Algebraic methods in random matrices and enumerative geometry

TL;DR

<3-5 sentence high-level summary>The paper surveys the symplectic invariant (Eynard–Orantin) framework for solving loop equations across matrix models and beyond, by associating to any regular spectral curve a hierarchy of invariants $F_g$ and correlators $\omega_n^{(g)}$ that are unified under symplectic transformations via a topological recursion. It develops the geometric toolkit (spectral curves, Bergmann kernels, Abel map, branch points, etc.), establishes key properties (homogeneity, modularity, integrability, Virasoro constraints, holomorphic anomaly), and links these invariants to matrix models (1- and 2-matrix, chains, external fields) as well as to problems in enumerative geometry and nonperturbative string theory. The work highlights deep connections to KP/Hirota integrability, Kontsevich-type structures, and BCOV-type holomorphic anomaly, and demonstrates broad applicability to maps enumeration, topological strings, and Brownian motion problems. Overall, it provides a comprehensive, broadly applicable toolkit for computing high-genus corrections and understanding the algebraic structure underlying diverse problems in mathematical physics.

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,...

Figures (3)

• Figure 1: Example of one starting point at 0 and two ending points at +1 and -1 half of the movers going to +1 and the other half to -1( the particles go from the left to the right). In the large $N$ limit, the Brownian movers fill the sector of the space time delimited by the figure. One sees that before $t={1 \over 2}$, they are distributed along a unique segment which splits into two disjoint segments for $t>{1 \over 2}$.
• Figure 2: For large times, the spectral curve is composed of $q+1$ sheets linked by $q$ real cuts $[z_{2i-1},z_{2i}]$. These cuts can be seen as the section of the support of the Brownian movers at fixed time. In the example depicted in figure \ref{['brown1']}, it corresponds to times greater than ${1 \over 2}$.
• Figure 3: For intermediate times some of the real cuts have merged and created imaginary cuts linking two non-physical sheets. In this example, $z_2$ and $z_3$ have collapsed and given rise to the imaginary cut $[z,\overline{z}]$.

Theorems & Definitions (75)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 3.1
• Theorem 3.1
• Lemma 3.1