The Yang-Mills functional and Laplace's equation on quantum Heisenberg manifolds

TL;DR

This paper explores the Yang-Mills functional on quantum Heisenberg manifolds, constructing critical points via noncommutative geometry and linking solutions to Laplace's equation in this setting.

Contribution

It introduces a family of connections on quantum Heisenberg manifolds that are critical points of the Yang-Mills functional, connecting them to solutions of Laplace's equation.

Findings

Constructed a family of Yang-Mills critical points on quantum Heisenberg manifolds.

Linked solutions of the Yang-Mills functional to Laplace's equation in the quantum setting.

Applied noncommutative geometric methods to analyze gauge theories on quantum manifolds.

Abstract

In this paper, we discuss the Yang-Mills functional and a certain family of its critical points on quantum Heisenberg manifolds using noncommutative geometrical methods developed by A. Connes and M. Rieffel. In our main result, we construct a certain family of connections on a projective module over a quantum Heisenberg manifold that give rise to critical points of the Yang-Mills functional. Moreover, we show that this set of solutions can be described as a set of solutions to Laplace's equation on quantum Heisenberg manifolds.