensional. Any state of the representation (2.17) is characterized by two variables (l(j), m), one can denote any state of (2.17) as follows where l(j) denotes the subspace (V j or V j− 1 2 ) to which the state (2.18) belongs, thus l(j) takes two values (2.20) In the sequel we will use the notation The generators of the osp(1|2) algebra in a superspin-j representation will be denoted as follows with the commutation relations (2.2) 8 . The generators (2.22) act on the states 9 (2.18) as follows [67,70] L 0 |l(j), m = m|l(j), m , where [x] represents the integer part of the number x (2x ∈ Z 10 ), and if l 1 (j) = j then l 2 (j) = j − 1 2 or vice-versa. 8 To distinguish conformal algebra and CS gauge algebra generators we use bold and regular fonts respectively. 9 In the sequel, the superscript i over generators labels superspins-ji representation. 10 Z is the set of integers.
The second fact is related to the structure of the Clebsch-Gordan coefficients (CGC) of osp(1|2). The tensor product of two representations (R j 1 , R j 2 ) decomposes in the direct sums any state in the tensor product can be expressed as follows where are the osp(1|2) Clebsch-Gordan coefficients, from (2.25) it follows that j 3 in l 3 = l 3 (j 3 ) can take values In [70][71][72] has been shown that the CGC (2.27) can be factorized as follows where C sl(2) (l 1 , m 1 ; l 2 , m 2 |l 3 , m 3 ) are the CGC of sl(2) (3.27) and is termed the symmetrical scalar factor. The scalar factor (2.30) does not depend on the projections (m 1 , m 2 , m 3 ), it has been computed in [72]. The singlet state (the state with the total superspin and projection equal to zero) in the tensor product of superspin representations will be denoted as follows |0, 0 = |s . (2.31) In the case of the tensor product of two representations (2.25) with the same superspin j 1 = j 2 = j the CGC of the singlet are given by [72] C(l 1 , m 1 ; where λ is the parity of j. Similarly to sl(2), in osp(1|2) it is defined the super 3 − j symbol given by the following formula [72] j 1 j 2 j 3 l 1 m 1 l 2 m 2 l 3 m 3 = j 1 j 2 j 3 l 1 l 2 l 3 where on the rhs the second factor is the Wigner 3−j symbol of sl(2) algebra. Using the super 3 − j symbol we can express the singlet state of the tensor product of three representations as follows

Wilson line construction
In this section we will compute using the Wilson lines formulation the CBs of the components of the two-point, three-point correlation functions on the sphere and one-point correlation function on the torus.

Two-and three-point spherical conformal blocks
Here we show that the CBs of each component of the two-point correlation function (2.8) and three-point correlation function (2.11) can be obtained by an ansatz similar to (1.3). We introduce the following notation As discussed in [20,26,48,49,58] the Wilson line description of CBs implies relations between conformal dimensions h i and spins j i (in our case superspins j i ) as well as relations between primary fields ϕ i and states of R j i , in the case of osp(1|2) we find the following relations 11 (the arrows stand for corresponds) On the sphere, by taking into account the relations (3.2) and using the notations (3.1) we will have that the components of the correlation functions (2.7, 2.9) can be expressed by the following formula 3) 11 For the other two components of (1.8) the relation isψi(z) → |ji, ji and ϕi(z) → |ji − 1 2 , ji − 1 2 .
where, n = 2 (resp. 3) in the case of two-point (resp. three-point) correlation function, c is a constant irrelevant to our discussions, s| is a singlet state in the tensor product of superspin representations R j i of osp(1|2), L (i) 1 is the lowering generator acting on the states of R j i , |j i − k i /2, j i − k i /2 (for k i = 0, 1) is a state in R j i according to the notation (2.18) and z b is an arbitrary point (the CBs do not depend on it). To see that indeed (3.3) does not depend on z b let us choose other point z b in (3.3) and see that it gives the same result, thus we have where we used the property of the singlet s| that it is annihilated by the action of the element 1 because the singlet has total superspin and projection equal to zero in the tensor product, and hence the singlet s| is invariant under the action of the group element The two-point correlation function. By using the relations (3.2) and the formula (3.3) we find that the CBs of the two components (2.8) can be computed by where s| is the singlet of the tensor product R j 1 ⊗ R j 1 . Choosing z b = z 2 and considering the coefficients of the singlet (2.32), the rhs of (3.5) reduces (up to irrelevant constant) to where z ij = z i − z j , by applying the generator L 1 to the states in R j 1 according to (2.23) we obtain which confirms (3.5). Similarly for (3.6) we obtain The three-point correlation function. According to (3.3) and the singlet (2.34), we have in this case the following equation this expression can be shown [48] to be proportional to z S 1 +S 2 −S 3 where C(j i , k i , λ i ) is a constant. Notice that (3.11) does not vanish only when S 1 + S 2 + S 3 is an integer that is the condition when the structure constant C 123 (or C 123 ) does not vanish.

One-point torus conformal block
In this section we will compute the lower and upper superblocks (2.15, 2.16) for the case where the conformal dimensions (h, h 1 ) are nonpositive integers or half-integers (these superblocks are associated with finite-dimensional representations). We will denote them by where the subscript f denotes finite-dimensional representation. As mentioned above the superblocks (2.15, 2.16) have poles when h is a nonpositive integer or nonpositive half-integer, hence we can not obtain B 0 (h, h 1 ) f , B 1 (h, h 1 ) f by substituting h with integer or half-integer nonpositive values. In order to compute them we substitute h with integer or half-integer nonpositive values into (2.15, 2.16) and subtract the infinite part 12 , the resulting expressions correspond to the superblocks associated with finite-dimensional representations, thus these superblocks are computed as follows (3.14) By computing these differences and taking into account only the first 4j + 1 terms 13 in the expansion in q we obtain the following expressions where F(−j, −j 1 , q) is the sl(2) one-point torus block given by (B.2). Now, we want to prove that superblocks (3.15, 3.16) can be computed by the following expression (up to an irrelevant constant c(j, j 1 , k)) where k = 0, 1. |j 1 − k 2 , −j 1 + k 2 is a state of the superspin representation R j 1 of osp(1|2), Tr j denotes the trace over the supermodule V j (2.17) of the superspin representation R j , z b is an arbitrary point in the bulk, z 1 is a point corresponding to the position of the fields (ϕ 1 , ψ 1 ) at the boundary 14 (the one-point blocks do not depend on z 1 ). W a [z 1 , z 2 ] are the Wilson line operators here L 1 and L −1 are the generators of the osp(1|2) gauge algebra which act on the states of the respective representation R a . I j;j,j 1 in (3.17) is the intertwining operator. In general one can have the intertwining operator I j 3 ;j 1 ,j 2 associated with representations (R j 1 , R j 2 , R j 3 ), it acts as follows I j 3 ;j 1 ,j 2 : and satisfies the following defining condition 13 Because there are only 4j + 1 states in finite-dimensional representations. 14 The boundary corresponds to ρ = ∞ (for details, see [44,46,47]).
where U j i (for i = 1, 2, 3) are the elements of the OSP (1|2) group in representations R j i . By taking into account these defining properties of the intertwining operator we have the following matrix elements in terms of the CGC m 3 , l 3 I j 3 ;j 1 ,j 2 l 1 , m 1 ⊗ l 2 , m 2 = C(l 1 , m 1 ; l 2 , m 2 |l 3 , m 3 ), (3.21) where the CGC are given by (2.29). The proof of the relation (3.17) is based on the structure (2.29) of the CGC and the Wilson lines formulation of the sl(2) torus block (see [58]). We decompose the trace T r j in (3.17) into two subtraces according to (2.17), thus the lhs of (3.17) becomes where By taking into account (2.29, 3.21) and that the action of the operator L 1 of the Wilson line operators W j , W j 1 on the states |l i , m i do not change the values of l i (see 2.23), we can rewrite (3.22) as follows where (F 1 , F 2 ) are two factors (2.30) and I sl 2 j 3 ;j 1 ,j 2 in (3.25) acts as follows (for fixed values of l i ) l 3 , m 3 |I sl 2 j 3 ;j 1 ,j 2 |l 1 , m 1 ⊗ |l 2 , m 2 = C sl(2) (l 1 , m 1 ; l 2 , m 2 |l 3 , m 3 ) = = (−1) −j 1 +j 2 −m 3 2j 3 + 1 j 1 j 2 j 3 m 1 m 2 −m 3 .

(3.27)
As the spherical CBs (3.25) does not depend on z b and due to the translational invariance of the one-point torus block we can choose z 1 = z b = 0. In order to simplify the computation of (3.25) we diagonalize W j [z b , z b + 2πτ ] = e 2πτ (L 1 + 1 4 L −1 ) expressing it in the following way where U j = e i 2 L −1 e −iL 1 e −iπL 0 is a group element in R j . Placing (3.28) in (3.25) and due to the property (3.20) of the intertwining operator, after this replacement, (3.25) remains unchanged except the boundary state |j 1 − k 2 , −j 1 + k 2 which transforms where β s are some coefficients depending on (j 1 , k, s), in fact, we will need only the state with projection s = 0. By taking into account (2.23, 3.27, 3.28, 3.29) we have that (3.25) becomes Due to the property of Wigner 3 − j symbol the nonzero terms correspond to 15 s = 0, by taking into account this and collecting a common factor c we have (3.30) is (3.31) Each sum in (3.31) can be shown [58] to be proportional to sl(2) one-point torus block for finite-dimensional representations (B.2) denoted by F. By computing 16 these sums, (3.31) can be expressed as follows.

Conclusions
In this work we studied the Wilson lines formulation of the osp(1|2) CBs. We considered CBs associated with finite-dimensional representations and light Φ i primary superfields. We used the decomposition of superfields into ordinary primary fields (1.8) and we concentrated only on the (ϕ i , ψ i ) sectors, thus we expressed the correlation functions of superfields in terms of ordinary fields, (eqs. (1.11, 1.12)). On the sphere, we showed that the CBs of the components of the two and three-point correlation functions can be computed by the Wilson lines formulation is very similar to the ansatz of the ordinary Virasoro case, and its proof essentially reduces to the sl 2 case. This reduction is related to the facts that the state space (2.17) of a superspin representation of osp(1|2) splits into two normal subspaces of two spin representations of sl 2 and that the CGC of osp(1|2) factorizes into the CGC of the sl 2 and the scalar factor as in (2.29). We also found in the super-Virasoro case that different components (ϕ i , ψ i ) of the superfield Φ i are related to different states of a superspin representation of osp(1|2) according to (1.15, 3.2), which make the dual construction more general.
According to [73] there are seven classes of superalgebras that involve extra internal degrees of freedom and realize the AdS 3 gravity in the CS formulation. In this work we have covered one of those cases, the osp(1|2). It might be interesting to extend the explicit construction of CBs in the Wilson lines context to the remaining classes. Another natural generalization consists in formulating the dual construction in the full N = 1 NS super-Virasoro CFT. In particular, this requires the study of quantum corrections (for related consideration, see [24]). It is also interesting to consider CBs of sl 3 algebra in higher genus topologies. sl 3 algebra is relevant for W 3 CFT in the large central limit. This consideration might be useful for a better understanding of CBs which are not completely fixed by the symmetry algebra in W 3 CFT (see, e.g. [74]). ity held at the Leibniz University Hannover and the Israeli Physical Society conference 2022 held at the Ben-Gurion University for giving the opportunity to participate and report this work.