Calderón problem for systems via complex parallel transport

We consider the Calderón problem for systems with unknown zeroth and first order terms, and improve on previously known results. More precisely, let $(M,g)$ be a compact Riemannian manifold with boundary, let $A$ be a connection matrix on $E=M\times {\mathbb{C}}^r$ and let $Q$ be a matrix potential. Let ${\mathrm{\Lambda }}_{A,Q}$ be the Dirichlet-to-Neumann map of the associated connection Laplacian with a potential. Under the assumption that $(M,g)$ is isometrically contained in the interior of $({\mathbb{R}}^2\times M_0,c(e\oplus g_0))$, where $(M_0,g_0)$ is an arbitrary compact Riemannian manifold with boundary, $e$ is the Euclidean metric on ${\mathbb{R}}^2$, and $c>0$, we show that ${\mathrm{\Lambda }}_{A,Q}$ uniquely determines $(A,Q)$ up to natural gauge invariances. Moreover, we introduce new concepts of complex ray transform and complex parallel transport problem, and study their fundamental properties and relations to the Calderón problem.