Extended gravity theories from dynamical noncommutativity
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Extended gravity theories from dynamical noncommutativity Paolo Aschieri · Leonardo Castellani
Received: 20 July 2012 / Accepted: 24 October 2012 / Published online: 9 November 2012 © Springer Science+Business Media New York 2012
Abstract In this paper we couple noncommutative vielbein gravity to scalar fields. Noncommutativity is encoded in a -product between forms, given by an abelian twist (a twist with commuting vector fields). A geometric generalization of the Seiberg– Witten map for abelian twists yields an extended theory of gravity coupled to scalars, where all fields are ordinary (commutative) fields. The vectors defining the twist can be related to the scalar fields and their derivatives, and hence acquire dynamics. Higher derivative corrections to the classical Einstein–Hilbert and Klein–Gordon actions are organized in successive powers of the noncommutativity parameter θ AB . Keywords Noncommutative gravity · Seiberg–Witten map · Abelian twist · Dynamical noncommutativity · Higher derivatives gravity 1 Introduction In this paper we study the coupling of scalar fields to the noncommutative gravity theory constructed in [1] and further developed in [2–4]. A noncommutative action is found, and generalizes the classical Einstein–Hilbert + Klein–Gordon actions. It is invariant under diffeomorphims and noncommutative local Lorentz transformations. i AB The noncommutativity is governed by an abelian twist, F = e− 2 θ X A ⊗X B , and the corresponding -product between forms reads:
P. Aschieri (B) · L. Castellani Dipartimento di Scienze e Innovazione Tecnologica, INFN Gruppo collegato di Alessandria, Università del Piemonte Orientale, Viale T. Michel 11, 15121 Alessandria, Italy e-mail: [email protected] L. Castellani e-mail: [email protected]
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τ ∧ τ ≡
∞ n i n=0
2
θ A1 B1 . . . θ An Bn ( X A1 . . . X An τ ) ∧ ( X B1 . . . X Bn τ )
i = τ ∧ τ + θ AB ( X A τ ) ∧ ( X B τ ) 2 1 i 2 A1 B1 A2 B2 + θ θ ( X A1 X A2 τ ) ∧ ( X B1 X B2 τ ) + · · · 2! 2
where the mutually commuting vector fields X A act on forms via the Lie derivatives X A . This product is associative, and the above formula holds also for τ or τ being 0-forms (i.e. functions).1 A different study of a Klein–Gordon action in a curved background is presented in [7,8], based on the metric formulation of twist noncommutative gravity [9], where noncommutative local Lorentz symmetry is absent. Use of the geometric generalization [3] of the Seiberg–Witten map [10] between noncommutative and commutative local Lorentz symmetry allows to reinterpret the noncommutative (NC) vielbein gravity coupled to scalar fields as a theory with ordinary fields on commutative spacetime, invariant under diffeomorphisms and usual local Lorentz rotations. It is a particular higher derivative extension of Einstein gravity coupled to scalar fields. The commuting vectors X A present in the twist also enter the action, but they can be related to the scalar fields, so that the resulting th
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