Vladislav G. Kupriyanov

In this paper, we consider the problem of defining the field strength of abelian potentials in Poisson Electrodynamics. Considering a local symplectic groupoid as a symplectic realization of the Poisson spacetime, it is possible to define several field strengths with different transformation properties. The introduction of gauge invariant momenta allows to find the relations that exist among all of them: in particular, they all vanish simultaneously. A Poisson Chern–Simons model is then proposed and its equations of motion are shortly discussed.

We investigate Snyder spacetime and its generalizations, including Yang and Snyder–de Sitter spaces, which constitute manifestly Lorentz-invariant noncommutative geometries. This work initiates a systematic study of gauge theory on such spaces in the semi-classical regime, formulated as Poisson gauge theory. As a first step, we construct the symplectic realizations of the relevant noncommutative spaces, a prerequisite for defining Poisson gauge transformations and field strengths. We present a general method for representing the Snyder algebra and its extensions in terms of canonical phase-space variables, enabling both the reproduction of known representations and the derivation of novel ones. These canonical constructions are employed to obtain explicit symplectic realizations for the Snyder–de Sitter space and to construct the deformed partial derivative which differentiates the underlying Poisson structure. Furthermore, we analyze the motion of freely falling particles in these backgrounds and comment on the geometry of the associated spaces.

We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative U(1) gauge theory. Our framework is based on an integrating symplectic groupoid for the underlying Poisson brackets, which we interpret as the classical phase space of a point particle on noncommutative spacetime. In this picture gauge fields arise as bisections of the symplectic groupoid while gauge transformations are parameterized by Lagrangian bisections. We provide a geometric construction of a gauge invariant action functional which minimally couples a dynamical charged particle to a background electromagnetic field. Our constructions are elucidated by several explicit examples, demonstrating the appearances of curved and even compact momentum spaces, the interplay between gauge transformations and spacetime diffeomorphisms, as well as emergent gravity phenomena.

We explicitly construct an L$_\infty$ algebra that defines U$_{\star}(1)$ gauge transformations on a space with an arbitrary non-commutative and even non-associative star product. Matter fields are naturally incorporated in this scheme as L$_\infty$ modules. Some possibilities for including P$_\infty$ algebras are also discussed.

We explicitly construct an L ∞ algebra that defines U ⋆ ( 1 ) gauge transformations on a space with an arbitrary noncommutative and even nonassociative star product. Matter fields are naturally incorporated in this scheme as L ∞ modules. Some possibilities for including P ∞ algebras are also discussed.