On Conservative Statistical Riemann Surfaces

TL;DR

This paper links information geometry and gauge theory, showing that conservative statistical structures on surfaces relate to harmonic forms, holomorphic differentials, and moduli spaces parametrized by complex geometric data.

Contribution

It introduces a novel correspondence between conservative statistical structures, Higgs bundles, and solutions to the Tzitzéica equation, extending the geometric understanding of statistical manifolds.

Findings

Chebyshev 1-form is harmonic for conservative structures

Traceless Amari--Chentsov tensor descends to a holomorphic cubic differential

Moduli space parametrized by a holomorphic vector bundle over Teichmüller space

Abstract

We establish a correspondence between information geometry and gauge theory. First, we define an important class of statistical manifolds, that is normalized and satisfies a conservation field equation. Second, we prove that for a conservative statistical structure on an orientable surface, the Chebyshev 1-form is constrained to be harmonic, and the traceless part of the Amari--Chentsov tensor descends to a holomorphic cubic differential. Then, we demonstrate that normalized conservative statistical structures are geometrically generated by solutions to the scalar Tzitz\'eica equation on Higgs bundles with general linear holonomy, generalizing the Labourie-Loftin correspondence. Finally, we prove that the moduli space of normalized conservative statistical structures on a closed orientable surface of genus at least 2 is completely parameterized by a holomorphic vector bundle over the Teichm\"uller space, consisting of Abelian differentials and cubic differentials.