We show how to identify a potential $V$ or a connection $\nabla^X=d+iX$ up to gauge on a complex vector bundle from boundary measurements (Cauchy data on the boundary) on a fixed Riemann surface with boundary. This problem consists in showing the injectivity of the nonlinear map $V\to N_V$ (or $X\to N_X$) where $N_V$ and $N_X$ are the Dirichlet-to-Neumann operators associated to the elliptic operator $P=\Delta+V$ or $P=(\nabla^X)^*\nabla^X$. The proof, following ideas of Bukhgeim, is based on the construction of particular complex geometric optics solutions $u=e^{\Phi(z)/h}(1+remainder)$ of $Pu=0$ with holomorphic phases $\Phi$ having isolated critical points.This is joint work with L.Tzou (Helsinki & MSRI).