Gauged Hyperinstantons and Monopole Equations

TL;DR

This paper explores the mathematical structure of hyperinstantons and monopole equations within topological gauge theories, revealing their connections to Seiberg-Witten equations and their deformation under gravity and gauge couplings.

Contribution

It introduces the general form of hyperinstanton equations coupled to gravity and gauge fields, extending previous work on topological twists and monopole equations.

Findings

Gauged hyperinstantons reduce to Seiberg-Witten equations for U(1) gauge group.

Deformation of self-duality conditions parallels previous gravitational deformations.

Hyperinstantonic equations are formulated with gravity and gauge couplings.

Abstract

The monopole equations in the dual abelian theory of the N=2 gauge-theory, recently proposed by Witten as a new tool to study topological invariants, are shown to be the simplest elements in a class of instanton equations that follow from the improved topological twist mechanism introduced by the authors in previous papers. When applied to the N=2 sigma-model, this twisting procedure suggested the introduction of the so-called hyperinstantons, or triholomorphic maps. When gauging the sigma-model by coupling it to the vector multiplet of a gauge group G, one gets gauged hyperinstantons that reduce to the Seiberg-Witten equations in the flat case and G=U(1). The deformation of the self-duality condition on the gauge-field strength due to the monopole-hyperinstanton is very similar to the deformation of the self-duality condition on the Riemann curvature previously observed by the authors when the hyperinstantons are coupled to topological gravity. In this paper the general form of the hyperinstantonic equations coupled to both gravity and gauge multiplets is presented.