Rectangular W-algebras of types $so(M)$ and $sp(2M)$ and dual coset CFTs

We examine rectangular W-algebras with $so(M)$ or $sp(2M)$ symmetry, which can be realized as the asymptotic symmetry of higher spin gravities with restricted matrix extensions. We compute the central charges of the algebras and the levels of $so(M)$ or $sp(2M)$ affine subalgebras by applying the Hamiltonian reductions of $so$ or $sp$ type Lie algebras. For simple cases with generators of spin up to two, we obtain their operator product expansions by requiring the associativity. We further claim that the W-algebras can be realized as the symmetry algebras of dual coset CFTs and provide several strong supports. The analysis can be regarded as a check of extended higher spin holographies including full quantum corrections. We also extend the analysis by introducing $\mathcal{N}=1$ supersymmetry.


Introduction and summary
It has been argued that higher spin gravity is useful to understand the tensionless limit of superstring theory [1]. In particular, the matrix extended version of Vasiliev theory is expected to explain higher Regge trajectories of superstrings, see, e.g., [2] including higher tensor extensions. In [3], it was proposed that classical 3d Prokushkin-Vasiliev theory with M × M matrix valued fields constructed in [4] is dual to 2d Grassmannian-like model at a large N limit, see also [5][6][7] for related works. Setting M = 1, the proposal essentially reduces to the Gaberdiel-Gopakumar duality [8] and its N = 2 version becomes the same as that of [9]. With N = 3 enhanced supersymmetry, the relation to superstring theory on AdS 3 ×M 7 has been discussed [10,11], where M 7 is a 7-dimensional manifold. It would be useful to introduce N = 4 supersymmetry as in [12,13], which enables us to relate to superstring theory on AdS 3 ×S 3 ×S 3 ×S 1 or AdS 3 ×S 3 ×T 4 . However, it seems difficult to introduce the matrix degrees of freedom in a way compatible with the N = 4 supersymmetry.
In [14], the asymptotic symmetry of the extended higher spin gravity was examined including full quantum corrections. 1 Three dimensional higher spin gravity can be constructed by a Chern-Simons gauge theory based on a higher rank gauge algebra g. Without matrix extension, the gauge algebra of the Prokushkin-Vasiliev theory is given by hs [λ], which can be truncated to sl(n) at λ = n (n = 2, 3, . . .). Setting g = sl(n), the higher spin gravity includes the gauge fields of spin s = 2, 3, . . . , n. We can extend hs[λ] by multiplying M × M matrix degrees of freedom [3,12], and the extended algebra may be denoted as hs M [λ]. The gauge algebra is truncated to sl(M n) at λ = n, and it may be decomposed as [14,17] sl(M n) sl(M ) ⊗ 1 n ⊕ 1 M ⊗ sl(n) ⊕ sl(M ) ⊗ sl(n) .
(1. 2) In particular, the higher spin gravity has sl(M ) (or su(M )) gauge sector. 2 The gravitational sector is identified with the principally embedded sl(2) in 1 M ⊗sl(n). Applying the general prescription of [18,19], the asymptotic symmetry can be identified as the rectangular Walgebra with su(M ) symmetry. The central charge c of the algebra and the level of su(M ) currents were computed and the operator product expansions (OPEs) among generators of spin up to two were obtained by requiring associativity of the OPEs. Based on the holography of [3], it was conjectured that the rectangular W-algebra can be realized by the symmetry algebra of (1.1) even with finite c, and several important checks have been provided. The analysis of symmetry with finite c would allow us to obtain some information on quantum effects of higher spin gravity.
In this paper, we generalize the analysis by considering the restricted matrix extensions of higher spin gravity. There are two ways to truncate the degrees of freedom [4], and the higher spin gravity includes so(M ) or sp(2m) (for M = 2m) gauge sector depending on the way of truncation. Without matrix extensions, the gauge algebra hs [λ] can be truncated to hs e [λ] with only even spin generators. The holography with the truncated higher spin gravity was proposed in [20,21] and the quantum asymptotic symmetry were analyzed in [22]. The even spin algebra hs e [λ] can be further truncated when λ takes an integer value, but the truncation depends on whether the integer number is even or odd. For λ = 2n + 1 and λ = 2n (n = 1, 2, . . .), hs e [λ] can be reduced to so(2n + 1) and sp(2n), respectively [23]. In a similar way, we can show that there are two types of truncation with integer λ for each restricted matrix extension, and hence we deal with four types of higher spin gravity with truncated spins in total.
For the truncated cases, we consider the Chern-Simons gravities with g = so(M (2n + 1)), sp(2M n), sp(2m(2n+1)), so(4mn). We further decompose the gauge algebras like (1.2) but with subalgebra so(M ) or sp(2m), see table 1. With specified gravitational sectors, the asymptotic symmetries of the corresponding higher spin gravities are identified as the W-algebras obtained by the Hamiltonian reductions of g. We may call the corresponding W-algebras as type g. We compute the central charges c and the levels of so(M ) or sp(2m) currents from the Hamiltonian reductions. The W-algebras of type sp(2M n), so(4mn) with n = 1 include only generators of spin up to two, and we obtain the OPEs among the generators by requiring the associativity. We further conjecture that the rectangular W-algebras with so(M ) or sp(2m) symmetry can be realized as the symmetry algebras of the cosets in table 2. We provide the maps of parameters by comparing the central charges c and the levels of the so(M ) or sp(2m) currents. Moreover, we construct symmetry generators up to spin two and reproduce the OPEs obtained from the associativity.
The analysis can be generalized by introducing N = 1 supersymmetry. As in the bosonic cases, we can construct the restricted matrix extensions of higher spin supergravity with so(M ) or sp(2m) symmetry. See [6] for the holography with the higher spin supergravities. Without matrix extensions, the holography with N = 1 truncated higher spin supergravity was proposed in [24] and the quantum asymptotic symmetry was examined in [25]. For the cases with truncated spins, we use the Chern-Simons supergravities based on superalgebras g = osp(M (2n + 1)|2M n), osp(M (2n − 1)|2M n), osp(4mn|2m(2n + 1)), osp(4mn|2m(2n − 1)) and decompose like (1.2) but with subalgebra so(M ) ⊕ so(M ) or sp(2m)⊕sp(2m) as in table 1. The asymptotic symmetries of the higher spin supergravities are the W-algebras obtained as the Hamiltonian reductions as in the bosonic cases, and the corresponding W-algebras are again called as type g. The central charges c and the levels 1 , 2 of so(M ) or sp(2m) currents are computed, and the OPEs among generators are obtained for the simple cases with currents of spin up to two by requiring associativity of the OPEs. We also conjecture that the cosets in table 2 realize the N = 1 rectangular  Table 2. Our proposals on the coset models whose symmetries are realized by the rectangular W-algebras. These W-algebras are obtained by the Hamiltonian reductions of the gauge (super)algebras and with the sl(2) (or osp(1|2)) embeddings listed in table 1.
W-algebras as the symmetry algebras. We give the maps of parameters by comparing the central charges c and the levels 1 , 2 of the affine symmetries and explicitly construct low spin generators of the symmetry algebras of the cosets.
In the companion paper [26], two of the current authors extend the previous analysis of [14] on the rectangular W-algebras with su(M ) symmetry in several ways. Firstly, we examine the OPEs among low spin generators with removing the restriction of n = 2. Moreover, we study the degenerate representations of the rectangular W-algebra with M = n = 2 and examine the relations to states of the coset (1.1) and conical defect geometry of Chern-Simons gravity in [27] (see also [15,28,29]). It should be possible to apply these analyses to the current examples with restricted matrix extensions. In [26], we also extend the previous analysis of [14] by introducing the N = 2 supersymmetry. It is important to consider the cases with more extended supersymmetry as in [10,11,30]. This paper is organized as follows. In the next section, we introduce the higher spin (super)algebras, which are used to construct higher spin (super)gravities. With the decompositions as listed in table 1, we identify sl(2) subalgebras as the gravitational sectors. In section 3, we examine the properties of the rectangular W-algebras with finite c for the bosonic cases. We explain the spin contents of the W-algebras and write down the central charges c and the levels of the so(M ) or sp(2m) currents. We then study the OPEs among generators for the simple examples with generators of spin up to two. We further conjecture that the rectangular W-algebras can be realized as the symmetry algebras of the cosets listed in table 2 and provide some supports. In section 4, we extend the analysis by introducing N = 1 supersymmetry. In appendix A, we provide the derivations of the central charges and the levels of so(M ) or sp(2m) currents for the rectangular W-algebras in the both cases with and without supersymmetry. In appendix B, we propose alternative coset descriptions of the rectangular W-(super)algebras by swapping the dual coset models in table 2 and provide several computations that support our proposal.

Higher spin (super)algebras
Three dimensional higher spin (super)gravity can be described by Chern-Simons gauge fields A,Ā. Here A andĀ take values in a higher rank gauge (super)algebra g. For g = sl(2), the Chern-Simons gauge theory describes the pure gravity on AdS 3 [31,32]. For generic g, we should specify the gravitational sl(2) subsector. In this section, we introduce the restricted matrix extensions of hs[λ] and its N = 1 supersymmetric extensions introduced in [4]. We then argue that the (super)algebras can be truncated when λ takes an integer number just as the algebra hs M [λ] can be reduced to sl(M n) at λ = n (n = 2, 3, . . .) [14]. Using the decomposition similar to (1.2), we identify an sl(2) subalgebra as the gravitational sector, see table 1. We examine the bosonic W-algebras in the next subsection and then generalize the analysis to the N = 1 W-algebras in subsection 2.2.

Restricted higher spin algebras
Without any matrix extensions and restrictions, the gauge algebra for the bosonic truncation of Prokushkin-Vasiliev theory of [4] is given by hs [λ]. In order to introduce the higher spin algebra, we define B[λ] by the universal enveloping algebra of sl(2) with removing an ideal as Here C 2 is the Casimir of sl (2). From this decomposition, we can see that the algebra can be truncated to sl(M n) as sl(M n) sl(M ) ⊗ 1 n ⊕ 1 M ⊗ sl(n) ⊕ sl(M ) ⊗ sl(n) (2.5) at λ = n. We use the principally embedded sl(2) in 1 M ⊗ hs[λ] or 1 M ⊗ sl(n) as the gravitational sector.
We can further generalize the gauge algebras by putting restrictions on the extra matrix degrees of freedom, and it is known that there are two ways to do so [4], see also [6]. For one type, we decompose the M × M matrix A M as : Here Ω 2m is a symplectic form expressed by a 2m × 2m matrix. We can decouple the trace part as A Ω,s where the generators of hs[λ] with λ = 2n + 1 could be given by traceless (2n + 1) × (2n + 1) matrices. Thus, at λ = 2n + 1, we have the truncated algebras

11)
A Ω,a (2.12) The first algebra includes so(M ) and so(2n + 1) as subalgebras, while the second one includes sp(2m) and so(2n + 1) as subalgebras. The gravitational sl (2) is identified with the one principally embedded in the so(2n + 1

13)
A Ω,a 14) The first algebra includes so(M ) and sp(2n) as subalgebras, and the second one includes sp(2m) and sp(2n) as subalgebras. The gravitational sl (2) is identified with the one principally embedded in the sp(2n).
The expressions of restricted algebras in (2.11), (2.12), (2.13) and (2.14) look complicated. However, as seen below, they can be regarded as decompositions of Lie algebras so(M L) and sp(2mL) with L = 2n or L = 2n + 1, see table 1. We first consider so(M L) and express the generators by T M L , which satisfy We then choose to express T M L by tensor products A M ⊗ B L . With this expression, the condition (2.15) can be realized by Here we have assumed that M L is even. As above, T M L are expressed by tensor products A M ⊗ B L and the condition (2.18) can be realized by and with = ±1 and η = ±1. Here η = −1 is possible if M = 2m, and the case corresponds to (2.12). Similarly, η = +1 is possible if L = 2n, and the case corresponds to (2.13).

Restricted higher spin superalgebras
The gauge algebra for the N = 2 Prokushkin-Vasiliev theory of [4] is given by shs[λ] without matrix extensions and restrictions. As in the case of hs[λ], we introduce sB[λ] by the universal enveloping algebra of osp(1|2) divided by an ideal as Here C 2 is the Casimir operator of osp(1|2). Decoupling the identity element 1, we define a higher spin superalgebra shs[λ] as [33] sB[λ] = shs[λ] ⊕ 1 . (2.22) The higher spin superalgebra satisfies a relation shs[λ] = shs [1 − λ]. At λ = −n, the superalgebra can be truncated to sl(n + 1|n). Multiplying the M × M matrix algebra, we define shs M [λ] by (see [3,12]) (2.23) The superalgebra can be decomposed as which can be truncated to at λ = −n. We can describe N = 1 supergravity on AdS 3 by osp(1|2) Chern-Simons gauge theory [31], and it is important to identify the supergravity sector in order to define a higher spin supergravity. In the current case, we use the principal embedding of osp(1|2) The superalgebra shs[λ] has a symmetry under a Z 2 action σ as closely explained in [25], and the Z 2 truncation of shs[λ] is denoted as shs σ [λ]. There are the sectors of integer and half-integer spins, and the truncated algebra includes only even spin generators for the integer spin sector. As explained in [25], shs σ [λ] can be truncated to osp(2n + 1|2n) at λ = −2n and osp(2n − 1|2n) at λ = −2n + 1 (n = 1, 2, . . .). Just as in the bosonic case, there are two types of truncation for the matrix extension as We would like to discuss the truncation of extended algebras at λ = −2n or λ = −2n + 1, where shs σ [λ] reduces osp(2n + 1|2n) or osp(2n − 1|2n). For this, we start by reviewing some properties of supermatrices and Lie superalgebra osp(m|2n), see, e.g., [34]. We express the generators of osp(m|2n) by even supermatrices of the form where A m , B m,2n , C 2n,m , D 2n are m × m, m × 2n, 2n × m, 2n × 2n matrices, respectively. We define the supertrace and the supertranspose as . (2.29) The condition for the generators of osp(m|2n) is then given by which can be rewritten as From this expression, we can see that A m and D 2n generates so(m) and sp(2n), respectively. We express a supermatrix satisfying the condition of osp(m|2n) by B σ m|2n . We then schematically decompose a supermatrix B m|2n as With this terminology, the expressions in (2.26) can be truncated to

Rectangular W-algebras
In subsection 2.1, we constructed four types of restricted matrix extensions for the gauge algebra and the gravitational sl(2) sectors are specified, see table 1. In order to construct a Chern-Simons gravity theory, we need to assign an asymptotic AdS condition. We choose the asymptotic AdS condition such that Here ρ is the radial coordinate of AdS 3 and the AdS boundary is located at ρ → ∞. Moreover, A AdS denotes the configuration of gauge fields describing the AdS background. With this setup, the asymptotic symmetry is given by the Hamiltonian reduction of the gauge algebra with the sl(2) embedding [18,19]. In this way, we now have four types of rectangular W-algebras, and we may call them as types so(M (2n + 1)), sp(2M n), sp(2m(2n + 1)), so(4mn). In the next subsection, we explain the basic properties of the W-algebras including spin contents, central charges, and the levels of affine so(M ) or sp(2m) algebra. In subsection 3.2, we compute OPEs for simple examples involving only generators up to spin two. In subsection 3.3, we propose the dual cosets whose symmetries are the same W-algebras as listed in table 2 and provide the maps of parameters. In subsection 3.4, we explicitly construct the generators of the symmetry algebras up to spin two and compare the OPEs obtained in subsection 3.2.

Basic properties of the algebras
In this subsection, we examine the basic properties of the rectangular W-algebras obtained by quantum Hamiltonian reductions of the gauge algebras listed in table 1. We first obtain the spin contents of the W-algebras and then write down the central charges c of the algebras and the levels of the so(M ) or sp(2m) currents. We explain the derivations of c and in appendix A.1.
Type so(M (2n + 1)) We set the gauge algebra as g = so(M (2n + 1)) and decompose it as (see (2.11)) Here (sym) L denotes the symmetric representation of so(L), which are generated by the matrices A s L . We use the principally embedded sl(2) in 1 M ⊗so(2n+1) as the gravitational sl (2). With the identification of gravitational sector, spin s for the embedded sl(2) can be directly related to the space-time spin. We denote the elements of sl(2) by the triplet With the sl(2) action, we can decompose so(2n + 1) and (sym) 2n+1 in (3.2) as where ρ s denotes the spin s representation of sl (2). Using the dimensions of so(M ) and the gauge algebra so(M (2n + 1)) can be decomposed as We can see that the modes with positive eigenvalues ofx diverge for ρ → ∞, thus the asymptotic AdS condition (3.1) requires that the positive modes vanish. Moreover, we can set the other modes to be zero as well except for one element in each ρ s . Since the gauge fields A = A µ dx µ have already one space-time index, the element has total space-time spin s + 1 for the one from ρ s . Therefore, the algebra includes M (M − 1)/2 odd spin currents with s = 1, 3, . . . , 2n + 1 and M (M + 1)/2 even spin currents with s = 2, 4, . . . , 2n after the Hamiltonian reduction with the sl(2) embedding. If we set M = 1, then only even spin currents with s = 2, 4, . . . , 2n are left. The spin content reproduces the one in [22] without matrix extension.
The M (M −1)/2 spin one currents form so(M ) affine algebra, and the spin two current in the singlet of so(M ) is the energy-momentum tensor. In appendix A.1, the level of so(M ) currents is computed as and the central charge is obtained as Here t is the level of so(M (2n + 1)) affine algebra. There could be several constructions of the same W-algebra, and the label t is not an intrinsic parameter. We can obtain the relation between c and as by removing the parameter t.
Type sp(2M n) The gauge algebra is set as g = sp(2M n) and the decomposition is (see (2.13)) Here (asym) 2n denotes the anti-symmetric representation of sp(2n) generated by matrices A Ω,s 2n . As the gravitational sector, we use the principally embedded sl(2) in 1 M ⊗ sp(2n).
With the sl(2), we decompose sp(2M n) as Here we have used the decompositions of sp(2n) and (asym) 2n in (3.10) as as well as ( 13) and the central charge is obtained as (3.14) Here the level of sp(2M n) affine algebra is denoted as t. The central charge can be written in terms of as by removing t.
There appears no analogous algebra in [22] since the matrix extension is essential in this case.

OPEs among generators
In the previous subsection, we have obtained the spin contents of the four types of Walgebras. In order to determine the algebras, we need the information of OPEs among generators as well. In this subsection, we compute the OPEs by requiring their associativity for the simplest non-trivial examples with spin one and two generators as was done in [14]. For types so(M (2n + 1)) and sp(2m(2n + 1)), the minimal cases are with n = 1, where the maximal spin is three. Therefore, we focus on the types sp(2M n) and so(4mn) with n = 1. are the anti-symmetric and traceless symmetric matrices, respectively. We introduce the invariant tensors by and use the convention g AB = δ AB . With this notation of indices, we use J a for the spin one currents, T for the energy-momentum tensor, and Q α for the charged spin two currents. We would like to obtain OPEs among J a , T , and Q α . For the spin one currents J a , we use the OPEs as Here we have set κ ab = 1 2 g ab for the normalization of level to be the conventional one. OPEs involving T are Here c is the central charge of the algebra. As in (3.15), there would be a relation between and c, but we do not require the relation for a while. The currents Q α have spin two and transform as in the symmetric representation of so(M ). Using (3.27), we require the OPEs Non-trivial OPEs left are Q α (z)Q β (0), and we obtain them by requiring associativity of the OPEs. The operator products Q α (z)Q β (0) would generate (composite) operators up to spin three, so we classify possible operators primary w.r.t. the Virasoro generator T . In order to make composite operators well defined, we adopt the normal ordering prescription (3.31) Spin one primary operators are J a . For spin two, there are along with Q α . Here the brackets (a 1 , a 2 , . . . , a p ) and [a 1 , a 2 , . . . , a p ] mean the symmetric and anti-symmetric indices, respectively, with the factor 1/(p!). For spin three, composite primary operators are (3.33) We expand the operator products Q α (z)Q β (0) in terms of these (composite) primary operators with several indices. In order to express the coefficients, we need invariant tensors consisting of traces of sl(M ) generators as classified in [14]. With two and three indices, we use the tensors in (3.27). The invariant tensors with four indices are In addition, we also need invariant tensors with five indices as With the preparations, the ansatz for the OPEs of Q α (z)Q β (0) can be written as Requiring associativity of the OPEs, we can uniquely fix the coefficients of our ansatz in terms of as 3 , , , c 74 = 0 up to an overall normalization ± √ c 1 for the spin two currents Q α . The associativity also fixes the central charge as which reproduces (3.15) with n = 1.
Type so(4mn) with n = 1 We then move to the W-algebra with sp(2m) symmetry obtained as the Hamiltonian reduction of so(4m). The spin one currents generate sp(2m) affine algebra. A spin two current is the energy-momentum tensor, and the others are in the anti-symmetric representation of sp(2m). In order to express the adjoint and anti-symmetric representations simultaneously, we again use the sl(2m) generators t A 2m expressed by 2m × 2m matrices. In this case, we decompose the indices as A = (a, α) with a = 1, 2 . . . , 2m 2 + m and α = 2m 2 + m + 1, . . . , 4m 2 − 1 such that We introduce the invariant tensors as in (3.27) with M = 2m. In particular, we raise or lower indices by g AB or its inverse. With these indices, the spin one and two currents are denoted as J a , T , and Q α . For OPEs among J a and T , we use (3.28) and (3.29). Here we set κ ab = g ab for the convention of to be as usual. Again, we do not specify the relation between and c for a while. We set Q α such that the OPEs (3.30) are satisfied. For the ansatz of the OPEs of Q α (z)Q β (0), we use the same form as in (3.36) with the same expressions of composite operators (3.32), (3.33). Notice that the explicit expressions differ since the invariant tensors (3.27), (3.34), (3.35) are constructed from the different set of generators ). Requiring associativity of the OPEs, we can uniquely fix the coefficients of our ansatz as 4 , c 53 = − c 1 6 (2 + 1)(2 + m + 2) , , c 74 = 0 up to the overall normalization ± √ c 1 of Q α . The central charge is also determined as which is the same as the expression of (3.26) with n = 1.

Structure of null vectors
In subsection 2. . In a coset model similar to ours, these two types of truncation were discussed in subsection 3.4 of [36], see also [12,26,37].

Comparison with dual coset models
As in [14], we would like to relate the W-algebras with so(M ) or sp(2m) symmetry to the symmetry algebras of some coset models. The corresponding coset models should be the holographic duals of the restricted matrix extensions of 3d Prokushkin-Vasiliev theory. The symmetry algebras become quite simple at the limit when the parameter λ in the gauge algebras becomes λ → 0 or λ → 1. At the limit, we can realize the rectangular W-algebras as orbifolds of free bosons and fermions [38,39], and we can easily obtain the cosets which reduce to the same free systems at the large level limit [40,41], see also [3,6]. Our proposal on the dual coset CFTs can be found in table 2. 5 A difficult issue is on the maps of parameters including quantum corrections. In this subsection, we achieve this by making use of the levels of so(M ) or sp(2m) affine symmetries and the central charges c of the algebras.
Type so(M (2n + 1)) We start from the W-algebra with so(M ) symmetry realized as the Hamiltonian reduction of so(M (2n + 1)). As a natural guess, we propose that the corresponding coset is given by The coset has the so(M ) affine symmetry and the level is k. The central charge is computed as where the dual Coxeter number of so(L) is given by h ∨ = L − 2. At the large k limit, the coset reduces to M N real bosons with so(N ) invariance in addition to the so(M ) currents. Compared with the level (3.7) and the central charge (3.8), the correspondence is realized with = k and λ = 2n + 1, where the 't Hooft parameter is given by The existence of two 't Hooft parameters implies a duality relation of the coset (3.42). The same kind of duality exists in the W-algebra with su(M ) symmetry, and it was analyzed in [14] to some extent.
Type sp(2M n) We then consider the W-algebra with so(M ) symmetry obtained from the Hamiltonian reduction of sp(2M n). We propose that the algebra can be realized by the coset The symmetry algebra of the coset includes so(M ) affine Lie algebra with level −2k and moreover it has m(m + 1)/2 generators at conformal weights 2, 4, . . . , 2N and m(m − 1)/2 generators at conformal weights 1, 3, . . . , 2N + 1 by Theorem 3.15 of [42] together with the theory of cosets of [38,39]. The central charge of the coset is There are two correspondences and this implies a duality of the coset (3.45).
Type sp(2m(2n + 1)) We move to the W-algebra with sp(2m) symmetry realized as the Hamiltonian reduction of sp(2m(2n + 1)). We propose that the corresponding coset is The level of the sp(2m) affine algebra is −k/2, and the central charge of the coset is For the dual Coxeter number of osp(N |2m), we have used h ∨ = N − 2m − 2 in this case. At the large k limit, the coset reduces to 2mN Majorana fermions with so(N ) invariance in addition to the sp(2m) currents. Compared with the expressions (3.19) and (3.20), we find that the correspondence happens at = −k/2, λ = 2n + 1 with Again, the possibility of two choices implies a duality of the coset (3.48).
Type so(4mn) Finally, we examine the W-algebra with sp(2m) symmetry obtained by the Hamiltonian reduction of so(4mn). The symmetry algebra is proposed to be realized by The coset includes sp(2m) affine symmetry with level k. The central charge is computed as At the large level limit, the coset reduces to 4mN real bosons with sp(2N ) invariance as well as the sp(2m) currents. Compared with (3.24) and (3.25), we find that the correspondence is realized at = k, λ = 2n with There should be a duality of the coset (3.51) as in the other cases.

Symmetry of dual coset models
In subsection 3.2, we have obtained OPEs for two simple examples with generators of spin up to two. In this subsection, we construct the generators of the restricted rectangular Walgebras in terms of dual cosets and check that the OPEs obtained above can be reproduced.
Type sp(2M n) with n = 1 We have proposed above that the W-algebra with so(M ) symmetry obtained from sp(2M ) can be realized as the symmetry algebra of (3. and require the conditions see (2.30) and (2.31). As generators, we use Here t a M and t p 2N are so(M ) and sp(2N ) generators, respectively. For t i M,2N we use a M × 2N matrix where only one element is non-zero and equal to one. We also need where t α M are generators for the symmetric representation of so(M ). Denoting and the invariant tensors Here str is defined in (2.29). For t a M , t α M , we use the same generators as in subsection 3.2. In particular, the normalizations are such that g ab = δ ab and g αβ = δ αβ . The osp(M |2N ) currents include bosonic ones J a , J p and fermionic onesJ i . The non-trivial OPEs are (3.60) With these preparations, we write down the generators of the W-algebra with so(M ) symmetry up to spin two. For the spin one generators, we have J a with level −2k. There are spin two generators, such as, the energy-momentum tensor T and the charged spin two currents. The energy-momentum tensor is obtained by the standard coset construction [44]. The charged spin two generators are given by linear combinations aŝ Here the relative coefficients are fixed so as to satisfy the OPEs (3.30). Using the OPEs in (3.60), we can explicitly evaluate the OPEs ofQ α (z)Q β (0). In fact, we find 6 with extra spin three currents P α at k = −2(N + 1). Since we can show P α (z)P β (0) ∼ O(z −5 ), the new operators P α are regarded as null vectors. Therefore, we can decouple P α from the rest by setting P α = 0 and identify the OPEs ofQ α and Q α by (3.62). 7 Type so(4mn) with n = 1 We consider the W-algebra with sp(2m) symmetry obtained from so(4m). In the previous subsection, we have proposed that the algebra can be realized as the symmetry algebra of the coset (3.51) at k = −2(N + 1) (or k = −2/3(N + m + 1)). In order to express the sp(2m + 2N ) currents, we decompose the generators of sp(2m + 2N ) in the following way. We use (2m + 2N ) × (2m + 2N ) matrices and realize the condition of sp(2m + 2N ) as Here Ω 2m and Ω 2N are symplectic forms satisfying For generators, we use where t a 2m and t p 2N are the generators of sp(2m) and sp(2N ) subalgebras, respectively. Moreover, each t i 2m,2N has only one non-zero element, which is equal to one. We also need where t α 2m are generators for the anti-symmetric representation of sp(2m). With t P 2m+2N = (t a 2m+2N , t α 2m+2N , t p 2m+2N , t i 2m+2N ), we introduce the metric as in (3.58) and the invariant tensors as in (3.59). We use the same expressions for t a 2m , t α 2m as in subsection 3.2 in order to make comparison easier. The OPEs of sp(2m + 2N ) currents are given by with J X = (J a , J p , J i ).
We construct the generators of the rectangular W-algebra up to spin two in the coset language. Spin one generators are J a with level k. As a spin two current, the energymomentum tensor T can be constructed in the standard way [44]. The other spin two currents are given byQ which satisfy the OPEs (3.30). Using (3.68), the OPEs ofQ α (z)Q β (0) can be evaluated as (3.62) with extra spin three currents P α at k = −2(N + 1). 8 We can show that P α have zero norm as P α (z)P β (0) ∼ O(z −5 ). Thus, we decouple P α and identify the OPEs ofQ α and Q α by (3.62).

N = 1 rectangular W-algebras
The gauge algebras for Chern-Simons supergravities have been analyzed in subsection 2.2, and it was found that there are four types of restricted matrix extensions, see table 1. We assign the asymptotic AdS boundary condition as in (3.1) for higher spin supergravity theories. Under this condition, the asymptotic symmetries are again given by the Hamiltonian reductions of superalgebras with the osp(1|2) embeddings [9,45,46]. In this way, we obtain four types of N = 1 rectangular W-algebras. In the next subsection, we explain their basic properties, such as, spin contents, central charges, and the levels of affine symmetries. In subsection 4.2, we show that the associativity gives strong constraints on OPEs for simple examples with generators of spin up to two. In subsection 4.3, we argue that the N = 1 W-algebras could be realized as the symmetry algebras of coset models listed in 2. In subsection 4.4, we explicitly construct generators up to spin two in terms of the cosets and check that the OPEs can be reproduced for several examples.

Basics properties of the superalgebras
In this subsection, we study the basic properties of the four types of N = 1 W-algebras obtained by the Hamiltonian reductions of the superalgebras listed in table 1. Specifically, we examine the spin contents of these superalgebras and write down the levels 1 , 2 of two so(M ) or sp(2m) affine subalgebras and the central charges c of the algebras, see appendix A.2 for the details of computations.
Type osp(M (2n + 1)|2M n) We start from the case with g = osp(M (2n + 1)|2M n). We decompose the superalgebra  The M (M − 1) spin one currents generate two so(M ) affine algebras and one of spin two currents is the energy-momentum tensor. In appendix A.2, we obtain the levels of two so(M ) affine algebras as and the total central charge as Here t is the level of osp(M (2n + 1)|2M n) affine algebra.  where t is the level of osp(M (2n − 1)|2M n) affine algebra.
The 2(2m 2 + m) spin one currents generate two sp(2m) affine algebras and one of spin two currents is the energy-momentum tensor. In appendix A.2, we find the levels of two sp(2m) affine algebras as (4.11) and the total central charge as where t is the level of osp(4mn|2m(2n − 1)) affine algebra.

OPEs among generators
In this subsection, we compute OPEs among generators of the N = 1 rectangular Walgebras only with generators of spins s = 1, 3/2, 2. 9 There are two types of them; one is type osp(M (2n − 1)|2M n) with n = 1, which has so(M ) symmetry. The other is type osp(4mn|2m(2n − 1)) with n = 1, which has sp(2m) symmetry. We first study the N = 1 W-algebra with so(M ) symmetry, and then briefly discuss the one with sp(2m) symmetry. Since expressions become quite complicated compared with the bosonic cases, we only provide the outlines of what we have done.  (3.27) and the normalization of generators is set as g AB = δ AB . The W-algebra includes two so(M ) currents J a , K a . With κ ab = 1 2 tr(t a M t b M ), the OPEs of J a , K a can be written as The W-algebra also includes the energy-momentum tensor T satisfying (4.14) We do not specify the relations among the levels 1 , 2 and the central charge c for a moment. The OPEs between the energy-momentum tensor and the so(M ) currents are The algebra includes fermionic generators G A with spin 3/2. There are three types, such as, G = G 0 in the singlet, G a in the adjoint, and G α in the symmetric representations of so(M ). For spin two generators, we have the energy-momentum tensor T and charged spin two currents Q α transforming as the symmetric representation under the so(M ) action. We choose the basis of generators to satisfy such that the currents are primary w.r.t. the Virasoro algebra. We obtain other OPEs among generators by requiring their associativity. For similar analysis without matrix extensions but with supersymmetry, see [16,25,36]. The operator products would produce composite operators of spin up to three, and we need to list all independent ones. In order to avoid making expressions complicated, we use abbreviated notation for composite operators as where . . . denote terms needed to satisfy the condition primary w.r.t. the Virasoro algebra.
Here we use the normal ordering prescription given in (3.31) to make the products of operators well defined. The definitions of composite operators are not unique in general and this fact will be utilized to make the form of OPEs simpler below. For spin one and 3/2, there are no primary operators other than the fundamental ones, J a , K a , and G A . We have primary operators for spin two and 5/2 along with Q a . We also find the composite primaries for spin three.
In the following, we consider the specific examples with M = 3, 4, 5 except for the OPEs of Q a × Q b . Here we have used the symbolic form to express operator products. The operator product of spin s 1 and s 2 operators produces (composite) generators of spin up to s 1 + s 2 − 1, so it is convenient to start from the cases with smaller s 1 + s 2 . The smallest case is with s 1 = s 2 = 1, but the OPEs were already given in (4.13). The next simplest case is with s 1 = 1 and s 2 = 3/2. Denoting I P = (J a , K a ), we examine the associativity of I P × I Q × G A . Up to the normalizations of G A , the associativity turns out to fix the OPEs of I P × G A uniquely. Firstly, we set (4.20) where the OPEs determine the normalization of G a relative to G. The OPEs of I P × G a are given by (4.21) Here we have fixed the normalization of G α relative to G by the OPEs. With the normalizations, the OPEs of

(4.22)
In the above analysis, we used the fact that only the non-trivial invariant tensors are among f ABC and d ABC . We next examine the OPEs of I P × Q α . We use the ansatz schematically of the forms We examine the associativity of I P × I Q × Q α and find that there might be several discrete solutions of a 1 and b 1 . The charge assignments of Q α w.r.t. J a and K a can be read off from the Hamiltonian reduction of osp(M |2M ), and the possible choice is a 1 = 0 and b 1 = 0 or a 1 = 0 and b 1 = 0. We use the former one by breaking the symmetry under the exchange of J a and K a . Moreover, there are ambiguities by redefining the spin two currents Q α such that as mentioned above. Using z 1 , z 2 , z 3 , we set a 21 = a 22 = b 22 = 0 to make the OPEs simpler. With this setup, solutions to the constraints from I P × I Q × Q α are given by In a similar manner, we can determine the OPEs of G A × G B and G A × Q α in terms of one parameter 1 up to the normalizations of G and Q α (or z 0 in (4.25)). Here we have used the associativity of I P × G A × G B , I P × G A × Q α , and G A × G B × G B . In particular, we have found the expressions of c and 2 in terms of 1 for M = 3, 4, 5 as which can be obtained from (4.5) and (4.6) as well. We have also determined the OPEs of Q α × Q β in terms of one parameter 1 up to the normalizations of G and Q α for the simplest non-trivial example with M = 3.
Type osp(4mn|2m(2n − 1)) with n = 1 We then consider the N = 1 W-algebra with sp(2m) symmetry obtained from the Hamiltonian reduction of osp(4m|2m). The situation is quite similar to the previous case, so we explain it only briefly. As above, we use t A 2m as the generators of sl(2m) and express them by 2m × 2m matrices. We decompose the indices as A = (a, α) with a = 1, 2 . . . , 2m 2 + m and α = 2m 2 + m + 1, . . . , 4m 2 − 1 such that (3.39) are satisfied. The invariant tensors are introduced as in (3.27) with M = 2m, and indices are raised or lowered by g AB or its inverse. There are two sp(2m) currents J a , K a and their levels are denoted as 1 , 2 , respectively, as in (4.13) but with κ ab = g ab . There are fermionic generators G A with spin 3/2, which are G = G 0 in the singlet, G a in the adjoint, and G α in the anti-symmetric representations of sp(2m). For spin two generators, we have the energy-momentum tensor T and charged spin two currents Q α transforming as the anti-symmetric representation under the sp(2m) action. We choose the basis primary w.r.t. the Virasoro algebra.
We determine OPEs by requiring their associativity. Here we mainly consider the examples with sp(4) and sp (6). For the OPEs of the form I P × G A with I P = (J a , K a ), we find along with (4.20) and (4.22). The composite operators generated by operator products are the same as before as in (4.18) and (4.19). We can apply the same arguments as before for the OPEs of I P × Q α , and we arrive at the same expressions as in (4.26). The OPEs of G A × G B and G A × Q α can be determined in terms of one parameter, say, 1 up to the overall normalizations of G and Q α . The central charge c and the other level 2 can be written in terms of 1 as c = − m 4 1 m + 1 (6 1 + 11) + 2m 2 + 5m + 2 with m = 2, 3. The same expressions can be obtained from (4.11) and (4.12). We checked that the OPEs of Q α × Q β are uniquely fixed in terms of 1 up to the normalizations of G and Q α for the example of sp(4).

Comparison with dual coset models
In this subsection, we propose cosets realizing the four types of N = 1 rectangular Walgebras as their symmetry algebras as listed in table 2 and identify the maps of parameters. 10 The cosets can be regarded as the holographic duals of the N = 1 matrix extensions of higher spin supergravities. The dual cosets were proposed in [6] by generalizing the holographic duality in [3] with N = 2 supersymmetry. However, the analysis was done in large c limit and with generic 't Hooft parameters. We are interested in the cases with finite c and integer λ, and, in particular, the truncations depend on whether λ is even or odd. For this, we work with the four cases contrary to the two cases analyzed in [6].
Type osp(M (2n + 1)|2M n) We first consider the N = 1 W-algebra with so(M ) symmetry obtained as the Hamiltonian reduction of osp(M (2n+1)|2M n). We propose that the algebra is realized as the symmetry of the coset where so(N M ) 1 can be described by N M Majorana fermions. The same coset was considered in [6] and also in [25] with M = 1. The symmetry algebra includes two sets of so(M ) currents. One comes from so(N + M ) and with level k. The other can be constructed from the free fermions and with level N . The central charge is computed as Compared with (4.2) and (4.3), we find a map The two choices may be understood as a duality of the coset (4.30) by exchanging two sets of so(M ) currents.
Type osp(M (2n − 1)|2M n) We then examine the N = 1 W-algebra with so(M ) symmetry given by the Hamiltonian reduction of osp(M (2n−1)|2M n). We propose that the algebra is realized as the symmetry of the coset where sp(2N M ) −1/2 is described by 2N M symplectic bosons. For M = 1, the coset reduces to the one proposed in [25]. The symmetry algebra includes affine so(M ) with level −2k from osp(M |2N ) k and affine so(M ) with level −2N from the symplectic bosons. The central charge is The comparison with (4.5) and (4.6) leads to a map The existence of two choices implies a duality of the coset (4.34).
Type osp(4mn|2m(2n + 1)) One of the N = 1 algebras with sp(2m) symmetry can be constructed by the Hamiltonian reduction of osp(4mn|2m(2n + 1)). We propose that the algebra can be identified as the symmetry of the coset The symmetry algebra includes two affine sp(2m) with level −k/2 and level −N/2 as subalgebras. The central charge is Compared with (4.8) and (4.9), we find a map The two choices imply a duality of the coset (4.38) as above.
Type osp(4mn|2m(2n − 1)) The other N = 1 algebra with sp(2m) symmetry is constructed by the Hamiltonian reduction of osp(4mn|2m(2n − 1)). Our proposal is that the algebra is realized as the symmetry of the coset The same coset was considered in [6]. The symmetry algebra includes two affine sp(2m) with level k and level N as subalgebras. The central charge is We compare them with (4.11), (4.12) and find that the correspondence happens at We again expect a duality of the coset (4.42)

Symmetry of dual coset models
In subsection 4.2, we obtained OPEs for the N = 1 W-algebras with generators up to spin two. In this subsection, we explicitly construct generators of the W-algebras in terms of dual coset models and reproduce the OPEs among generators. (4.46) Using the symplectic bosons, we can construct so(M ) −2N and sp(2N ) −M/2 currents as In particular, the sp(2N ) k−M/2 factor in the coset (4.34) is given bŷ In terms of these currents, we write down generators of the N = 1 W-algebra with so(M ) symmetry up to spin two. For spin one generators, we have J a and K a = J a f with levels −2k and −2N , respectively. For spin 3/2 generators, we use products ofJ j and ϕ i such as to be regular w.r.t.Ĵ p . Using suitable invariant tensors, we have where the overall factors are chosen in order to match with the OPEs (4.20), (4.21), and (4.22). There are spin two generators, such as, the energy-momentum tensor T and charged spin two currents Q α . The energy-momentum tensor can be obtained by the standard coset construction [44]. For the charged spin two generators, we find that are Virasoro primaries and satisfy the OPEs (4.26). They are actually the same as those for the bosonic case in (3.61), and this is because we use the definition of spin two currents such as to satisfy the same OPEs. Setting as in (4.36) with n = 1, we have checked for several examples that other OPEs among generators are reproduced up to null vectors.
Type osp(4mn|2m(2n − 1)) with n = 1 We then examine the N = 1 W-algebra with sp(2m) symmetry obtained from osp(4m|2m), which is supposed to be realized as the symmetry of the coset (4.42). In order to describe the currents in the coset (4.42), we use the generators of sp(2m + 2N ) as in (3.66), (3.67) and the invariant tensors as in (3.58), (3.59). With the notation, the OPEs of sp(2m + 2N ) currents and the free fermions ψ i from so(4mN ) 1 are given by (3.68) and (4.52) We can construct sp(2m) N and sp(2N ) m currents from the free fermions as The sp(2N ) k+m currents in the coset (4.42) are given bŷ Next we construct generators of the N = 1 W-algebra up to spin two in the coset language. Spin one generators are J a and K a = J a f . Spin 3/2 generators are constructed as as in the case with so(M ). As a spin two current, the energy-momentum tensor T can be constructed in the standard way [44]. The other spin two currents are given by

A Computations on central charges and levels
In this appendix, we write down some details of computations on the central charges of rectangular W-algebras and the levels of so(M ) or sp(2m) affine subalgebras. Let us consider a vertex algebra W t (g,x,f ). Here g denotes a Lie (super)algebra with a suitable norm (.|.). The label t is the level of the universal affine vertex algebra V t (g), which is used to construct W t (g,x,f ). Furthermore,x,f are even elements of g, and we require that they form a sl(2) algebra (3.3) with an additional elementê. We decompose g by the eigenvalue of adjoint action adx as We choose a basis {u α } α∈S j for g j and denote S + = j>0 S j . The formula of central charge for the vertex algebra W t (g,x,f ) may be found in (2.3) of [47] as Here h ∨ represents the dual Coxeter number of g and p(α) denotes the parity of u α . Moreover, (m α , 1−m α ) are the conformal dimensions of ghost system and we set m α = 1−j.
There is no universal formula for the levels of affine subalgebras, but they can be evaluated for each example as below.

A.1 Rectangular W-algebras
As explained in subsection 3.1, there are four types of bosonic rectangular W-algebras obtained from the Hamiltonian reductions of the gauge algebras listed in table 1. Here we compute the central charges of the algebras by applying the formula (A.2) and moreover obtain the levels of so(M ) or sp(2m) affine subalgebras.
The central charge of the algebra can be computed by applying the formula (A.2). The dimension of so(M (2n + 1)) is M (2n + 1)(M (2n + 1) − 1)/2 and the dual Coxeter number Type sp(2M n) We then examine the W-algebra with so(M ) symmetry obtained as the Hamiltonian reduction of sp(2M n). We decompose sp(2M n) as in (2.13) or (3.10) and use the sl (2) principally embedded in 1 M ⊗ sp(2n).
We compute the central charge from (A.2). The dimension of sp(2M n) is M n(2M n+1) and the dual Coxeter number is M n + 1. Using the convention (ŷ|ẑ) = tr(ŷẑ), we find (x|x) = 2M n(4n 2 − 1)/12. We decompose sp(2n) by the principally embedded sl (2)  The total central charge is then evaluated as in (3.14). The level of so(M ) affine algebra can be obtained as in the case of (3.7). The ghosts transform in the symmetric and adjoint representations of so(M ), and the numbers of sets of ghosts are 2n 2 and 2n(n − 1), respectively. From this, the level of so(M ) can be found as (3.13). The factor 2 in the first term of the right hand side comes from the difference of convention for the levels of so(L) and sp(2K).
The which lead to (4.2).

B Alternative proposals of dual coset models
In this appendix, we provide alternative proposals on the coset models dual to higher spin gravities with restricted matrix extensions as listed in table 3. Compared with the previous proposals in table 2, the cosets are simply swapped.

B.1 Comparison with dual coset models
We summarize the map of parameters for each dual coset model with a rectangular W-(super)algebra as its symmetry.  Table 3. Alternative proposals on the coset models whose symmetries are realized by the rectangular W-(super)algebras.
Type so(M (2n + 1)) As the first example, we study the W-algebra with so(M ) symmetry obtained from the Hamiltonian reduction of so(M (2n + 1)). We propose that the symmetry can be realized by the coset (3.45) along with (3.42). The central charge of the coset is (3.46) and the level of so(M ) affine symmetry is −2k. Compared with the central charge (3.8) and the level (3.7) for the rectangular W-algebra, we find that the correspondence happens at = −2k and λ = 2n + 1 with λ = k + 1 k + N + 1 , λ = − k + 1 k + N − M/2 + 1 . (B.1) Type sp(2M n) We move to the W-algebra with so(M ) symmetry obtained from the Hamiltonian reduction of sp(2nM ). The symmetry is proposed to be realized by the coset (3.42) along with (3.45). The central charge of the coset is (3.43) and the level of so(M ) affine symmetry is k. Compared with (3.14) and (3.13), the correspondence is realized with = k and λ = 2n with Type sp(2m(2n + 1)) We then consider the W-algebra with sp(2m) symmetry obtained from the Hamiltonian reduction of sp(2m(2n + 1)). The dual coset is proposed as (3.51) in addition to (3.48).
The central charge of the coset is (3.52) and the level of sp(2m) affine symmetry is k. Compared with (3.20) and (3.19), we find the map of parameters as = k and λ = 2n + 1 with Type so(4mn) As the final example of bosonic W-algebras, we examine the one with sp(2m) symmetry from the Hamiltonian reduction of so(4mn). We propose the dual coset as (3.48) along with (3.51). The central charge of the coset is (3.49) and the level of sp(2m) affine symmetry is −k/2. Comparison with (3.25) and (3.24) leads to = −k/2 and λ = 2n with Type osp(M (2n + 1)|2M n) As the fist example of N = 1 W-algebras, we study the one with so(M ) symmetry obtained from the Hamiltonian reduction of osp(M (2n + 1)|2M n). We propose the dual coset as Type osp(4mn|2m(2n + 1)) We then consider the N = 1 W-algebra with sp(2m) symmetry as the Hamiltonian reduction of osp(4mn|2m(2n + 1)). The symmetry is proposed to be given by the coset (4.42) as well as (4.38). The central charge of the coset is (4.43) and the levels of sp(2m) affine symmetries are k and N . Comparison with (4.9) and (4.8) leads to 1 = k , 2 = N , λ ≡ Type osp(4mn|2m(2n − 1)) As the final example, we examine the N = 1 W-algebra with sp(2m) symmetry obtained from the Hamiltonian reduction of osp(4mn|2m(2n − 1)). The symmetry is proposed to be given by the coset (4.38) as well as (4.42). The central charge of the coset is (4.39) and the levels of sp(2m) affine symmetries are −k/2 and −N/2. Compared with (4.12) and (4.11), we find a map as

B.2 Symmetry of dual coset models
We check the rectangular W-algebras with generators of spin up to two can be realized from the proposed cosets by explicitly constructing symmetry generators.
Type sp(2M n) with n = 1 We consider the W-algebra with so(M ) symmetry obtained from the Hamiltonian reduction of sp(2M ). We construct symmetry generators in the coset (3.42) and check that they generate the OPEs obtained in subsection 3. Here t i 2m,N has one non-zero element, which is equal to one. With t α 2m as the generators of anti-symmetric representation of sp(2m), we introduce