On Duality, Legendre Bundles and Deformations

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-Kähler structure.