Infinite Conformal Algebras in Supersymmetric Theories on Four Manifolds

Abstract

We study a supersymmetric theory twisted on a Kähler 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 $\stackrel{¸}{.}$ 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.

NBI-HE-94-34, Latex, 57 pages

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• A 2d (0,2) appetizer
• Twisted characters and holomorphic symmetries
• On the six-dimensional origin of the AGT correspondence
• Holomorphic Currents and Duality in N=1 Supersymmetric Theories
• Evidence for an Algebra of ${\mathbf{G}}_2$ Instantons
• Comments on chiral algebras and $\Omega$-deformations
• Toward Null State Equations in $d>2$
• A domain wall in twisted M-theory
• Holomorphic field theories and higher algebra