Discrete Lagrangian field theories on Lie groupoids

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 set-up 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.