  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 $q$.
