Inverse problem for the Yang-Mills equations

TL;DR

This paper demonstrates that a connection in Yang-Mills equations can be uniquely recovered from source-to-solution data in Minkowski space, using advanced symbol analysis and a non-abelian light ray transform approach.

Contribution

It introduces a novel method to invert the Yang-Mills equations up to gauge from nonlinear wave interactions, applicable to any compact Lie group.

Findings

Successful recovery of connections from source-to-solution data

Reduction of the inverse problem to a broken non-abelian light ray transform

Applicability to any compact Lie group

Abstract

We show that a connection can be recovered up to gauge from source-to-solution type data associated with the Yang-Mills equations in the four dimensional Minkowski space. Our proof analyzes the principal symbols of waves generated by suitable nonlinear interactions and reduces the inversion to a broken non-abelian light ray transform. The principal symbol analysis of the interaction is based on a delicate calculation that involves the structure of the Lie algebra under consideration and the final result holds for any compact Lie group.