An inverse problem for fractional connection Laplacians

Consider a fractional operator $P^s$, $0<s<1$, for connection Laplacian $P:={\mathrm{\nabla }}^{\ast }\mathrm{\nabla }+A$ on a smooth Hermitian vector bundle over a closed, connected Riemannian manifold of dimension $n\ge 2$. We show that local knowledge of the metric, Hermitian bundle, connection, potential, and source-to-solution map associated with $P^s$ determines these structures globally. This extends a result known for the fractional Laplace-Beltrami operator.