Background independent noncommutative gravity from Fedosov quantization of endomorphism bundle

Abstract

Model of noncommutative gravity is constructed by means of Fedosov deformation quantization of endomorphism bundle. The fields describing noncommutativity -symplectic form and symplectic connection -are dynamical, and the resulting theory is coordinate covariant and background independent. Its interpretation in terms of Seiberg-Witten map is provided.Also, new action for ordinary (commutative) general relativity is given, which in the present context appears as a commutative limit of noncommutative theory.

Field Theory with Coordinate Dependent Noncommutativity

“…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). Here we outline the approach to curved noncommutative space which was recently put forward by M. Dobrski [44] who discussed the case of pure gravity following a series of related works by the same author, notably [43]: this formulation appears to fit nicely with the one that we considered here for flat noncommutative space-time. In a separate work (in preparation), we further discuss star products on curved manifolds and in particular different approaches to the description of tensor fields and differential forms on noncommutative manifolds.…”

“…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).…”

“…Different conditions or relations for the connection coefficients • Γ λ µν and Γ λ µν can be envisaged [44,98]. The strongest condition which consists of equating both connections is very stringent since this condition entails that ω is covariantly constant with respect to the Levi-Civita connection ∇: in four dimensions this implies that the metric locally decomposes into a sum of two-dimensional metrics [44,99]. A weaker compatibility condition was considered by M. Dobrski who referred to it as a weakly compatible metric/Fedosov structure:…”

“…The set of equations (6.8) can be combined in the following manner [44]. By applying δ λ ν • ∇µ to equation (6.8b) and then substituting equation (6.8c), we get an expression for • ∇µ s µ = ∇ µ s µ : substitution of the latter into equation (6.8a) leads to the result…”

“…In Section 6, we address the formulation of classical field theories on curved noncommutative space-time by generalizing the description considered in flat space. More precisely, we follow the star product approach to gravity which was recently put forward by M. Dobrski [44] and which appears to fit nicely into the framework discussed in flat space-time. To conclude, we comment on the quantum theory in Section 7.…”