On Duality, Legendre Bundles and Deformations

TL;DR

This paper introduces the Legendre bundle, a geometric structure capturing duality in Hessian manifolds, unifying exponential families in information geometry and certain quantum field theories within a single framework.

Contribution

It presents the Legendre bundle as a new geometric structure with a canonical para-Kähler form, linking information geometry and quantum field theories.

Findings

Legendre bundle encodes duality in Hessian manifolds.

Exponential families and Hessian QFTs are realizations of the Legendre bundle.

The Legendre bundle has a canonical para-Kähler structure.

Abstract

We introduce the Legendre bundle, a geometric structure encoding the essential duality of dually flat (Hessian) manifolds, and demonstrate that both exponential families in information geometry and a natural class of quantum field theories -- which we term Hessian QFTs -- arise as distinct realisations of this single framework. The Legendre bundle is shown to carry a canonical para-K\"ahler structure.