Algebraic methods in random matrices and enumerative geometry

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,...

review article, Latex, 139 pages, many figures

References in corpus (5)

• A Perturbative Window into Non-Perturbative Physics
• Les Houches lectures on matrix models and topological strings
• Matrix Quantum Mechanics and Two-dimensional String Theory in Non-trivial Backgrounds
• Large N expansion of the 2-matrix model, multicut case
• The 2-matrix model, biorthogonal polynomials, Riemann-Hilbert problem, and algebraic geometry