Identification of a Connection from Cauchy Data on a Riemann Surface with Boundary

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.