Geometry of the Gauge Algebra in Noncommutative Yang-Mills Theory

TL;DR

This paper explores the complex structure of the gauge algebra in noncommutative Yang-Mills theory, providing multiple mathematical descriptions and physical interpretations of the infinite-dimensional Lie algebra of star-gauge transformations.

Contribution

It offers a comprehensive analysis of the gauge algebra's geometry and various algebraic representations in noncommutative Yang-Mills theory.

Findings

Multiple equivalent descriptions of the gauge algebra are provided.

Connections between algebraic structures and physical interpretations are elucidated.

The role of automorphisms and symplectic diffeomorphisms is clarified.

Abstract

A detailed description of the infinite-dimensional Lie algebra of star-gauge transformations in noncommutative Yang-Mills theory is presented. Various descriptions of this algebra are given in terms of inner automorphisms of the underlying deformed algebra of functions on spacetime, of deformed symplectic diffeomorphisms, of the infinite unitary Lie algebra, and of the algebra of compact operators on a quantum mechanical Hilbert space. The spacetime and string interpretations are also elucidated.