Rectangular W algebras and superalgebras and their representations

Abstract

We study W-algebras obtained by quantum Hamiltonian reduction of sl(M n) associated to the sl(2) embedding of rectangular type. The algebra can be realized as the asymptotic symmetry of higher spin gravity with M × M matrix valued fields. In our previous work, we examined the basic properties of the W-algebra and claimed that the algebra can be realized as the symmetry of Grassmannianlike coset even with finite central charge based on a proposal of holography. In this paper, we extend the analysis in the following ways. Firstly, we compute the operator product expansions among low spin generators removing the restriction of n = 2. Secondly, we investigate the degenerate representations in several ways, and see the relations to the coset spectrum and the conical defect geometry of the higher spin gravity. For these analyses, we mainly set M = n = 2. Finally, we extend the previous analysis by introducing N = 2 supersymmetry. *

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].…”

“…(A. 26) With this description, the subalgebras sp(2m) ⊗ ½ 2n+1|2n and ½ 2m ⊗ osp(2n + 1|2n) are manifestly realized. We principally embed osp(1|2) into the ½ 2m ⊗ osp(2n + 1|2n) algebra.…”

On extensions of $$ \mathfrak{gl}\widehat{\left(\left.m\right|n\right)} $$ Kac-Moody algebras and Calabi-Yau singularities

“…It should be straightforward to find a proposal for all OPEs of W m|n×∞ by correctly recovering these (−1) # factors in the formulas of [48]. Algebras of type m|n = m|0 have been recently also studied in a different basis for example in [27,49,53,54] who used names rectangular W-algebras or W m+∞ . Furthermore, the algebra W 1+∞ is well-known to be isomorphic to the affine Yangian of gl(1) [16,28].…”

The matrix-extended $$ {\mathcal{W}}_{1+\infty } $$ algebra

“…It turns out that there is unique algebra (depending now on three parameters -the central charge c and the analogous parameter to λ as well as the matrix rank m) with this spin content. This algebra and its cousins were studied before in [22][23][24][25][26][27], where it was sometimes referred to as 'rectangular W-algebra'. Moreover, its geometric realizations are described in [28].…”

“…In this paper, we study a matrix-extended version of W 1+∞ [λ]. This algebra was studied previously in [23][24][25] and generalizations were considered in [26,27]. The purpose of this paper is to expose the structure of this algebra and to study its representation theory.…”