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Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry

Bertrand Eynard, Olivier Marchal

Abstract

In this article, we define a non-commutative deformation of the "symplectic invariants" of an algebraic hyperelliptical plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some algebraic geometry quantities. In particular our non-commutative Bergmann kernel satisfies a Rauch variational formula. Those non-commutative invariants are inspired from the large N expansion of formal non-hermitian matrix models. Thus they are expected to be related to the enumeration problem of discrete non-orientable surfaces of arbitrary topologies.

Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry

Abstract

In this article, we define a non-commutative deformation of the "symplectic invariants" of an algebraic hyperelliptical plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some algebraic geometry quantities. In particular our non-commutative Bergmann kernel satisfies a Rauch variational formula. Those non-commutative invariants are inspired from the large N expansion of formal non-hermitian matrix models. Thus they are expected to be related to the enumeration problem of discrete non-orientable surfaces of arbitrary topologies.

Paper Structure

This paper contains 30 sections, 9 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Definitions, kernel and Bethe ansatz
  3. The problem
  4. Analytical structure of the kernel $G$
  5. Bethe ansatz and monodromies
  6. Schroedinger equation
  7. Classical limit
  8. Definition of correlators and free energies
  9. Definition of correlators
  10. Properties of correlators
  11. Definition of free energies
  12. Classical limit and WKB expansion
  13. Classical limit
  14. WKB expansion of the wave function
  15. $\hbar$ expansion of correlators and energies
  16. ...and 15 more sections

Key Result

Theorem 3.1

Each $W_n^{(g)}$ is a rational function of all its arguments. It has poles only at the $s_i$'s (except $W_2^{(0)}$, which also has a pole at $x_1=x_2$). In particular it has no poles at the $\alpha_i$'s. Moreover, it vanishes as $O(1/x_i)$ when $x_i\to\infty$.

Theorems & Definitions (18)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 8 more