Gauge Theories on Noncommutative Euclidean Spaces

We consider gauge theories on noncommutative euclidean space . In particular, we discuss the structure of gauge group following standard mathematical definitions and using the ideas of hep-th/0102182 . AMS classification: 81T13, 81T75 The goal of this note is to consider gauge theories on noncommutative euclidean space and, in particular, to study the structure of gauge group. This group was analyzed by J.Harvey in recent paper [1]. It was suggested in this paper that the definition of the gauge group ”presumably can be derived from the first principles”. We would like to analyze the relation of Harvey’s definition to the standard mathematical definition using as a starting point some ideas of [2], in particular, the idea that the theory becomes more transparent if along with simple modules A we consider more complicated modules Frn. (The central point of [2]-the suggestion to work with unitized algebras-is mentioned only in passing at the very end.) Mathematical definition of a gauge field is based on a notion of connection on a module E over associative algebra A. There exist different versions of this notion (see [4] for details, [5] for more general treatment). Our consideration does not depend on these subtleties. We can use, for example, the very first definition [3]; in this definition linear operators ∇1,...,∇d specify a connection on right A-module E if they satisfy Leibniz rule: ∇α(ea) = (∇αe) · a+ e∂αa where ∂1, ..., ∂d are derivations on A, e ∈ E, a ∈ A. One assumes that these derivations (i. e. infinitesimal automorphisms) constitute a basis of a Lie algebra. By definition a gauge field is a unitary connection (i. e. ∇α should be anti Hermitian operators). It is supposed usually that A is a unital Banach algebra over C and E is a Hilbert A-module (i. e. E is equipped with A-valued Hermitian inner product < , >; then the condition of unitarity of connection takes the form