Abstract
We define a new homotopy algebraic structure, that we call a braided L_infty -algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel’d twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein–Cartan–Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.
Highlights
In this paper, we will present a new perspective on symmetries of noncommutative field theories, and develop first principles which enable systematic constructions of new examples of field theories with noncommutative gauge symmetries
To achieve this we develop a new notion of homotopy algebra that is suited to organize the symmetries and dynamical content of noncommutative gauge theories; we call this new mathematical object a ‘braided L∞-algebra’
The braided homotopy relations guarantee the braided representation properties, braided covariance of the field equations, and braided gauge invariance of the action functional. These symmetries differ from the star-gauge symmetries that are usually employed in noncommutative gauge theories, which close as classical Lie algebras
Summary
We will present a new perspective on symmetries of noncommutative field theories, and develop first principles which enable systematic constructions of new examples of field theories with noncommutative gauge symmetries. The noncommutative extension of the bootstrap approach has been employed by [21,39] as a means for constructing new examples of noncommutative gauge theories In this approach the L∞-algebra framework encodes the symmetries and dynamics of the field theory, and the general failure of the deformed exterior algebra from forming a differential graded algebra and, in the case of nonassociative gauge symmetries, the failure of the Jacobi identities. The braided homotopy relations guarantee the braided representation properties, braided covariance of the field equations, and braided gauge invariance of the action functional These symmetries differ from the star-gauge symmetries that are usually employed in noncommutative gauge theories, which close as classical Lie algebras.
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