Noncommutative gauge theory using a covariant star product defined between Lie-valued differential forms

Abstract

We develop an internal gauge theory using a covariant star product. The space-time is a symplectic manifold endowed only with torsion but no curvature. It is shown that, in order to assure the restrictions imposed by the associativity property of the star product, the torsion of the space-time has to be covariant constant. An illustrative example is given and it is concluded that in this case the conditions necessary to define a covariant star product on a symplectic manifold completely determine its connection.

Tensor calculus on noncommutative spaces

“…Covariant star product were also studied in [9,10,12,13,14,15]. The aim of these works was to make the star products as general as possible.…”

“…More recently, a manifestly covariant universal star product was constructed [9]. Further generalizations of the star product consist in its' extension to the exterior algebra of differential forms [10,11,12], Lie algebra valued differential forms [13], and tensor valued differential forms [14]. Gauge theories with covariant star products were considered in [15].…”

Field Theory with Coordinate Dependent Noncommutativity

“…With the description of gravity in mind, the formulation of noncommutative field theories (and in particular of gauge theories) on generic symplectic manifolds with curvature and/or torsion has been addressed by various authors using diverse approaches, e.g., see [3,4,5,11,12,24,28,29,31,32,33,43,44,46,51,59,63,64,81,94,104,105,106] as well as [86,95] for some nice introductions and overviews of the literature up to the year 2010. In relationship with the main subject of the present work (in particular the conservation laws for field theories on flat noncommutative space-time) we note that it should also be possible to obtain the energymomentum tensor (EMT) of matter fields in flat space-time by coupling these fields to a metric tensor field: the EMT is then given by the flat space limit of the curved space EMT defined as the variational derivative of the matter field action with respect to the metric tensor (see [20] and references therein for a justification of this procedure).…”

Is covariant star product unique?

“…The product (2.6) is not covariant at the second order of the expansion in h since it contains usual derivatives of the Poisson tensor. Over last years much progress has been achieved in covariantization of the Kontsevich procedure [9,1] and in extending star products to differential forms [17,6,21], to Lie algebra valued differential forms [5], and in covariant holomorphic products [7].…”