The holonomy inverse problem

Abstract

Let $(M,g)$ be a smooth Anosov Riemannian manifold and ${\mathcal{C}}^{♯}$ the set of its primitive closed geodesics. Given a Hermitian vector bundle $\mathcal{E}$ equipped with a unitary connection ${\nabla }^{\mathcal{E}}$, we define ${\mathcal{T}}^{♯}(\mathcal{E},{\nabla }^{\mathcal{E}})$ as the sequence of traces of holonomies of ${\nabla }^{\mathcal{E}}$ along elements of ${\mathcal{C}}^{♯}$. This descends to a homomorphism on the additive moduli space $\mathbb{A}$ of connections up to gauge ${\mathcal{T}}^{♯}:(\mathbb{A},\oplus )\to {\ell }^{\infty }({\mathcal{C}}^{♯})$, 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 ${\mathcal{T}}^{♯}$ is locally injective near generic points of $\mathbb{A}$ when $\dim (M)\ge 3$. 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 $\mathbb{A}$ to the Pollicott–Ruelle resonance near zero of a certain natural transport operator.