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Invariants of algebraic curves and topological expansion

Bertrand Eynard, Nicolas Orantin

TL;DR

The paper introduces an infinite family of invariants F^{(g)} for any algebraic curve, defined via residues on the curve’s spectral geometry, and shows these invariants reproduce the topological expansion of various matrix-integral models. It establishes a formal tau-function structure through Hirota equations and builds a diagrammatic calculus (W_k^{(g)} with F^{(g)}) that depends only on the curve’s local branch behavior, yet matches the large-N expansion of matrix models when the curve is the spectral curve. It proves powerful symplectic and modular invariances, analyzes singular limits (double scaling and minimal models), and demonstrates applications to Kontsevich integrals and external-field matrix models. The framework unifies geometric, combinatorial, and integrable aspects, providing new proofs and general tools for computing topological expansions across multiple models. The results bridge algebraic geometry with mathematical physics, offering a versatile approach to understand and compute higher-genus contributions in diverse settings, including CFT minimal models and topological string theory.

Abstract

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a tau function attached to an algebraic curve. These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e. the (p,q) minimal models of conformal field theory. As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV tau-function.

Invariants of algebraic curves and topological expansion

TL;DR

The paper introduces an infinite family of invariants F^{(g)} for any algebraic curve, defined via residues on the curve’s spectral geometry, and shows these invariants reproduce the topological expansion of various matrix-integral models. It establishes a formal tau-function structure through Hirota equations and builds a diagrammatic calculus (W_k^{(g)} with F^{(g)}) that depends only on the curve’s local branch behavior, yet matches the large-N expansion of matrix models when the curve is the spectral curve. It proves powerful symplectic and modular invariances, analyzes singular limits (double scaling and minimal models), and demonstrates applications to Kontsevich integrals and external-field matrix models. The framework unifies geometric, combinatorial, and integrable aspects, providing new proofs and general tools for computing topological expansions across multiple models. The results bridge algebraic geometry with mathematical physics, offering a versatile approach to understand and compute higher-genus contributions in diverse settings, including CFT minimal models and topological string theory.

Abstract

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a tau function attached to an algebraic curve. These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e. the (p,q) minimal models of conformal field theory. As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV tau-function.

Paper Structure

This paper contains 80 sections, 33 theorems, 434 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results of this article
  3. Definitions
  4. Remarkable properties
  5. Some applications, Kontsevitch's integral
  6. Algebraic curves, reminder and notations
  7. Some properties of algebraic curves
  8. Sheets
  9. Branch points and conjugated points
  10. Genus, cycles, Abel map
  11. Theta-functions and prime forms
  12. Bergmann kernel
  13. 3rd type differentials
  14. Modified set of cycles:
  15. Modified Bergmann kernel
  16. ...and 65 more sections

Key Result

Theorem 4.1

The correlation function $W_3^{(0)}$ is worth:

Figures (2)

  • Figure 1: Example of an algebraic curve with two $x$-branch points $a_1$ and $a_2$ and a three sheeted structure ($x$ has three preimages). One can see that the map $p \to \overline{p}$ is not globally defined, for instance when $q \to p$, we have $\overline{q} \to p^{(2)}$. The notion of conjugated point depends on the branch point.
  • Figure 2: Example of canonical cycles and the corresponding fundamental domain in the case of the torus (genus ${\mathbf g \!\!\! /\;\;}=1$).

Theorems & Definitions (46)

  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Remark 4.1
  • Definition 4.4
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5
  • ...and 36 more