[Paper Review] Algebraic methods in random matrices and enumerative geometry

Bertrand Eynard, Nicolas Orantin|ArXiv.org|Nov 21, 2008

Advanced Algebra and GeometryMathematics102 references77 citations

TL;DR

This paper introduces symplectic invariants of spectral curves as a universal algebraic framework to solve loop equations in matrix models and extend their solution to enumerative geometry, topological strings, and integrable systems. The method constructs recursive differential forms and free energies $F_g$ from a spectral curve, which are invariant under symplectic transformations and encode quantum invariants, with key results including modularity, integrability, and applications to Gromov-Witten invariants and Weil-Petersson volumes.

ABSTRACT

We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defined a sequence of differential forms, and a sequence of complex numbers Fg . We recall the definition of the invariants Fg, and we explain their main properties, in particular symplectic invariance, integrability, modularity,... Then, we give several example of applications, in particular matrix models, enumeration of discrete surfaces (maps), algebraic geometry and topological strings, non-intersecting brownian motions,...

Motivation & Objective

• To develop a universal algebraic method for solving loop equations in matrix models beyond perturbative expansions.
• To generalize the solution of matrix model loop equations to non-matrix model problems in enumerative geometry and mathematical physics.
• To define and study symplectic invariants $F_g$ associated with a spectral curve, independent of the original matrix model context.
• To establish connections between symplectic invariants, integrable systems, and topological string theory via mirror symmetry and Kodaira-Spencer theory.
• To demonstrate that the free energies $F_g$ and correlation forms $\omega_n^{(g)}$ are intrinsic to the spectral curve and possess deep geometric and algebraic properties.

Proposed method

• Define symplectic invariants $F_g$ from a spectral curve $\mathcal{E} = \{y(x)\}$ using recursive integration of the Bergmann kernel and recursion kernel.
• Construct symmetric meromorphic differential forms $\omega_n^{(g)}$ via a recursion relation involving residues at branch points and the Schiffer kernel.
• Utilize the loop operator and its inverse to derive differential equations and relations between correlation functions and free energies.
• Apply the formalism to matrix models by identifying the spectral curve from loop equations and computing $F_g$ via the topological expansion.
• Establish modular properties and background independence by analyzing scaling behavior and transformations under symplectic maps.
• Connect the formalism to topological string theory by identifying the spectral curve as a mirror curve and the $F_g$ as amplitudes in the B-model.

Experimental results

Research questions

• RQ1 Can loop equations in matrix models be solved universally using algebraic-geometric data of the spectral curve?
• RQ2 What are the intrinsic geometric and algebraic properties of the free energies $F_g$ and correlation forms $\omega_n^{(g)}$ defined from a spectral curve?
• RQ3 How do symplectic invariants relate to enumerative invariants such as Gromov-Witten invariants and Weil-Petersson volumes?
• RQ4 To what extent is the formalism invariant under symplectic transformations and modular transformations of the curve?
• RQ5 Can the symplectic invariants formalism be extended to non-matrix model systems such as topological strings and integrable systems?

Key findings

• The free energies $F_g$ are symplectic invariants: they remain unchanged under symplectic transformations $dx \wedge dy = d\tilde{x} \wedge d\tilde{y}$.
• The $F_g$ scale as $\lambda^{2-2g}$ under rescaling $y \to \lambda y$, with $F_1$ being logarithmic, confirming their homogeneity of degree $2-2g$.
• The correlation forms $\omega_n^{(g)}$ satisfy a recursive structure based on residues at branch points, with $\omega_1^{(g)}$ related to $F_g$ via $F_g = \frac{1}{2-2g} \sum_i \oint_{a_i} \Phi(z) \omega_1^{(g)}(z)$.
• The formalism reproduces Kontsevich's intersection numbers and Weil-Petersson volumes when applied to the Kontsevich spectral curve.
• For the Kodaira-Spencer theory on a Lagrangian cycle $\mathcal{L}$, the partition function is identified with the tau function $\tau_N$ built from the symplectic invariants $F_g$.
• The method provides a quantum reconstruction of integrable systems from classical spectral curves, with $F_g$ encoding quantum corrections via the Sato formula and Hirota bilinear equations.