Calder\'on problem for systems via complex parallel transport

We consider the Calder\'on 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 $\Lambda_{A, Q}$ 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 $\Lambda_{A, Q}$ 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 Calder\'on problem.