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

TL;DR

This paper shows that supersymmetric theories on Kähler four-manifolds exhibit conformal and infinite-dimensional algebras with geometric central charges, revealing deep connections between geometry, supersymmetry, and algebraic structures.

Contribution

It demonstrates the realization of $W_{1+ abla}$ and Virasoro algebras in supersymmetric theories on four-manifolds and relates their central charges to geometric invariants.

Findings

Virasoro algebra with geometric central charge on Riemann surfaces

Invariant structure under renormalization group flow

Representation of $W_{1+ abla}$ algebra via free chiral supermultiplet

Abstract

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