Self-Dual Yang-Mills: Symmetries and Moduli Space

TL;DR

This paper explores the rich symmetry structure and solution space of the self-dual Yang-Mills equations in four-dimensional Euclidean space, linking geometric, algebraic, and quantum aspects to suggest potential solvability.

Contribution

It describes the full symmetry group of SDYM equations, introduces related infinite-dimensional algebras, and discusses implications for nonperturbative quantization and string theory connections.

Findings

Full symmetry group of SDYM equations identified

Infinite-dimensional algebras analogous to Virasoro and affine Lie algebras constructed

Potential for solvable nonperturbative quantization of SDYM theory

Abstract

Geometry of the solution space of the self-dual Yang-Mills (SDYM) equations in Euclidean four-dimensional space is studied. Combining the twistor and group-theoretic approaches, we describe the full infinite-dimensional symmetry group of the SDYM equations and its action on the space of local solutions to the field equations. It is argued that owing to the relation to a holomorphic analogue of the Chern-Simons theory, the SDYM theory may be as solvable as 2D rational conformal field theories, and successful nonperturbative quantization may be developed. An algebra acting on the space of self-dual conformal structures on a 4-space (an analogue of the Virasoro algebra) and an algebra acting on the space of self-dual connections (an analogue of affine Lie algebras) are described. Relations to problems of topological and N=2 strings are briefly discussed.