The holonomy inverse problem

  • Mihajlo Cekić

    Universität Zürich, Switzerland
  • Thibault Lefeuvre

    Sorbonne Université, Paris, France
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Abstract

Let be a smooth Anosov Riemannian manifold and the set of its primitive closed geodesics. Given a Hermitian vector bundle equipped with a unitary connection , we define as the sequence of traces of holonomies of along elements of . This descends to a homomorphism on the additive moduli space of connections up to gauge , which we call the primitive trace map. It is the restriction of the well-known Wilson loop operator to primitive closed geodesics.

The main theorem of this paper shows that the primitive trace map is locally injective near generic points of when . We obtain global results in some particular cases: flat bundles, direct sums of line bundles, and general bundles in negative curvature under a spectral assumption which is satisfied in particular for connections with small curvature. As a consequence of the main theorem, we also derive a spectral rigidity result for the connection Laplacian.

The proofs are based on two new ingredients: a Livšic-type theorem in hyperbolic dynamical systems showing that the cohomology class of a unitary cocycle is determined by its trace along closed primitive orbits, and a theorem relating the local geometry of to the Pollicott–Ruelle resonance near zero of a certain natural transport operator.

Cite this article

Mihajlo Cekić, Thibault Lefeuvre, The holonomy inverse problem. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1409