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.
Highlights
We extend the analysis on the asymptotic symmetry of the 3d higher spin gravity beyond the classical limit
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
In subsection 2.2, we examine the asymptotic symmetry of the higher spin gravity at the classical limit applying the method of [15,16,17,18]
Summary
The main conjecture of this work is that the coset (1.3) at level k is isomorphic to the simple rectangular W -algebra of sl(M n) at level −t where the levels are related via k = −tn + M n(n − 1), if nN n(N + M ). We verify that central charges of theories and levels of current algebras agree. We prove a uniqueness result of this type of W-algebra for the case n = 2, and various N, M. This means that a simple chiral algebra of this type with strong generators only in weight one and two is completely determined by the level of the current algebra and the central charge.
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