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

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Cited by in corpus (25)

• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems
• Matrix Models for Random Partitions
• 3d $\mathcal{N}=4$ OPE Coefficients from Fermi Gas
• Modular invariant holographic correlators for $\mathcal{N}=4$ SYM with general gauge group
• An analysis of the intermediate field theory of $T^4$ tensor model
• A short overview of the "Topological recursion"
• Universal scaling limits of matrix models, and (p,q) Liouville gravity
• The Real Topological String on a local Calabi-Yau
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• Topological Recursion in The Ramond Sector
• Deformed Cauchy random matrix ensembles and large $N$ phase transitions
• Tracy-Widom GUE law and symplectic invariants
• Seiberg-Witten equations and non-commutative spectral curves in Liouville theory
• Super Topological Recursion and Gaiotto Vectors For Superconformal Blocks
• FZZT Brane Relations in the Presence of Boundary Magnetic Fields
• Large genus behavior of topological recursion
• A Givental-like Formula and Bilinear Identities for Tensor Models
• $\mathcal{N}=1$ Super Topological Recursion
• Topological Recursion, Airy structures in the space of cycles
• A Riemann-Hilbert problem for equations of Painlevé type in the one matrix model with semi-classical potential
• The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants
• A New Class of Higher Quantum Airy Structures as Modules of $\mathcal{W}({\mathfrak{gl}}_r)$-Algebras
• New Algebraic Structures from Hermitian One-Matrix Model
• Topological expansion of unitary integrals and maps
• Quantum chaos in 2D gravity