Realization of $W_{1+\infty}$ and Virasoro Algebras in Supersymmetric Theories on Four Manifolds

Abstract

We demonstrate that a supersymmetric theory twisted on a Kähler four manifold $M=Σ_1 \times Σ_2 ,$ where $Σ_{1,2}$ are 2D Riemann surfaces, possesses a "left-moving" conformal stress tensor on $Σ_1$ ($Σ_2$) in the BRST cohomology. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic of the $Σ_2$ ($Σ_1$) surface. This structure is shown to be invariant under renormalization group. We also give a representation of the algebra $W_{1+\infty}$ in terms of a free chiral supermultiplet.