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Hirota, Fay and Geometry

Bertrand Eynard, Soufiane Oukassi

Abstract

This is a review of the relationship between Fay identities and Hirota equations in integrable systems, reformulated in a geometric language compatible with recent Topological Recursion formalism. We write Hirota equations as trans-series, and Fay identities as spinor functional relations. We also recall several constructions of how some solutions to Fay/Hirota equations can be built from Riemann surface geometry.

Hirota, Fay and Geometry

Abstract

This is a review of the relationship between Fay identities and Hirota equations in integrable systems, reformulated in a geometric language compatible with recent Topological Recursion formalism. We write Hirota equations as trans-series, and Fay identities as spinor functional relations. We also recall several constructions of how some solutions to Fay/Hirota equations can be built from Riemann surface geometry.
Paper Structure (30 sections, 29 theorems, 234 equations)

This paper contains 30 sections, 29 theorems, 234 equations.

Table of Contents

  1. Introduction: Tau functions and Hirota
  2. Tau functions as graded formal series
  3. Hirota Equation
  4. Turning Hirota to PDEs
  5. Insertion operator
  6. Reformulation as spinor trans-series
  7. Motivation
  8. Tau function as trans-series
  9. Hirota with Divisors
  10. Fay Identities
  11. Miwa Jimbo
  12. Conclusion on formal ${\cal T}$-functions
  13. Tau functions and Riemann Surfaces
  14. Introduction and motivation
  15. Riemann surfaces
  16. ...and 15 more sections

Key Result

Theorem 1.1

The Tau functions of the KdV, KP, Toda hierarchies satisfy the Hirota equation.

Theorems & Definitions (69)

  • Definition 1.1: Sato's vector
  • Remark 1.1
  • Definition 1.2: Divisors
  • Definition 1.3: Sato's vector for divisors
  • Remark 1.2
  • Definition 1.4: Hirota Equation
  • Theorem 1.1: Hirota
  • Theorem 1.2: Hirota as PDEs
  • Remark 1.3
  • Example 1.1
  • ...and 59 more