Locally Anisotropic Supergravity and Gauge Gravity on Noncommutative Spaces

Abstract

We outline the the geometry of locally anisotropic (la) superspaces and la-supergravity. The approach is backgrounded on the method of anholonomic superframes with associated nonlinear connection structure. Following the formalism of enveloping algebras and star product calculus we propose a model of gauge la-gravity on noncommutative spaces. The corresponding Seiberg-Witten maps are established which allow the definition of dynamics for a finite number of gravitational gauge field components on noncommutative spaces. *

Axiomatic formulations of modified gravity theories with nonlinear dispersion relations and Finsler–Lagrange–Hamilton geometry

“…Equation (64) can be integrated in general off-diagonal forms for configurations with D = D and, if necessary, for further restrictions to D | T =0 = ∇. Exact classical and quantum solutions with generalized connections and for nonholonomic configurations have been studied in a series of works [143,144,146,152,153,156,157,161,166,169,180,181]. Recently, such solutions were found for locally anisotropic interactions of gravitational fields and effective matter fields, EYMH systems in GR and string gravity, for nonholonomic quantization of gauge models of gravity etc., see [23,190,198,203,209] and references therein.…”

“…It is possible to elaborate a N-adapted Palatini formalism [104], quasi-classical extensions, noncommutative and nonassociative generalizations, deformation quantization, gauge like gravity theories etc., see Refs. [32][33][34]37,38,42,146,147,152,156,157,159,162,167,168,170,179,190,206].…”

“…A series of results on MGTs with local anisotropy and Finsler like modifications were reviewed and presented in [153,161,169,173,185]. It was shown that such constructions involve MDRs which can be derived for propagation of point mass classical particles and/or models with quantum variables and (non)commutative relations describing quantum fundamental field interactions [32,[59][60][61][62][63]156,157,168,174,179,216,217]. Various classes of locally anisotropic (non)commutative/associative classical and quantum field and spacetime theories can be described naturally in terms of nonholonomic variables as some models of commutative and/or noncommutative geometries.…”

On axiomatic formulation of gravity and matter field theories with MDRs and Finsler-Lagrange-Hamilton geometry on (co)tangent Lorentz bundles

“…Equations ( 64) can be integrated in general off-diagonal forms for configurations with D = D and, if necessary, for further restrictions to D | T =0 = ∇. Exact classical and quantum solutions with generalized connections and for nonholonomic configurations have been studied in a series of works [278,316,280,283,298,303,302,299,308,325,344,345]. Recently, such solutions were found for locally anisotropic interactions of gravitational fields and effective matter fields, EYMH systems in GR and string gravity, for nonholonomic quantization of gauge models of gravity etc., see [355,365,371,376,50] and references therein.…”

“…A series of results on MGTs with local anisotropy and Finsler like modifications were reviewed and presented in [299,308,325,333,349]. It was shown that such constructions involve MDRs which can be derived for propagation of point mass classical particles and/or models with quantum variables and (non)commutative relations describing quantum fundamental field interactions [302,303,119,121,122,123,124,321,67,334,343,387,388]. Various classes of locally anisotropic (non) commutative/ associative classical and quantum field and spacetime theories can be described naturally in terms of nonholonomic variables as some models of commutative and/or noncommutative geometries.…”