Rectangular W-algebras, extended higher spin gravity and dual coset CFTs

Abstract

We analyze the asymptotic symmetry of higher spin gravity with M × M matrix valued fields, which is given by rectangular W-algebras with su(M ) symmetry. The matrix valued extension is expected to be useful for the relation between higher spin gravity and string theory. With the truncation of spin as s = 2, 3, . . . , n, we evaluate the central charge c of the algebra and the level k of the affine currents with finite c, k. For the simplest case with n = 2, we obtain the operator product expansions among generators by requiring their associativity. We conjecture that the symmetry is the same as that of Grassmannian-like coset based on our proposal of higher spin holography. Comparing c, k from the both theories, we obtain the map of parameters. We explicitly construct low spin generators from the coset theory, and, in particular, we reproduce the operator product expansions of the rectangular W-algebra for n = 2. We interpret the map of parameters by decomposing the algebra in the coset description.

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

“…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 [10,30,11].…”

“…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 Type of W-algebra Dual coset model examine the OPEs among low spin generators with removing the restriction of n = 2.…”

“….) [14]. Using the decomposition similar to (1.2), we identify an sl(2) subalgebra as the gravitational sector, see table 1.…”

“…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, .…”

Rectangular W algebras and superalgebras and their representations

“…Based on the holography of [5], we claim that the N = 2 W-algebra is realized as the symmetry of the coset We check that the OPEs among generators can be reproduced from the coset (1.4) for several examples. 2 This paper is organized as follows. In the next section, we start by reviewing the results of [2].…”