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Structure of Dubrovin-Zhang free energy functions and universal identities

Sergey Shadrin, Zhe Wang

Abstract

We prove a structural theorem relating the higher genera free energy functions of the Dubrovin-Zhang hierarchies to the Witten-Kontsevich free energy function of the Korteweg-de Vries hierarchy. As an important application, for any given genus $g\geq 1$, we construct a set of universal identities valid for the free energy functions of any Dubrovin-Zhang hierarchy. In particular, we present some techniques that can be used to derive universal identities without relying on the geometry of the moduli space of stable curves of higher genus.

Structure of Dubrovin-Zhang free energy functions and universal identities

Abstract

We prove a structural theorem relating the higher genera free energy functions of the Dubrovin-Zhang hierarchies to the Witten-Kontsevich free energy function of the Korteweg-de Vries hierarchy. As an important application, for any given genus , we construct a set of universal identities valid for the free energy functions of any Dubrovin-Zhang hierarchy. In particular, we present some techniques that can be used to derive universal identities without relying on the geometry of the moduli space of stable curves of higher genus.
Paper Structure (15 sections, 24 theorems, 179 equations)

This paper contains 15 sections, 24 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Structure of free energy functions
  3. Givental theory
  4. Loop equation of the free energy function
  5. Decomposition of free energy functions
  6. Universal identities of free energy functions
  7. A useful set of operators
  8. Basic notation for trees
  9. Eguchi-Xiong operators and their generalizations
  10. Definition of operators
  11. Action on $v^{\beta,b}$
  12. The iterative structure of operators
  13. Universal identities
  14. Towards more general universal identities
  15. Conclusion

Key Result

Theorem 1.1

Given a semisimple Frobenius manifold (with a choice of calibration) of rank $N$, its higher genus free energy function $F_g$ admits the following decomposition: here $F_g^{KdV}$ is the genus $g$ free energy function ab of the Gromov-Witten theory of the point and the function $H_g$ satisfies the conditions

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 43 more