Yang-Mills connections on quantum Heisenberg manifolds

Abstract

We investigate critical points and minimizers of the Yang-Mills functional YM on quantum Heisenberg manifolds Dc µν, where the Yang-Mills functional is defined on the set of all compatible linear connections on finitely generated projective modules over Dc µν . 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 Dc µν: (i) Abadie’s module Ξ of trace 2µ and its submodules; (ii) modules Ξ ′ of trace 2ν; (iii) tensor product modules of the form P Ec µν ⊗Ξ, where Ec µν is Morita equivalent to Dc µν and P is a projection in Ec µν . 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 Dc µν via concrete examples. In particular, we show that every Yang-Mills connection ∇ on Ξ over Dc µν with constant curvature should have a certain form of the curvature such as Θ∇(X, Y ) = Θ∇(X, Z) = 0 and Θ∇(Y, Z) = πi µ IdE. 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 Dc µν 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 Dc µν . Consequently the critical points and minimizers of YM depend crucially on the geometric structure of Dc µν and of the projective modules over Dc µν . Furthermore, we construct the Grassmannian connection on the projective modules Ξ ′ with trace 2ν over Dc µν and compute its corresponding curvature. Finally, we construct tensor product connections on P Ec µν ⊗ Ξ whose coupling constant is 2ν and characterize the critical points of YM for this projective module.