Superconformal blocks from Wilson lines with loop corrections

We compute the $\mathcal{N}=1$ superconformal blocks from the networks of open Wilson lines in the $\text{osp}(1|2)$ Chern-Simons theory in the expansion of large central charge $c$. We first reproduce the $1/c$ correction of conformal weight from an open Wilson line by adopting the regularization prescription developed in our previous works. We then obtain the closed form expressions of superconformal blocks including $1/c$ corrections, which were not available before. We also examine heavy operators corresponding to supersymmetric conical spaces, and the geometry is quantized by utilizing the coadjoint orbits of the super Virasoro group. Superconformal blocks involving these operators are also examined.


Introduction
In [1,2], we computed the conformal blocks of the 2d W N minimal model from the bulk viewpoints in the large c expansion, where c is the central charge of the model. The large c regime of the model is supposed to be described by the 3d sl(N ) Chern-Simons gauge theory [3][4][5][6], and it was proposed that the conformal blocks can be computed from the networks of open Wilson lines in the Chern-Simons theory [7][8][9]. It is notoriously difficult to deal with quantum gravity effects generically. Nevertheless, we were able to offer a prescription to regularize divergences from loop diagrams associated with the Wilson line networks by making use of the boundary symmetry. Moreover, conformal blocks are known to be useful ingredients, e.g., for conformal bootstrap programs (see [10] for instance). Our previous works confirmed that the Wilson line method is a promising approach to compute conformal blocks in 1/c expansion. Related works may be found in [11][12][13][14].
In this paper, we consider a supersymmetric extension of the previous works. There are two main motivations for introducing supersymmetry. The first one is on the suppression of quantum effects and the second one is on the relation to superstring theory. We mainly focus on the N = 1 superconformal blocks in this paper, but we can discuss extensions to a theory with N = 2 W N +1 symmetry. Its large c regime is supposed to be described by the sl(N + 1|N ) Chern-Simons gauge theory [15,16], and there seem suppressions in the 1/c corrections of conformal weights, for instance. The relation between higher spin theory and superstring theory can be also argued with extended supersymmetry. As concrete examples, there are proposals on higher spin holographies with N = 3 supersymmetry in [17][18][19] and with N = 4 supersymmetry in [20,21]. One of the aim of this paper is to prepare for the analysis on these complicated cases.
As a simple and specific example, we examine the 2d N = 1 super Virasoro minimal model [22] from the bulk viewpoints. We examine the superconformal blocks of the model from the 3d osp(1|2) Chern-Simons gauge theory [23] in the large c expansion. We first examine the expectation value of an open Wilson line with divergences from loop diagrams regularized by adopting the prescription in [1,2]. From this computation, we read off the 1/c corrections to the conformal weight h of light operator with h = O(c 0 ). We then compute superconformal blocks involving these light operators from the networks of open Wilson lines including the 1/c corrections. We evaluate the identity four point blocks up to the 1/c 2 order, and consider three and general four point blocks up to the 1/c order. 1 In particular, we find the closed form expressions of the four point blocks with 1/c corrections. We then analyze the 1/c corrections for heavy operators with h = O(c), which correspond to supersymmetric conical spaces. For the purpose, we extend the analysis for the bosonic case in [24], which uses the quantization of the coadjoint orbits of the Virasoro group, see [25,26]. We further evaluate a heavy-heavy-light-light block with 1/c corrections from an open Wilson line in a supersymmetric conical space. See [27][28][29][30] for some recent works on the large c limit of N = 1 superconformal blocks.
The organization of this paper is as follows. In the next section, we summarize several basics of the N = 1 superconformal field theory. We first introduce the minimal model of N = 1 super Virasoro algebra and then study superconformal blocks. In section 3, we introduce an open Wilson line in the osp(1|2) Chern-Simons gauge theory and explain our prescription to regularize divergences arising from loop diagrams. Applying the prescription, we reproduce the 1/c corrections in the conformal weight of light operator. In section 4, we compute the 1/c corrections in the three and four point blocks from the networks of open Wilson lines. In particular, we obtain the closed form expressions for the four point blocks including 1/c corrections. In section 5, we examine the 1/c correction of heavy operator from the supersymmetric conical defect geometry in the osp(1|2) Chern-Simons gauge theory. We first compute the 1/c correction of the conformal weight by quantizing the coadjoint orbits of the super Virasoro group. We then study a heavy-heavy-light-light block from an open Wilson line in the supersymmetric conical space. Section 6 is devoted to conclusion and discussions. In appendix A, we include the correlators of superconformal currents and give a derivation of a superconformal Ward-Takahashi identity. In appendix B, we obtain the large c limit of four point blocks and the expressions of four point blocks with degenerate operators in the expansion of z and 1/c with the techniques of superconformal field theory.
2 Basics of N = 1 superconformal field theory We start with a review on the N = 1 super Virasoro minimal model. The N = 1 superconformal algebra is generated by energy-momentum tensor T (z) with dimension 2 and superconformal generator G(z) with dimension 3/2. In terms of mode expansions, the (anti-)commutation relations are [L m , L n ] = (m − n)L m+n + c 12 m(m 2 − 1)δ m+n , where r, u ∈ Z + 1/2 for the NS-sector and r, u ∈ Z for the R-sector. A superconformal primary state |h satisfies L 0 |h = h|h , L n |h = G r |h = 0 (2.2) with n, r > 0. The unitary minimal models are defined with the discrete values of central charge with k = 0, 1, 2, · · · . The conformal weight of primary state is [22] h m,n = ((k + 4)m − (k + 2)n) 2 − 4 8(k + 2)(k + 4) + 1 − (−1) m+n 32 (2.4) with m + n ∈ 2Z for the NS-sector and m + n ∈ 2Z + 1 for the R-sector. Here we are interested in an analytic continuation of the minimal models with large c. The expression of central charge in (2.3) suggests that 2 The conformal weight in (2.4) is expanded in 1/c as (2.6) 2 We can take k = −2 + O(c −1 ), but in that case we should replace m and n in (2.4).
We first study light states with conformal weight of order c 0 . This means that we should set n = 1. Rewriting m = 4j + 1, we have We only consider the NS-sector for light states, thus j ∈ Z/2. We describe these states in terms of bulk open Wilson lines. The conformal weights of other states are of order c, and we consider a particular set of operators with (s = 1, 2, 3, . . .) Here s is odd for the NS-sector and s is even for the R-sector. These state are conjectured to corresponds to supersymmetric conical spaces. In order to analyze correlation functions, it is convenient to work with superspace (z, θ), see, e.g., [31,32]. The generators of superconformal algebra are written in terms of a superfield T (z, θ) = T F (z) + θT B (z) with T (z) = T B (z) and G(z) = 2T F (z). For a superconformal primary, we define a superfield as In terms of superfield, the two point function can be written as With component fields, the two point functions are (2.12) Expanding the conformal weight in 1/c as the two point function becomes where only the holomorphic sector is expressed. We can also expand W W in 1/c in a similar manner.
The three point function of superfields depends on (3|3) supercoordinates, 3 while only (3|2) parameters can be fixed by using osp(1|2) subalgebra as the global part of super Virasoro algebra. Therefore, the three point function is given by a function of a fermionic superconformal invariant For spinless operators, we thus find with h 12 = h 1 + h 2 − h 3 and so on. In particular, we have and Later, we reproduce the 1/c corrections of these three point blocks from the networks of open Wilson lines.
Since the four point function of superfields depends on (4|4) supercoordinates, it is parametrized by one bosonic and two fermionic superconformal invariants. For them, we may choose (see, e.g., [33,34]) For spinless operators, the four point function is then expressed as This, in particular, means that there are four types of independent superconformal blocks, such as, The other types are obtained simply by the action of superconformal transformations. In this paper, we consider the 1/c corrections in the four point function V V V V , but the extension to the other cases should be straightforward. We compute superconformal blocks for the four point function from the bulk viewpoints. We denote the operator product expansions schematically as (2.24) In order to evaluate them, we need the forms of the operators C 12 (N, h) explicitly. They can be computed purely from the superconformal algebra but quite complicated in general. At the large c limit, however, the operators C 12 (N, h) becomes simplified as (see (4.23) of [34]) the large c limit of superconformal blocks can be computed as (see, e.g., [33,34]) With general c, we can obtain superconformal blocks by solving differential equations [33,36] in case that degenerate operators are involved. It is difficult to obtain their closed forms but not so to find out their expressions in z expansion, see appendix B.2 for the details. Another way is to use recursion relations as in [33], and the first few terms in z expansion were obtained. The recursion relations are generalizations of the bosonic counter parts obtained in [37], see also [38,39] for other types of recursion relations for the N = 1 superconformal blocks.

Conformal weight of primary operator
In this section, we evaluate a two point function from an open Wilson line in the osp(1|2) Chern-Simons gauge theory. In the next subsection, we give our strategy of Wilson line computation by focusing on how to renormalize divergences from loop diagrams developed in [1,2]. In subsection 3.2, we reproduce the conformal weight of primary operator up to 1/c 2 order in (2.7) with the bulk method.

Strategy of Wilson line computation
We examine the 2d N = 1 super Virasoro minimal model with large c in terms of the osp(1|2) ⊕ osp(1|2) Chern-Simons gauge theory, which describes a N = 1 supergravity theory on AdS 3 [23]. The action is given by The gauge fields A,Ã take values in the osp(1|2) Lie superalgebra, see, e.g., [40] for some basics of Lie superalgebra. We use the generators of the Lie superalgebra as V 2 n , V 3/2 r with n = ±1, 0, r = ±1/2. The generators V 2 n , V In a gauge, solutions to the equations of motion can be written as 4 The AdS solution can be given by a(z) = V 2 1 , which corresponds to the AdS metric ds 2 = dρ 2 + e 2ρ dzdz. Assigning that the configuration approaches to the AdS one for ρ → ∞, we can set [41,42] (see [43,44] for higher spin gravity and [16,45,46] for higher spin supergravity) It was shown that T (z) and G(z) generate the N = 1 superconformal algebra with the central charge This value is the same as the well known Brown-Henneaux one [47]. As in [1,2,12], we compute the two point function V V from Here the open Wilson line is defined as where P denotes the path ordering and ρ-dependence is removed by a gauge transformation. The finite dimensional representation of osp(1|2) is labeled by j = 0, 1/2, 1, · · · (see, e.g., [40]). In the above expression, |hw and |lw denote the highest and lowest weight states of the representation j, respectively. We also use as a bulk computation for ΛΛ . We evaluate the path integral for the expectation value of the open Wilson line by integrating over the gauge fields. As explained in [11,12], it can be effectively done by computing the products of N = 1 superconformal currents T (z), G(z) inside the Wilson line operator (3.7) with (3.4) in terms of correlators of a N = 1 superconformal theory. We should integrate over the position z of T (z), G(z) inside the Wilson line operator (3.7) with (3.4), and there would be divergences arises from the coincident points of at least two of T (z), G(z). Following [1,2], we regularize divergences by shifting the conformal weights of T (z), G(z) as 2 → 2 − , 3/2 → 3/2 − . Namely, we use the correlators of T (z), G(z), such as, see (A.2) for others. In order to remove terms diverging at → 0, we introduce parameters c s (s = 2, 3/2) with and insert them in the Wilson line operator as We claim that all divergences can be removed by renormalizing the parameters c s along with the normalization of the Wilson line such as Notice that there is no explicit relation between Z V and Z W since the introduction of the regular in (3.9) breaks the supersymmetry. We choose Z V , Z W , and c s (s = 2, 3/2) such that there are no divergences at the limit → 0. However, there are terms surviving at the limit → 0, and we have to decide how to deal with them. Since we can fix the -independent terms in Z V and Z W from (3.12), the problem is for c s . In fact, the authors in [12] failed to reproduce the 1/c 2 order corrections in the conformal weight of primary operators in the Virasoro minimal model due to this issue. In [1,2], we have offered a prescription to fix -independent terms in c s by requiring that the boundary correlators are consistent with the boundary symmetry.
Here we adopt the same prescription. Concretely, we fix the -independent terms in c 2 and c 3/2 such as to be consistent with the N = 1 superconformal Ward-Takahashi identities see appendix A for a derivation of the right equation. In [1], we have solved the problem on 1/c 2 order corrections in [9] and extended to the case with the W 3 minimal model. In the analysis of sl (2), it was convenient to work with the x-basis as in [1,2,7,11]. In the current case with osp(1|2), we introduce a bosonic parameter x and a fermionic one ξ as In terms of x, ξ, the osp(1|2) generators are represented as The lowest weight state is x, ξ|lw = 1, and the next one is proportional to V where the normalizations are chosen for convenience. The highest weight state is determined from the condition that it is annihilated by the actions of V 2 −1 and V with our choice of normalizations. With these expressions, the open Wilson line can be expanded in 1/c as Moreover, we defined

Two point function up to 1/c 2 order
In this subsection, we evaluate the expectation value of the open Wilson line W (1)j (z; 0) up to 1/c 2 order. We first evaluate W (1)j (z; 0) up to 1/c order. We reproduce the conformal weight (2.7) and fix Z V up to the order. We also examine W (2)j (z; 0) since we need the expression of Z W to determine c 3/2 . We then compute the three point function with T (z) or G(z) and fix c 2 and c 3/2 through the superconformal Ward-Takahashi identities (3.13). With the information of Z V , c 2 , and c 3/2 up to the order 1/c, we reproduce the conformal weight (2.7) up to the 1/c 2 order from the open Wilson line W (1)j (z; 0) . We start from the expectation value of the open Wilson line W (1)j (z; 0) . At the leading order in 1/c, we find as expected. At the next leading order, there are two contributions with s = 2, 3/2. Here J (s) and N s were introduced in (3.19). Moreover, we have used . Introducing the regulator as in (3.9), the sum of them is computed as Therefore, by setting we reproduce (3.12) with the correct shift of conformal weight as in (2.14) with (2.7). In the same manner, we have up to the term vanishing at → 0. Thus the normalization leads to (3.12) with the same shift of conformal weight. In order to fix c 2 and c 3/2 , we need to evaluate the three point function with T (y) or G(y) as in (3.13) with an open Wilson line. For the three point function with T (y), we compute as argued in [1,2]. At the leading order in 1/c, this becomes which is consistent with (3.13). At the next leading order in 1/c, there are three types of contribution. The first one comes from the above integral with the 1/c order term in The second type is with the insertions of two extra currents as with s = 2, 3/2. The third type is with the insertions of three extra currents as For the second and third types, we use the correlators of currents in (A.2). The sum of them can be written as up to the order 1/c. The factor Z 2 V obtained in (3.26) is included such that the operator V h (z) in the three point function (3.13) is normalized as in (3.12). The relation in (3.13) is realized if we choose the parameter c 2 as In this way, we fix the 1/c order term in c 2 along with removing the divergence from the 1/ order term in → 0. Next, we evaluate the three point function with G(y) in (3.13) using an open Wilson line. For this, we define Then, the three point function is evaluated from Here Z W and Z V obtained in (3.28) and (3.26) are included such that Λ h (z) and V h (0) in the three point function are normalized as in (3.12). At the leading order in 1/c, we have as in (3.13).
Just as in the case with H (2) (z), there are three types of contribution at the next leading order. The first one comes from (3.40) but with c The second one are with the insertion of two extra currents as with (s, s ) = (2, 3/2), (3/2, 2). The third one are with the insertion of three extra currents as The non-trivial contributions are with From the sum of these contributions, we find up to 1/c order. Choosing the parameter c 3/2 as as in (2.14). There are four types of contribution to this quantity. We have already evaluated the 1/c order term in W (1)j (z; 0) above as the sum of I (z) defined in (3.24). Therefore, the first contribution is with the 1/c order term in Z 2 V as For the computation of W (1)j (z; 0) at the order of 1/c, we have set c 2 = c 3/2 = 1. Therefore, the second type of contribution is The third type are with the insertion of three currents as The fourth type are with the insertion of four currents as For the correlators of currents, we use (A.2). Let us first sum over the third and fourth types of contribution as + 3j log(z) 6(2j + 1) 4j 2 + 2j − 3 log(z) + (2j − 1) 40j 2 + 44j + 1 , where we have kept only the term including log(z). From this, we observe that there is non-local divergence from the term proportional to log(z)/ . Moreover, the terms in front of log(z) and log 2 (z) do not reproduce (2.14) with (2.7). This corresponds to the problem in [12] for the bosonic case. The problem was solved in [1] by including Z V and c 2 in the current terminology. Similarly we arrive at after including the effects of Z V , c 2 , and c 3/2 . In the final expression, there is no non-local term and the 1/c 2 order correction of the conformal weight in (2.7) is correctly reproduced.

Superconformal blocks
From the networks of open Wilson lines, we study more generic correlators. As in [2,8,9,12], we evaluate for n-point blocks of the type n i=1 V . Here S represents a singlet state in the product representation j 1 ⊗ · · · ⊗ j n . We can construct a singlet from the product representation in many ways in general. There are also many conformal blocks for a correltor in general as well. As explained in [2,9], a different choice of S leads to a different conformal block. The open Wilson lines are connected by the vertex S| at z = z 0 , but the expectation value should not depend on z 0 . We evaluate the 1/c corrections for the Wilson line networks by following the analysis in [2]. When we consider correlators involving W (or Λ), then we should replace |hw by | hw .
As in [2], it is convenient to move to the conjugate form of (4.1) and to use the basis in terms of The singlet condition is now written as for all s = 2, 3/2 and m = −s + 1, · · · , s − 1. Here the generators V s m (X i ) are given in In the next subsection, we reproduce the three point blocks in (2.17) and (2.18) up to 1/c order. In subsection 4.2 and subsection 4.3, we consider the four point function with h q and h j in (2.7). We obtain the closed form expressions of the identity block up to 1/c 2 order and those of the general blocks up to 1/c order. We check that the results are consistent with those obtained from a N = 1 superconformal field theory.

Three point blocks
We consider the three point blocks in (2.17) and (2.18). In the expression of (4.2) with (3.16), we evaluate for (2.17) and 3 (j i |X i ) = ξ 123 X j 12 12 X j 13 13 X j 32

32
(4.7) with and j 12 = j 1 + j 2 − j 3 and so on. Noticing that we can reproduce the leading order expressions of (2.17) and (2.18) from (4.5) and (4.6), respectively. After these preparations, we move to the next leading order in 1/c and compute the contributions expressed by the diagrams of figure 1. We define where V s (z) is given in (3.22). The integrals to compute are where the two point functions of currents are given in (3.9). The contributions at the 1/c order are given by the sums of these integrals as which are evaluated as 13 log(z − 1)) (4.14) and so on. Here we have subtracted the terms proportional to the tree level expressions, which change only irrelevant overall constants. We can see that they reproduce the 1/c order terms in (2.17) and (2.18). Before going into the analysis of four point blocks, we would like to remark on a point, which will be important later. There are two independent singlets |S as in (4.7), and we chose one of them for (4.5) and the other for (4.6). The choice sounds obvious from the analysis at the leading order in 1/c, but it could be subtle at the higher orders. An open Wilson line includes the information of both V h (z) and Λ h (z), since it represents the propagation of an superfield. An insertion of G(z) exchange V h (z) and Λ h (z) as in (3.13) and (3.37), thus we should be careful on quantum effects associated with the insertions of G(z). For three point blocks at 1/c order, this does not cause any problems. This is because the number of G(z) inserted should be even and the even number of ξ i action does not exchange the two solutions in (4.7). However, this point becomes important for four point blocks as seen below. see also [11]. There are three types of contributions to the 1/c corrections as illustrated with the diagrams of figure 2. See figure 1 and figure 2 in [2] for the diagrams omitted here. The contributions coming from the top diagrams are of the self-energy type. We do not study the contributions here since we already knew that the net effect only shifts the conformal weight. The middle left diagram describes the exchange of a current. Noticing that only the energy momentum tensor (or an integer spin current) is exchanged, the contribution from the diagram is

Identity four point blocks
(1)q (z; 0; z 2 )f (1)j (∞; 1; z 1 ) T (z 2 )T (z 1 ) = 2h j h q c z 2q+2 2 F 1 (2, 2; 4; z) (4. 16) up to 1/c order, where h q = −q and h j = −j. With q = 1/2, the z expansion of the above expression reproduces the 1/c order term in (B.12). The middle diagrams in figure  2 describes a type of contributions, which can be included by using h q , h j in (4.16) with 1/c corrections as shown in [2]. Therefore, the 1/c 2 order contributions of this type can  Another type of contributions at the order of 1/c 2 come from the exchanges of two currents as in the bottom diagrams of figure 1. In the current case, the spin (or more precisely the statistic) of two currents should be the same. Thus the contribution is the sum of (s = 2, 3/2) The sum is computed as 3/2 (z) = 18jqz 2q−2 c 2 2z 2 (8j(q + 1) + 8q + 3) (4.20) From the sum s=h,2,3/2 K s (z), we obtain the identity block as We can check that the result matches with the expression in (B.12) by setting q = 1/2.

General four point blocks
In this subsection, we study the four point blocks for (4.4) but with the exchange of general operators. Using the expression in (4.2) with (3.16), we evaluate the networks of open Wilson lines as (4.23) Here we set j 1 = j 2 = q, j 3 = j 4 = j and (z 1 , z 2 , z 3 , z 4 ) = (z, 0, ∞, 1). As explained in section 2, there are two independent superconformal blocks, where the integer and half-integer level of descendant operators are exchanged. A main issue here is which solutions to the singlet equation (4.3) should be used. With (4.9), the leading order expressions of superconformal blocks in 1/c are obtained simply by replacing x i in the vertices with z i . Therefore, we know which solutions are used at the leading order in 1/c. Moreover, as discussed at the end of subsection 4.1, we may need to use a different vertex when the insertions of G(z) are involved. In this subsection, we only consider the corrections of 1/c order, thus there should be either no insertion or two insertions of G(z). The two insertions of G(z) would change the type of V V V V into that with two V 's replaced by two W 's. As explained in (2.20) in terms of superconformal invariants, there are two independent types which are not exchanged by superconformal transformations. We can change the z → 0 channel blocks for V V V V to those of by superconformal transformations. However, this is not possible for The leading order expressions of the four point blocks for (4.4) in 1/c were already given in (2.27). Thus the vertices of type V V V V can be obtained by replacing z i with x i . For our application, it is convenient to rewrite them as the integration over the products of two three point blocks as (see [2,48] with x = x . The integration contour is x ∈ (0, ∞) and the prefactors are The integration contour is x ∈ (0, ∞) and the prefactors are 5 For the rest of computation, we closely follow the analysis in [2]. The contributions to the 1/c corrections of four point blocks come from the diagrams in figure 4. For the top diagrams, we compute with a = 1, 2, 3 and s = 2, 3/2, where t s,α a (z, x) were defined in (4.11) with j 1 = j 2 = q, j 3 = p. We can show explicitly that the contributions from the bottom left diagram are proportional to the leading order expressions, so we neglect them. 5 The first factor in N For the other two diagrams, we define the functions with V s (z) in (3.22) and j 1 = j 2 = q, j 3 = p. Here we have set We further define the other functionŝ (4.34) with j 1 = j 2 = j and Here η s,α (ξ 3 ) is given in (4.33). For the spin 2 exchanges, we evaluate the integrals 3/2 (z; z a ; y 1 , x)û There could be terms proportional to the leading order expressions in the 1/c order contributions, but they can be subtracted (or tuned in a proper way). This is because we are interested in superconformal blocks, so the overall factors do not matter. Our normalizations are In order to express the results, it might be useful to introduce functions as We then find . We can check that they reproduce the 1/c order terms in (B.13) by setting q = 1/2. Moreover, the first few terms in z expansions with general h i and finite c were obtained in [33] by solving recursion relations. We also find agreements for

Superconformal blocks involving heavy operators
In the previous sections, we have considered light operators with conformal weight of order c 0 . In this section, we analyze the 1/c corrections for heavy operators with conformal weight of order c from the bulk viewpoints. We first construct supersymmetric conical spaces, which correspond to a set of heavy operators. We then examine the quantum corrections to the conical spaces by making use of the symmetry of the super Virasoro group, and we reproduce the 1/c corrections for the conformal weights of the corresponding heavy operators. Using the analysis, we compute a heavy-heavy-light-light block from an open Wilson line in the conical geometry.
We begin with classical supersymmetric conical spaces in the osp(1|2) Chern-Simons gauge theory. In order to discuss the supersymmetric geometry, it is convenient to work with a cylindrical coordinate w, which is related to the planer one z as z = e iw . With the coordinate, we use instead of (3.4). Asymptotic symmetry is given by the N = 1 superconformal algebra generated by T (w), G(w) [41] as mentioned in section 2. Here we put the phase factors with respect to T (z) and G(z) in the planer coordinate in order to match with the conventional notations. We consider a gauge configuration independent of w and corresponding to a geometry. This implies that Here any restriction is not yet assigned on s, and the condition for s will be obtained by requiring the geometry preserving supersymmetry below. In order to obtain the supersymmetric condition, we apply the method developed for the supersymmetric geometry in the sl(N + 1|N ) Chern-Simons theory as in [16,[49][50][51][52][53]. We look for spinors, which satisfy the Killing spinor equation, Formally, solutions can be written as with a constant spinorˆ . The problem is now whether the spinor satisfies the antiperiodic (or periodic) boundary condition in the NS-sector (or the R-sector). For the current configuration of gauge field, we find (w = φ + iτ ) with a matrix S. Rewriting the spinor as Therefore, the anti-periodic (or periodic) boundary condition leads to an odd (or even) integer s. Recall that the conical defect geometry corresponds to the integer s for the bosonic case [24], see also [4].
(5. 26) This implies that the expansion in terms of f r , η r corresponds to the quantization with the Planck constant as h ∼ 1/c, see [25] for the Virasoro group. For the R-sector, we use in addition to (5.24) and (5.26) expect for those involving η 0 and B 0 . The vacuum state and its dual are defined as for m > 0 and r > 0.
When we replace f m , η r by operators A m , B r , we should take care of the ambiguity related to the ordering of operators. This ambiguity exists for 0 →ˆ 0 , and we express it asˆ by introducing a parameter n 0 . Here n 0 corresponds to the 1/c correction for the conformal weight of the heavy operator dual to the supersymmetric conical geometry with label s. As in [24], we fix n 0 by requiring that the generators satisfy the (anti-)commutation relations of the superconformal algebra. Usinĝ we require that 6 This, in particular, means Using the (anti-)commutation relation, we find s 0|{ĝ s/2 ,ĝ −s/2 }|0 s = i

Open Wilson line in a supersymmetric conical space
In our previous work [2], we have developed a way to compute a heavy-heavy-light-light block from the bulk perspective, and in this subsection, we generalize the analysis to the supersymmetric case. As a heavy-heavy-light-light block, we consider whose expression in z expansion is obtained in appendix B.2. In the cylindrical coordinate, the block can be expressed as which is related to (5.36) as The phase factor is included for later convenience. Following the analysis in [2], we compute the superconformal block from an open Wilson line in a supersymmetric conical geometry with label s as where we have used (5.1). We examine the Wilson line operator in 1/c expansion. We defineĴ Here we have used We set c s = 1 in (5.41) since we consider only the next leading contributions in 1/c. The leading order in 1/c becomes which reproduces the z expansion of the leading order term of (B.15) in 1/c by using (5.38). In fact, it is the same as the bosonic case in (B.19), see [28] for general arguments. We then move to the 1/c corrections. The contributions can be divided into two types; one is with the insertions of one current and the other with two currents. A type of 1/c correction with a current insertion comes from

(5.45)
This is almost the same as that in the bosonic case but with a different value of n 0 . The difference is ∆n 0 = (5s + 1)(s − 1) 16 Another contribution is with the insertion of twoĴ (2) (w)'s, but it is exactly the same as the bosonic case as Therefore, we do not repeat here. There is also a contribution with the insertion of twô J (3/2) (w)'s (or two G(w)'s), which is new in the current case. Below we examine the integral for the contribution. We thus evaluate Let us first set s to be odd and then discuss the case with even s. We define the two point function of the superconformal current as which is computed as with w 21 = w 2 − w 1 . The integral (5.48) diverges due to the singular behavior of the two point function in (5.51). As in the bosonic case analyzed in [2,56], we redefine the integral (5.48) by removing the divergent term as (5.52) Here we have introduced new parameters as x = e iw , x 1 = e iw 1 , x 2 = e iw 2 , y = e isw , y 1 = e isw 1 , y 2 = e isw 2 .  (3)) , −294 x 7 − 1 log x 6 + x 5 + x 4 + x 3 + x 2 + z + 1 − 294x + 515 − 294 log (7) .

The divergent integral is
s,div (w) = 6iy −s/2 sc y 1 dy 2 Taking care of the coordinate transformation from y to w as in [2], we use Here a new parameter δ is introduced by using the ambiguity in the overall normalization at the order of 0 . Now we compare the result obtained above to (B.16) computed with the technique of superconformal field theory. As explained above, some of the contributions are the same as those for the bosonic case and the others are not. The additional part is given by the sum as s,div (w) . Just as in the odd s case, we define the integral by removing the divergent term as

(5.59)
We can perform the integrals for specific s = 2, 4, 6, · · · with no difficulty. The evaluation of the divergent integral is the same as in (5.56). In this way, we can compute the 1/c correction of the block with specific s = 2, 4, 6, · · · , though we do not go into the details here.

Conclusion and open problems
In this paper, we computed the N = 1 superconformal blocks form the networks of open Wilson lines in the osp(1|2) Chern-Simons theory in the large c expansion. This is a supersymmetric extension of the previous works in [1,2], and one of the aim is to show that the method developed in these papers is useful for examining other models as well. There were, however, several points we should elaborate in the current example as explained below. We reproduced the conformal weight of light operator in (2.7) from an open Wilson line up to the 1/c 2 order. For the purpose, we fixed the parameter c s introduced in (3.11) by making use of the superconformal Ward-Takahashi identities (3.13). Since the insertion of G(z) exchanges V h (z) and Λ h (z), we should take care of the renormalization of the two operators Z V , Z W in terms of open Wilson line as in (3.39). We also computed the 1/c corrections of superconformal blocks from the networks of open Wilson lines. At the connecting point z 0 , we should assign the vertex |S , which is given by a solution to the singlet conditions (4.3). There are several solutions, and we may need to use several ones, even when we consider only one type of superconformal blocks. This is because there could be the insertions of G(z) from open Wilson lines, and they can change the vertex we should use, see figure 3. Furthermore, we examined the 1/c corrections associated with a heavy operator from a supersymmetric conical space. A main point is to construct the symplectic form over the coadjoint orbits of the super Virasoro group in order to quantize the geometry. Fortunately, it was already obtained in [55]. Applying the result, the rest is rather straightforward by extending the analysis in [24] for the conformal weight and our previous one in [2] for a heavy-heavy-light-light block.
We would like to think about the following open problems. For light operators, we considered only the NS-sector, and it would be interesting to deal with also the R-sector. In particular, the correlators with the simplest degenerate operator in the R-sector satisfy the second order differential equations [57]. Therefore, there would be a simple way to express light operators in the R-sector even in terms of the osp(1|2) Chern-Simons gauge theory. For heavy operators, we analyzed the R-sector as well in terms of Wilson line operator. We have not compared the results with those from conformal field theory, but it should not be so difficult to do so. We would like to systematically analyze correlators involving operators in the R-sector from the viewpoint of superconformal field theory. In our previous paper [2], we studied only a simple example of general W 3 four point block from the bulk viewpoints, and the current work may be useful to extend the analysis for more general blocks.
In this paper, we considered the N = 1 super Virasoro minimal model since it is the simplest example with supersymmetry. However, the examples with extended supersymmetry are physically more interesting as mentioned in the introduction. For examples, we could similarly analyze the N = p superconformal blocks with p = 2, 3, 4 in terms of Chern-Simons gauge theory as in [44]. We also would like to study the blocks with respect to the N = 2 super W N +1 algebra and their extensions. These superconformal blocks should be useful as fundamental objects for the study of superstring theory and AdS/CFT correspondence. Furthermore, we could read off higher spin charges in the N = 2 W N +1 minimal model by using the N = 2 superconformal blocks as in [58]. It might be rather easier to use the N = 1 superconformal blocks obtained here in the N = 1 higher spin holographies of [59,60].
Here we have used

B.2 Superconformal blocks with degenerate operators
It is generically difficult to compute conformal blocks with primary operators. However, if one of the operators is a degenerate one, then we may be able to obtain the expressions by solving differential equations. Here we consider a four point function with a simplest degenerate operator V h 3,1 , see (2.4). We denote the four point function as Then, the differential equation for g was obtained as [33,36]  m . We can show that the leading order term of (B.15) in 1/c coincides with that of (B.17) or (B.19).