The attenuated ray transform for connections and Higgs fields

TL;DR

This paper proves the injectivity and uniqueness of the attenuated ray transform on simple surfaces with boundary, considering unitary connections and Higgs fields, using energy identities and gauge transformations.

Contribution

It establishes the injectivity and uniqueness results for the attenuated ray transform with connections and Higgs fields on simple surfaces, advancing inverse problem theory.

Findings

Injectivity of the transform modulo natural obstructions

Unique determination of connection and Higgs field from scattering data

Use of Pestov type energy identity and gauge transformations

Abstract

We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo a gauge transformation. The proofs involve a Pestov type energy identity for connections together with holomorphic gauge transformations which arrange the curvature of the connection to have definite sign.