Algebraic methods in random matrices and enumerative geometry

TL;DR

This paper reviews symplectic invariants in algebraic geometry and their applications to matrix models, enumerative geometry, and string theory, highlighting their properties and extending their use beyond traditional matrix model contexts.

Contribution

It introduces a unified approach using symplectic invariants for solving loop equations and applies this method to diverse fields like matrix models, geometry, and stochastic processes.

Findings

Symplectic invariants are defined for spectral curves and possess key properties like invariance and modularity.

The method extends beyond matrix models to applications in geometry and physics.

Examples include enumeration of maps, algebraic geometry, and non-intersecting Brownian motions.

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,...