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

We consider a connection ${\mathrm{\nabla }}^X$ 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) $L:={{\mathrm{\nabla }}^X}^{\ast }{\mathrm{\nabla }}^X+q$, with $q$ a complex valued potential, uniquely determines the connection up to gauge isomorphism, and the potential $q$.