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.

arXiv commentadded references; corrected typos. 32 pages

References in corpus (51)

• Integrability and Matrix Models
• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• Worldsheet Derivation of a Large N Duality
• The Topological CP^1 Model and the Large-N Matrix Integral
• Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
• On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour
• On KP-integrable Hurwitz functions
• Virasoro Symmetries of the Extended Toda Hierarchy
• Virasoro constraints and topological recursion for Grothendieck's dessin counting
• Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths
• The matrix model for dessins d'enfants
• Correlation functions of the KdV hierarchy and applications to intersection numbers over ${\overline{\mathcal{M}}}_{g,n}$
• Cut-and-Join operator representation for Kontsevich-Witten tau-function
• On the Genus Expansion in the Topological String Theory
• The local Gromov-Witten theory of CP^1 and integrable hierarchies
• On Hurwitz numbers and Hodge integrals
• Combinatorial solutions to integrable hierarchies
• The matrix model for hypergeometric Hurwitz numbers
• Integrable hierarchies and the mirror model of local CP1
• Ramifications of Hurwitz theory, KP integrability and quantum curves
• Quantum and Classical Aspects of Deformed $c=1$ Strings.
• Scalar products of symmetric functions and matrix integrals
• Generating functions for weighted Hurwitz numbers
• Weighted Hurwitz numbers and topological recursion
• Moments of random matrices and hypergeometric orthogonal polynomials
• Schur function expansions of KP tau functions associated to algebraic curves
• Emergent Geometry and Mirror Symmetry of A Point
• Algebro-geometric approach in the theory of integrable hydrodynamic type systems
• Generating series for GUE correlators
• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities
• Integer moments of complex Wishart matrices and Hurwitz numbers
• Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers
• Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials
• Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals
• On tau-functions for the Toda lattice hierarchy
• Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy
• Matrix resolvent and the discrete KdV hierarchy
• On Itzykson-Zuber Ansatz
• Hermitian One-Matrix Model and KP Hierarchy
• Dessins d'Enfants, Seiberg-Witten Curves and Conformal Blocks
• Emergent Geometry of Matrix Models with Even Couplings
• 'Universality' of the Ablowitz-Ladik hierarchy
• Grothendieck's Dessins d'Enfants in a Web of Dualities. II
• Genus Expansions of Hermitian One-Matrix Models: Fat Graphs vs. Thin Graphs
• Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method
• Generating weighted Hurwitz numbers
• 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