Discrete Lagrangian field theories on Lie groupoids

TL;DR

This paper develops a geometric framework for discrete classical field theories using Lie groupoids, deriving multisymplectic field equations, discrete Lie-Poisson equations, and connecting to lattice gauge theories.

Contribution

It introduces a novel Lie groupoid-based approach to discrete field theories, including new discrete Lie-Poisson equations and reduction methods.

Findings

Derived multisymplectic field equations from variational principles

Established discrete Lie-Poisson equations within the framework

Connected discrete field theories to lattice gauge theories

Abstract

We present a geometric framework for discrete classical field theories, where fields are modeled as "morphisms" defined on a discrete grid in the base space, and take values in a Lie groupoid. We describe the basic geometric setup and derive the field equations from a variational principle. We also show that the solutions of these equations are multisymplectic in the sense of Bridges and Marsden. The groupoid framework employed here allows us to recover not only some previously known results on discrete multisymplectic field theories, but also to derive a number of new results, most notably a notion of discrete Lie-Poisson equations and discrete reduction. In a final section, we establish the connection with discrete differential geometry and gauge theories on a lattice.