Abstract :
We study a supersymmetric theory twisted on a Kähler four-manifold M = Σ1 × Σ2, where Σ1,2 are two-dimensional Riemann surfaces. We demonstrate that it possesses a “left-moving” conformal stress tensor on Σ1 (Σ2) 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 χ of the Σ2 (Σ1) 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 χ. This structure is shown to be invariant under renormalization group. A representation of the algebra W1+∞ in terms of a free chiral supermultiplet is also given. We discuss the role of instantons and a possible relation between the dynamics of four-dimensional Yang-Mills theories and those of two-dimensional sigma models.
