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

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 $\text{AKMSA}pgl{44}_0$ symmetry at critical level and its $\text{AKMSA}psl{44}_0$ current-current perturbation is the nonlinear model. We find that the perturbation preserves ${\mathcal{W}}_4^{(2)}$-algebra symmetry at critical level. There is a topological algebra associated to $\text{AKMSA}pgl{44}_0$ 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 ${\mathcal{W}}_4^{(2)}$-algebra. The Zhu functor maps the linear model to a higher spin theory. We analyze its $\text{SLSA}psl44$ action and find finite dimensional short multiplets.