Twisted Self-duality

TL;DR

This paper generalizes self-duality equations in Yang-Mills theory using a non-trivial involution, leading to twisted self-duality concepts with applications in lower-dimensional theories and connections to exceptional field theory and gravitational instantons.

Contribution

It introduces a novel twisted self-duality framework for Yang-Mills theory with explicit solutions and links to exceptional field theory and gravitational instantons.

Findings

Constructed explicit solutions for twisted self-duality in $su(2) r su(2)$ gauge theory.

Derived lower-dimensional non-linear equations via dimensional reduction.

Connected twisted self-duality to E_7 exceptional field theory and Eguchi-Hanson instanton.

Abstract

We examine a generalisation of the usual self-duality equations for Yang-Mills theory when the colour space admits a non-trivial involution. This involution allows us to construct a non-trivial twist which may be combined with the Hodge star to form a twisted self-dual curvature. We will construct a simple example of twisted self-duality for $su(2) \oplus su(2)$ gauge theory along with its explicit solutions and then dimensionally reduce from four dimensions to obtain families of non-trivial non-linear equations in lower dimensions. This twisted self-duality constraint will be shown to arise in E_7 exceptional field theory through a Scherk-Schwarz reduction and we will show how an Eguchi-Hanson gravitational instanton also obeys the twisted self-duality condition.