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Symplectic Groupoids and Poisson Electrodynamics

Vladislav G. Kupriyanov, Alexey A. Sharapov, Richard J. Szabo

Abstract

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.

Symplectic Groupoids and Poisson Electrodynamics

Abstract

We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative 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.
Paper Structure (13 sections, 1 theorem, 106 equations, 2 figures, 1 table)

This paper contains 13 sections, 1 theorem, 106 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Symplectic groupoids
  3. Groupoids.
  4. Lie groupoids.
  5. Symplectic groupoids.
  6. Local symplectic groupoids.
  7. Bisections
  8. Bisections of Lie groupoids.
  9. Lagrangian bisections.
  10. Classical Poisson electrodynamics
  11. Minimal coupling.
  12. Field strength tensors.
  13. Action functional for the electromagnetic field.

Key Result

Proposition 3.5

For any multiplicative $k$-form $\alpha\in \mathsf{\Lambda}^k (G)$ define the pair of maps Then Furthermore, the maps (ST) are crossed homomorphisms from the groups of bisections to the corresponding modules: for all $\Sigma_1, \Sigma_2\in \mathscr{B}(G)$.

Figures (2)

  • Figure 1: Groupoid multiplication.
  • Figure 2: The left and right actions of bisections.

Theorems & Definitions (19)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.6
  • Example 2.7
  • Remark 2.16
  • Example 2.18
  • Example 2.19
  • Example 2.21
  • Definition 3.1
  • ...and 9 more