Gauge and Integrable Theories in Loop Spaces

TL;DR

This paper introduces an integral formulation for a broad class of field theories using loop space mathematics, enabling natural derivation of conservation laws and insights into global properties of integrable and gauge theories.

Contribution

It develops a novel integral approach based on non-abelian Stokes theorems in loop spaces, applicable to various dimensions and types of gauge and integrable theories.

Findings

Provides a unified integral framework for field equations

Derives conservation laws directly from the integral formulation

Enhances understanding of global properties in gauge theories

Abstract

We propose an integral formulation of the equations of motion of a large class of field theories which leads in a quite natural and direct way to the construction of conservation laws. The approach is based on generalized non-abelian Stokes theorems for p-form connections, and its appropriate mathematical language is that of loop spaces. The equations of motion are written as the equality of an hyper-volume ordered integral to an hyper-surface ordered integral on the border of that hyper-volume. The approach applies to integrable field theories in (1+1) dimensions, Chern-Simons theories in (2+1) dimensions, and non-abelian gauge theories in (2+1) and (3+1) dimensions. The results presented in this paper are relevant for the understanding of global properties of those theories.