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On Duality, Legendre Bundles and Deformations

N. C. Combe, P. G. Combe, H. K. Nencka

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-Kähler structure.

On Duality, Legendre Bundles and Deformations

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-Kähler structure.

Paper Structure

This paper contains 19 sections, 6 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Motivation and object of study.
  3. Main result.
  4. Structure of the paper.
  5. The Legendre bundle
  6. Legendre Duality
  7. Dually Flat Manifolds
  8. (Classical) Legendre bundle
  9. Equivalence with Dually Flat Manifolds
  10. Examples of Classical Legendre bundles
  11. Exponential Families
  12. Para-Kähler structures on Legendre bundles
  13. Hessian Quantum Field Theory
  14. Restricted Class and Formal Setup
  15. Families of Legendre Bundles
  16. ...and 4 more sections

Key Result

Proposition 1

Let $B$ be a smooth manifold. The following two structures are equivalent: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Definition 1
  • Remark 3
  • Definition 2: Legendre bundle
  • Proposition 1: Equivalence of structures
  • proof
  • Proposition 2
  • proof
  • Definition 3
  • ...and 11 more