Yang-Mills connections on quantum Heisenberg manifolds

Abstract

We investigate critical points and minimizers of the Yang-Mills functional YM on quantum Heisenberg manifolds Dμνc, where the Yang-Mills functional is defined on the set of all compatible linear connections on finitely generated projective modules over Dμνc. 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 Dμνc: (i) Abadie's module Ξ of trace 2μ and its submodules; (ii) modules Ξ′ of trace 2ν; (iii) tensor product modules of the form PEμνc⊗Ξ, where Eμνc is Morita equivalent to Dμνc and P is a projection in Eμνc. We present a characterization of critical points and minimizers of YM, and provide a class of new Yang-Mills connections with constant curvature on Ξ over Dμνc via concrete examples. In particular, we show that every Yang-Mills connection ∇ on Ξ over Dμνc with constant curvature should have a certain form of the curvature such as Θ∇(X,Y)=Θ∇(X,Z)=0 and Θ∇(Y,Z)=πiμIdE. Also we show that these Yang-Mills connections with constant curvature do not provide global minima but only local minima. We do this by constructing a set of compatible connections that are not critical points but their values are smaller than those of Yang-Mills connections with constant curvature. Our other results include: (i) an example of a compatible linear connection with constant curvature on Dμνc such that the corresponding connection on an isomorphic projective module does not have constant curvature, and (ii) the construction of a compatible linear connection with constant curvature which neither attains its minimum nor is a critical point of YM on Dμνc. Consequently the critical points and minimizers of YM depend crucially on the geometric structure of Dμνc and of the projective modules over Dμνc. Furthermore, we construct the Grassmannian connection on the projective modules Ξ′ with trace 2ν over Dμνc and compute its corresponding curvature. Finally, we construct tensor product connections on PEμνc⊗Ξ whose coupling constant is 2ν and characterize the critical points of YM for this projective module.