Yang-Mills connections on quantum Heisenberg manifolds

TL;DR

This paper studies Yang-Mills connections with constant curvature on quantum Heisenberg manifolds, characterizing critical points and minima, and providing explicit examples and constructions of such connections.

Contribution

It characterizes Yang-Mills connections with constant curvature on quantum Heisenberg modules and constructs explicit examples, revealing their local minimality and dependence on geometric structures.

Findings

Yang-Mills connections with constant curvature have a specific curvature form.

Such connections are local minima, not global minima, of the Yang-Mills functional.

Explicit constructions of Grassmannian and tensor product connections are provided.

Abstract

We investigate critical points and minimizers of the Yang-Mills functional YM on quantum Heisenberg manifolds $D^c_{\mu\nu}$, where the Yang-Mills functional is defined on the set of all compatible linear connections on finitely generated projective modules over the QHMs. A compatible linear connection which is both a critical point and minimizer of YM is called a Yang-Mills connection. In this paper, we investigate Yang-Mills connections with constant curvature. We are interested in Yang-Mills connections on the following classes of modules over the QHMs: (i) Abadie's module $\Xi$ of trace $2\mu$ and its submodules; (ii) modules $\Xi^\prime$ of trace $2\nu$; (iii) tensor product modules of the form $P E^c_{\mu\nu}\otimes \Xi$, where $E^c_{\mu\nu}$ is Morita equivalent to $D^c_{\mu\nu}$ and $P$ is a projection in $E^c_{\mu\nu}$. We present a characterization of critical points and minimizers of YM, and provide a class of new Yang-Mills connections with constant curvature via concrete examples. In particular, we show that every Yang-Mills connection $\nabla$ on $\Xi$ over $D^c_{\mu\nu}$ with constant curvature should have a certain form of the curvature such that $\Theta_\nabla(X,Y)=\Theta_\nabla(X,Z)=0$ and $\Theta(Y,Z)=\frac{\pi i}{\mu} Id_E$. Also we show that these Yang-Mills connections with constant curvature do not provide global minima but only local minima, and give two other examples which show that the critical points and minimizers of YM depend crucially on the geometric structure of the QHMs and of the projective modules over them. Furthermore, we construct the Grassmannian connection on the projective modules $\Xi^\prime$ with trace $2\nu$ over the QHMs and compute its corresponding curvature, and we construct a tensor product connections on $P E^c_{\mu\nu}\otimes \Xi$ whose coupling constant is $2\nu$ and characterize the critical points of YM for this projective module.