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The attenuated ray transform for connections and Higgs fields

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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.

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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, UK

    Gabriel P. Paternain

  2. Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

    Mikko Salo

  3. Department of Mathematics, University of Washington, Seattle, USA

    Gunther Uhlmann

  4. Department of Mathematics, University of California, Irvine, USA

    Gunther Uhlmann

Authors
  1. Gabriel P. Paternain
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  2. Mikko Salo
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  3. Gunther Uhlmann
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Correspondence to Mikko Salo.

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Paternain, G.P., Salo, M. & Uhlmann, G. The attenuated ray transform for connections and Higgs fields. Geom. Funct. Anal. 22, 1460–1489 (2012). https://doi.org/10.1007/s00039-012-0183-6

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  • DOI: https://doi.org/10.1007/s00039-012-0183-6

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