Infinite Conformal Algebras in Supersymmetric Theories on Four Manifolds

TL;DR

This paper explores the emergence of infinite conformal algebras, including Virasoro and Kac-Moody, in supersymmetric theories on four-manifolds, revealing geometric origins of central charges and invariance under renormalization.

Contribution

It demonstrates the realization of Virasoro and Kac-Moody algebras in BRST cohomology for twisted supersymmetric theories on four-manifolds, with geometric central charges and algebraic structures.

Findings

Virasoro algebra with geometric central charge derived from Euler characteristic

Kac-Moody algebra realization with level proportional to Euler characteristic

Invariant structure under renormalization group flow

Abstract

We study a supersymmetric theory twisted on a K\"ahler four manifold $M=\Sigma_1 \times \Sigma_2 ,$ where $\Sigma_{1,2}$ are 2D Riemann surfaces. We demonstrate that it possesses a "left-moving" conformal stress tensor on $\Sigma_1$ ($\Sigma_2$) 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 $\Sigma_2$ ($\Sigma_1$) 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.