Small coupling limit and multiple solutions to the Dirichlet Problem for Yang Mills connections in 4 dimensions - Part I

TL;DR

This paper investigates the structure of solutions to the small coupling limit of the Yang-Mills Dirichlet problem in four dimensions, proving the existence of multiple solutions and developing a Morse theory for this non-compact variational problem.

Contribution

It introduces a finite-dimensional reduction approach and establishes multiple solutions for the small coupling Yang-Mills Dirichlet problem, extending to general manifolds and Lie groups.

Findings

Existence of multiple solutions, including non-minimal ones.

Development of a Morse theory for the problem.

Reduction of the infinite-dimensional problem to a finite-dimensional one.

Abstract

In this paper (Part I) and its sequels (Part II and Part III), we analyze the structure of the space of solutions to the epsilon-Dirichlet problem for the Yang-Mills equations on the 4-dimensional disk, for small values of the coupling constant epsilon. These are in one-to-one correspondence with solutions to the Dirichlet problem for the Yang Mills equations, for small boundary data. We prove the existence of multiple solutions, and, in particular, non minimal ones, and establish a Morse Theory for this non-compact variational problem. In part I, we describe the problem, state the main theorems and do the first part of the proof. This consists in transforming the problem into a finite dimensional problem, by seeking solutions that are approximated by the connected sum of a minimal solution with an instanton, plus a correction term due to the boundary. An auxiliary equation is introduced that allows us to solve the problem orthogonally to the tangent space to the space of approximate solutions. In Part II, the finite dimensional problem is solved via the Ljusternik-Schirelman theory, and the existence proofs are completed. In Part III, we prove that the space of gauge equivalence classes of Sobolev connections with prescribed boundary value is a smooth manifold, as well as some technical lemmas used in Part I. The methods employed still work when the 4-dimensional disk is replaced by a more general compact manifold with boundary, and SU(2) is replaced by any compact Lie group.