Regularized polysymplectic geometry and first steps towards Floer theory for covariant field theories

TL;DR

This paper introduces a new geometric framework for covariant field theories that enables the development of Hamiltonian Floer theory analogues, demonstrating existence results for Floer curves in coupled particle-field systems.

Contribution

It presents a regularization method that overcomes degeneracy issues, allowing the definition of Floer theory for covariant field theories in a relativistic setting.

Findings

Established convergence of Floer curves to space-time periodic solutions.

Proved existence of Floer curves for certain coupled particle-field systems.

Developed a new geometric framework suitable for variational techniques in covariant field theories.

Abstract

It is the goal of this paper to present the first steps for defining the analogue of Hamiltonian Floer theory for covariant field theory, treating time and space relativistically. While there already exist a number of competing geometric frameworks for covariant field theory generalizing symplectic geometry, none of them are readily suitable for variational techniques such as Hamiltonian Floer theory, since the corresponding action functionals are too degenerate. Instead, we show how a regularization procedure introduced by Bridges leads to a new geometric framework for which we can show that the finite energy $L^2$-gradient lines of the corresponding action functional, called Floer curves, converge asymptotically to space-time periodic solutions. As a concrete example we prove the existence of Floer curves, and hence also of space-time periodic solutions, for a class of coupled particle-field systems defined in this new framework.