Fermionic Coset, Critical Level W^(2)_4-Algebra and Higher Spins

TL;DR

This paper explores the symmetries and algebraic structures of the fermionic coset limit of certain string models, revealing connections to higher spin algebras and topological BRST structures.

Contribution

It demonstrates the preservation of $ ext{W}^{(2)}_4$-algebra symmetry at critical level and maps the linear model to a higher spin theory via the Zhu functor.

Findings

Affine $ ext{pgl}(4|4)_0$ symmetry at critical level

Preservation of $ ext{W}^{(2)}_4$-algebra under perturbation

Identification of higher spin theory through Zhu functor

Abstract

The fermionic coset is a limit of the pure spinor formulation of the AdS5xS5 sigma model as well as a limit of a nonlinear topological A-model, introduced by Berkovits. We study the latter, especially its symmetries, and map them to higher spin algebras. We show the following. The linear A-model possesses affine $\AKMSA{pgl}{4}{4}_0$ symmetry at critical level and its $\AKMSA{psl}{4}{4}_0$ current-current perturbation is the nonlinear model. We find that the perturbation preserves $\mathcal{W}^{(2)}_4$-algebra symmetry at critical level. There is a topological algebra associated to $\AKMSA{pgl}{4}{4}_0$ with the properties that the perturbation is BRST-exact. Further, the BRST-cohomology contains world-sheet supersymmetric symplectic fermions and the non-trivial generators of the $\mathcal{W}^{(2)}_4$-algebra. The Zhu functor maps the linear model to a higher spin theory. We analyze its $\SLSA{psl}{4}{4}$ action and find finite dimensional short multiplets.