Grothendieck's Dessins d'Enfants in a Web of Dualities. III

Abstract

We identify the dessin partition function with the partition function of the Laguerre unitary ensemble (LUE). Combined with the result due to Cunden et al on the relationship between the LUE correlators and strictly monotone Hurwitz numbers introduced by Goulden et al, we then establish connection of dessin counting to strictly monotone Hurwitz numbers. We also introduce a correction factor for the dessin/LUE partition function, which plays an important role in showing that the corrected dessin/LUE partition function is a tau-function of the Toda lattice hierarchy. As an application, we use the approach of Dubrovin and Zhang for the computation of the dessin correlators. In physicists' terminology, we establish dualities among dessin counting, generalized Penner model, and ${\mathbb{P}}^1$-topological sigma model.

added references; corrected typos. 32 pages

References in corpus (16)

• On KP-integrable Hurwitz functions
• The matrix model for hypergeometric Hurwitz numbers
• Emergent Geometry and Mirror Symmetry of A Point
• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities
• Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials
• Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals
• Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy
• Emergent Geometry of Matrix Models with Even Couplings
• Dessins d'Enfants, Seiberg-Witten Curves and Conformal Blocks
• Grothendieck's Dessins d'Enfants in a Web of Dualities. II
• Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method
• Integrable structures of specialized hypergeometric tau functions
• Grothendieck's Dessins d'Enfants in a Web of Dualities
• On Equivariant Gromov--Witten Invariants of Resolved Conifold with Diagonal and Anti-Diagonal Actions
• GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back