On the L${}_{\infty }$ structure of Poisson gauge theory

Abstract

The Poisson gauge theory is a semi-classical limit of full non-commutative gauge theory. In this work we construct an L${}_{\infty }^{full}$ algebra which governs both the action of gauge symmetries and the dynamics of the Poisson gauge theory. We derive the minimal set of non-vanishing $\ell$-brackets and prove that they satisfy the corresponding homotopy relations. On the one hand, it provides new explicit non-trivial examples of L${}_{\infty }$ algebras. On the other hand, it can be used as a starting point for bootstrapping the full non-commutative gauge theory. The first few brackets of such a theory are constructed explicitly in the text. In addition we show that the derivation properties of $\ell$-brackets on L${}_{\infty }^{full}$ with respect to the truncated product on the exterior algebra are satisfied only for the canonical non-commutativity. In general, L${}_{\infty }^{full}$ does not have a structure of P${}_{\infty }$ algebra.

30 pages, matches with a published version

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Cited by in corpus (4)

• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• Hamiltonian analysis in Lie-Poisson gauge theory
• Light Propagation in $\kappa$-Minkowski Space-Time: Gauge Ambiguities and Invariance
• Generalized symmetries as homotopy Lie algebras