Infinite Conformal Algebras in Supersymmetric Theories on Four Manifolds

Abstract

We study a supersymmetric theory twisted on a Kähler four manifold $M=Σ_1 \times Σ_2 ,$ where $Σ_{1,2}$ are 2D Riemann surfaces. We demonstrate that it possesses a "left-moving" conformal stress tensor on $Σ_1$ ($Σ_2$) in a BRST cohomology, which generates the Virasoro algebra with the conventional commutation relations. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic $\c$ of the $Σ_2$ ($Σ_1$) surface. It is shown that this construction can be extended to include a realization of a Kac-Moody algebra in BRST cohomology with a level proportional to the Euler characteristic $\c .$ This structure is shown to be invariant under renormalization group. A representation of the algebra $W_{1+\infty}$ in terms of a free chiral supermultiplet is also given. We discuss the role of instantons and a possible relation between the dynamics of 4D Yang-Mills theories and those of 2D sigma models.