A uniqueness result for the Calderón problem for $U(N)$-connections coupled to spinors

In this paper we define a Dirichlet-to-Neumann map for a twisted Dirac Laplacian acting on bundle-valued spinors over a spin manifold. We show that this map is a pseudodifferential operator of order 1 whose symbol determines the Taylor series of the metric and connection at the boundary. We go on to show that if two real-analytic connections couple to a spinor via the Yang–Mills–Dirac equations with appropriate boundary conditions, and have equal Dirichlet-to-Neumann maps, then the two connections are globally gauge equivalent in the smooth category. In the abelian case, the global gauge equivalence is in the real-analytic category.