Identification of a connection from Cauchy data on a Riemann surface with boundary

TL;DR

This paper proves that the Cauchy data of a magnetic Laplacian on a Riemann surface with boundary uniquely determines the connection and potential, advancing inverse boundary value problem theory.

Contribution

It establishes the uniqueness of recovering a connection and potential from boundary measurements on a Riemann surface, extending inverse problem results to complex line bundles.

Findings

Cauchy data space determines the connection up to gauge

The potential q is uniquely recoverable from boundary data

Results apply to magnetic Laplacians on Riemann surfaces

Abstract

We consider a connection $\nabla^X$ on a complex line bundle over a Riemann surface with boundary $M_0$, with connection 1-form $X$. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) $L:={\nabla^X}^*\nabla^X + q$, with $q$ a complex valued potential, uniquely determines the connection up to gauge isomorphism, and the potential $q$.