The Calder\'on problem for the fractional Dirac operator

TL;DR

This paper proves that the source-to-solution map for the fractional Dirac operator uniquely determines the geometric and bundle structures of a Riemannian manifold, with implications for physics and related fields.

Contribution

It establishes a unique inverse result for the fractional Dirac operator on manifolds, linking boundary data to the full geometric and bundle structures.

Findings

Unique determination of manifold structure from source-to-solution map

Reconstruction of Riemannian metric and bundle data

Potential applications in physics and other disciplines

Abstract

We show that knowledge of the source-to-solution map for the fractional Dirac operator acting over sections of a Hermitian vector bundle over a smooth closed connencted Riemannian manifold of dimension $m\geq 2$ determines uniquely the smooth structure, Riemannian metric, Hermitian bundle and connection, and its Clifford modulo up to a isometry. We also mention several potential applications in physics and other fields.