Anti-self-dual Yang-Mills equations on noncommutative spacetime

TL;DR

This paper extends anti-self-dual Yang-Mills equations to noncommutative spacetime using the star-product, preserving many integrability properties and enabling new insights into noncommutative gauge theories.

Contribution

It constructs a noncommutative analogue of ASDYM equations with preserved integrability structures, extending the twistorial interpretation to noncommutative geometry.

Findings

Preservation of zero-curvature representation in noncommutative setting

Extension of twistorial interpretation to noncommutative spacetime

Breakdown of some finite-dimensional linear algebra structures

Abstract

By replacing the ordinary product with the so called $\star$-product, one can construct an analogue of the anti-self-dual Yang-Mills (ASDYM) equations on the noncommutative $\bbR^4$. Many properties of the ordinary ASDYM equations turn out to be inherited by the $\star$-product ASDYM equation. In particular, the twistorial interpretation of the ordinary ASDYM equations can be extended to the noncommutative $\bbR^4$, from which one can also derive the fundamental strutures for integrability such as a zero-curvature representation, an associated linear system, the Riemann-Hilbert problem, etc. These properties are further preserved under dimensional reduction to the principal chiral field model and Hitchin's Higgs pair equations. However, some structures relying on finite dimensional linear algebra break down in the $\star$-product analogues.