A Uniqueness Result for the Calderon Problem for $U(N)$-connections coupled to spinors

TL;DR

This paper establishes a boundary determination result for the Calderon problem involving $U(N)$-connections coupled with spinors, showing that the Dirichlet-to-Neumann map uniquely determines the connection up to gauge equivalence.

Contribution

It introduces a Dirichlet-to-Neumann map for twisted Dirac operators and proves its ability to determine connections uniquely under certain conditions.

Findings

Dirichlet-to-Neumann map is a pseudodifferential operator of order 1.

The map's symbol encodes the Taylor series of the metric and connection at the boundary.

Connections coupled via Yang--Mills--Dirac equations are gauge equivalent if their Dirichlet-to-Neumann maps agree.

Abstract

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 locally gauge equivalent. In the abelian case, the connections are globally gauge equivalent.