Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The $τ$-function of the universal Whitham hierarchy, matrix models and topological field theories

I. M. Krichever

TL;DR

This work defines and analyzes the τ-function for the universal Whitham hierarchy, exposing deep links between dispersionless integrable systems, topological field theories, and gravity. It shows that algebraic orbits of the Whitham flows yield exact solutions to WDVV equations and derives truncated Virasoro/W-constraints that align Whitham dynamics with gravitational descendants in simple cases. The framework is extended from genus zero to arbitrary genus, introducing a generating potential S and a comprehensive set of times that encode moduli-space geometry and Frobenius-algebra deformations. Overall, the results propose a universal tree-level topological theory connected to matrix-model structures and gravitational couplings via the Whitham hierarchy.

Abstract

The universal Witham hierarchy is considered from the point of view of topological field theories. The $τ$-function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity.

The $τ$-function of the universal Whitham hierarchy, matrix models and topological field theories

TL;DR

This work defines and analyzes the τ-function for the universal Whitham hierarchy, exposing deep links between dispersionless integrable systems, topological field theories, and gravity. It shows that algebraic orbits of the Whitham flows yield exact solutions to WDVV equations and derives truncated Virasoro/W-constraints that align Whitham dynamics with gravitational descendants in simple cases. The framework is extended from genus zero to arbitrary genus, introducing a generating potential S and a comprehensive set of times that encode moduli-space geometry and Frobenius-algebra deformations. Overall, the results propose a universal tree-level topological theory connected to matrix-model structures and gravitational couplings via the Whitham hierarchy.

Abstract

The universal Witham hierarchy is considered from the point of view of topological field theories. The -function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity.

Paper Structure

This paper contains 7 sections, 21 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Whitham hierarchy. Zero genus case
  3. $\tau$-function
  4. Truncated Virasoro and W-constraints
  5. Primary rings of the topological field theories
  6. Generating form of the Whitham equations
  7. Arbitrary genus case

Key Result

Theorem 2.1

The zero-curvature form ((1.11)) of the Whitham hierarchy in zero-genus case is equivalent to the Sato-form that is a compatible system of evolution equations

Theorems & Definitions (21)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.1
  • Corollary 2.2
  • Theorem 3.1
  • Corollary 3.1
  • Theorem 4.1
  • Theorem 5.1
  • Corollary 5.1
  • ...and 11 more