Dimensional reduction over the quantum sphere and non-abelian q-vortices

TL;DR

This paper extends equivariant dimensional reduction to quantum spaces, leading to q-deformed gauge theories and vortex equations, with explicit examples showing altered stability and moduli space properties.

Contribution

It introduces a framework for dimensional reduction over quantum spheres, deriving q-deformed quiver gauge theories and instanton equations, with detailed analysis and examples.

Findings

q-deformed vortex and instanton equations on quantum spaces

Moduli spaces are better behaved but more constrained by q-deformation

Explicit examples of non-abelian vortices and q-instantons

Abstract

We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a Kaehler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU_q(2), and also of invariant gauge connections on these bundles. The reduction of Yang-Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-Kaehler quotient construction.