Yang-Mills connections and sigma-models on quantum Heisenberg manifolds

TL;DR

This paper constructs spectral triples on quantum Heisenberg manifolds, linking non-linear sigma models and Yang-Mills theory, and analyzes energy bounds related to topological charges and curvature in this noncommutative setting.

Contribution

It extends spectral triple constructions to quantum Heisenberg manifolds and explores their connection to sigma-models and Yang-Mills theory, revealing new geometric insights.

Findings

Derived a lower bound for the energy functional linked to topological charge.

Analyzed the curvature dependence of the energy bounds for specific projections.

Uncovered an interplay between sigma models and Yang-Mills theory in the quantum Heisenberg context.

Abstract

We construct a spectral triple on a quantum Heisenberg manifold, which generalizes the results of Chakraborty and Shinha, and associate to it an energy functional on the set of projections, following the approach of Mathai-Rosenberg to non-linear sigma models. The spectral triples that we construct extend the We derive a lower bound for this energy functional that is linked on the topological charge of the projection which depends on the curvature of a compatible connection. A detailed study of this lower bound is given for the Kang projection in quantum Heisenberg manifolds. These results display an intriguing interplay between non-linear sigma models and Yang-Mills theory on quantum Heisenberg manifolds, unlike in the well-studied case of noncommutative tori.