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## Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics

International Mathematics Research Notices. 2015. Vol. 2015. No. 8. P. 2082-2110.

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial constants and find recurrence relations for them. These results are used to obtain new formulas for the genus 0 double Hurwitz numbers. They can also serve as a starting point for a constructive approach to the Riemann mapping problem and the inverse potential problem in 2D.

Keywords: числа ГурвицаHurwitz numbersконформное отображениеToda hierarchyconformal mapsИерархия Тоды

Publication based on the results of:

Journal of Geometry and Physics 2012 Vol. 62 P. 148-155

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues ...

Added: September 19, 2012

Advances in Mathematics 2015 Vol. 279 P. 67-103

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of ...

Added: September 24, 2015

Теоретическая и математическая физика 2011 Т. 166 № 1 С. 3-27

We define cut-and-join operators in Hurwitz theory for merging two branch points of an arbitrary type. These operators have two alternative descriptions: (1) the GL characters are their eigenfunctions and the symmetric group characters are their eigenvalues; (2) they can be represented as W-type differential operators (in particular, acting on the time variables in the ...

Added: November 24, 2012

Успехи математических наук 2015 Т. 70 № 3 С. 70-106

This paper reviews modern approaches to the construction of formal solutions to integrable hierarchies of mathematical physics whose coefficients are answers to various enumerative problems. The relationship between these approaches and the combinatorics of symmetric groups and their representations is explained. Applications of the results to the construction of efficient computations in problems related to ...

Added: September 21, 2015

Journal of Physics A: Mathematical and Theoretical 2013 Vol. 46 No. 18 P. 185203

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in a transverse direction. Similarly to the Laplacian growth in radial geometry, this problem can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. However, the relevant solution ...

Added: April 29, 2013

European Journal of Combinatorics 2010 Vol. 31 No. 1 P. 129-144

Consider factorizations into transpositions of an n-cycle in the symmetric group Sn. To every such factorization we assign a monomial in variables wij that retains the transpositions used, but forgets their order. Summing over all possible factorizations of n-cycles we obtain a polynomial that happens to admit a closed expression. From this expression we deduce ...

Added: November 7, 2012

Journal of Knot Theory and Its Ramifications 2014 Vol. 23 No. 6 P. 1-16

The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The corresponding Cardy-Frobenius algebra is induced by arbitrary Young diagrams and arbitrary bipartite graphs. It ...

Added: April 2, 2014

Theoretical and Mathematical Physics 2020 Vol. 204 No. 3 P. 1166-1194

To obtain a generating function of the most general form for Hurwitz numbers with arbitrary base surfaceand arbitrary ramification profiles, we consider a matrix model constructed according to a graph on anoriented connected surfaceΣwith no boundary. The vertices of this graph, called stars, are small discs,and the graph itself is a clean dessin d’enfants. We ...

Added: September 27, 2020

Combinatorics of Bousquet-M\'elou--Schaeffer numbers in the light of topological recursion / Cornell University. Series arXiv "math". 2019.

In this paper we prove, in a purely combinatorial way, a structural quasi-polynomiality property for the Bousquet-M\'elou--Schaeffer numbers. Conjecturally, this property should follow from the Chekhov-Eynard-Orantin topological recursion for these numbers (or, to be more precise, the Bouchard-Eynard version of the topological recursion for higher order critical points), which we derive in this paper from ...

Added: October 8, 2019

Функциональный анализ и его приложения 2015 Т. 49 № 2 С. 1-6

The investigation of decompositions of a permutation into a product of permutations
satisfying certain conditions plays a key role in the study of meromorphic functions or, equivalently,
branched coverings of the 2-sphere; it goes back to A. Hurwitz' work in the late nineteenth century.
In 2000 M. Bousquet-Melou and G. Schaeffer obtained an elegant formula for the number ...

Added: July 18, 2015

Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type / Cornell University. Series math "arxiv.org". 2020. No. 2012.14723.

We study the n-point differentials corresponding to Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions), with an emphasis on their ℏ2-deformations and expansions.
Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We ...

Added: April 20, 2022

On Hurwitz--Severi numbers / Cornell University. Series math "arxiv.org". 2016. No. 06935.

For a point p in a complex projective plane and a triple (g,d,l) of non-negative
integers we define a plane Hurwitz number of the Severi variety
W_{g,d,l} consisting of all reduced irreducible plane curves of
genus g and degree d+l having an l-fold node at p and at
most ordinary nodes as singularities at the other points. In the ...

Added: July 5, 2016

Explicit closed algebraic formulas for Orlov-Scherbin n-point functions / Cornell University. Series "Working papers by Cornell University". 2020.

We derive a new explicit formula in terms of sums over graphs for the n-point correlation functions of general formal weighted double Hurwitz numbers coming from the Orlov-Scherbin partition functions. Notably, we use the change of variables suggested by the associated spectral curve, and our formula turns out to be a polynomial expression in a certain ...

Added: October 6, 2020

Journal of Physics: Conference Series 2016 Vol. 670 P. 1-6

We describe all formal symmetric solutions of dispersionless 2D Toda hierarchy. This classication we use for solving of two classical problems: 1) The calculation of conformal mapping of an arbitrary simply connected domain to the standard disk; 2) Calculation of 2-Hurwitz numbers of genus 0. ...

Added: February 10, 2016

American Mathematical Society Translations 2014 Vol. 234 P. 1-12

In 2001 Ivanov and Kerov associated with the infinite permutation group S∞ certain commutative associative algebra A∞ called the algebra of conjugacy classes of partial elements. A standard basis of A∞ islabeled by Yang diagrams of all orders. Mironov, Morozov, Natanzon, 2012, have proved that the completion of A∞ is isomorphic to the direct product ...

Added: April 2, 2014

European Journal of Combinatorics 2020 Vol. 90 P. 103184

In this paper we prove, in a purely combinatorial-algebraic way, a structural quasi-polynomiality property for the Bousquet-Mélou–Schaeffer numbers. Conjecturally, this property should follow from the Chekhov–Eynard–Orantin topological recursion for these numbers (or, to be more precise, the Bouchard–Eynard version of the topological recursion for higher order critical points), which we derive in this paper from ...

Added: September 22, 2020

Journal of Geometry and Physics 2019 Vol. 137 P. 1-6

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of ...

Added: February 20, 2019

Annales Academiae Scientiarum Fennicae Mathematica 2018 Vol. 43 P. 349-363

In this article, to each generic real meromorphic function (i.e., having only simple branch points in the appropriate sense) we associate a certain combinatorial gadget which we call the park of a function. We show that the park determines the topological type of the generic real meromorphic function and the set of parks produce an stratification ...

Added: March 4, 2018

Journal of Physics A: Mathematical and Theoretical 2012 No. 45 P. 1-10

We construct partition functions that are tau-functions of integrable hierarchies. ...

Added: September 19, 2012

Journal of London Mathematical Society 2015 Vol. 92 No. 3 P. 547-565

In this paper, we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard and Orantin, where the main new step compared to the existing proofs is ...

Added: November 16, 2015

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 2019 Vol. XIX No. 1 P. 155-167

For a point p of the complex projective plane and a triple (g,d,l) of non-negative integers we define a Hurwitz--Severi number H(g,d,l) as the number of generic irreducible plane curves of genus g and degree d+l having an l-fold node at p and at most ordinary nodes as singularities at the other points, such that the ...

Added: April 14, 2017

Springer, 2019

This book collects papers based on the XXXVI Białowieża Workshop on Geometric Methods in Physics, 2017. The Workshop, which attracts a community of experts active at the crossroads of mathematics and physics, represents a major annual event in the field. Based on presentations given at the Workshop, the papers gathered here are previously unpublished, at ...

Added: November 12, 2018

Hurwitz numbers and matrix integrals labeled with chord diagrams / . 2018. No. 5.

Abstract: We shall consider the product of complex random matrices from the independent complex Ginibre ensembles. The product includes complex matrices
Zi,Z†i,i=1,…,n and 2n sources (complex matrices Ci and C∗i). Any such product can be represented by a chord diagram that encodes the order of the matrices in the product. We introduce the Euler characteristic E∗ of ...

Added: November 12, 2018

Функциональный анализ и его приложения 2019 Т. 53 № 1 С. 16-30

В данной работе мы получили новые формулы для степеней стратов пространств Гурвица рода 0, отвечающих функциям с двумя непростыми критическими значениями с предписанными разбиениями кратностей прообразов. При этом один из прообразов имеет произвольную кратность, а другой кратность коразмерности 1. В качестве следствий мы получили новые выражения для серий двойных чисел Гурвица. ...

Added: October 20, 2016