On the L∞ structure of Poisson gauge theory

The Poisson gauge theory is a semi-classical limit of full non-commutative gauge theory. In this work we construct an L ∞ 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 ℓ-brackets and prove that they satisfy the corresponding homotopy relations. On the one hand, it provides new explicit non-trivial examples of L∞ 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 ℓ-brackets on L ∞ full with respect to the truncated product on the exterior algebra are satisfied only for the canonical non-commutativity. In general, L ∞ full does not have a structure of P∞ algebra.