Yang-mills for Quantum Heisenberg Manifolds

We consider the Yang-Mills problem for a quantum Heisenberg manifold, which is a C∗-algebra defined by the (strict) deformation quantization of the ordinary Heisenberg manifold, in the setting of non-commutative differential geometry following Connes and Rieffel [Co] [Co1]. 1. Preliminaries Classical Yang-Mills theory is concerned with the set of connections (i.e. gauge potentials) on a vector bundle of a smooth manifold. The Yang-Mills functional YM measures the “strength” of the curvature of a connection. The Yang-Mills problem is determining the nature of the set of connections where YM attains its minimum, or more generally the nature of the set of critical points for YM. Since there is a well-developed non-commutative analogue of this setting. we can define a non-commutative Yang-Mills problem as follows [CoRie]: Let (A,G, α) be a C-dynamical system, where G is a Lie group. It is said that x in A is C-vector if and only if g → αg(x) from G to the normed space is of C . Then A = {x ∈ A| x is of C } is norm dense in A. In this case we call A the smooth dense subalgebra of A. Since a C-algebra with a smooth dense subalgebra is an analogue of a smooth manifold, finitely generated projective A-modules are the appropriate generalizations of vector bundles over the manifold. By Connes [Co1], a finitely generated projective A-module Ξ always exists if there is a finitely generated projective A-module Ξ under G-invariant maps. In addition, an hermitian structure on Ξ is given by a A-valued positive definite inner product < ξ, η >∈ A for ξ, η ∈ Ξ. Let L be the Lie-algebra of (unbounded) derivations of A given by the representation δ from the Lie-algebra g of G where δX(x) = limt→0 1 t (αexp(tX)(x)− x) for X ∈ g. Definition 1.1. Given Ξ, a connection on Ξ is a linear map ∇ : Ξ → Ξ ⊗ L such that, for all X ∈ g, ξ ∈ Ξ and x ∈ A one has ∇X(x · ξ) = x · ∇X(ξ) + δX(x) · ξ. We shall say that ∇ is compatible with the hermitian metric if and only if < ∇X(ξ), η > + < ξ,∇X(η) >= δX(< ξ, η >) for all ξ, η ∈ Ξ, X ∈ g. Date: May 15, 2009. 2000 Mathematics Subject Classification. Primary:46L05, Secondary:70S15.