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## Integrability of Hurwitz Partition Functions

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

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

Research target:
Mathematics

Language:
English

Mironov A., Morozov A., Natanzon S. M., 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

Burman Y. M., Zvonkine D., 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

А.Д. Миронов, А.Ю. Морозов, С.М. Натанзон, 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

Natanzon S. M., Orlov A. Y., 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

Zabrodin A., 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

Kazaryan M., Lando S., 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

Dunin-Barkowski P., Lewanski D., Popolitov A. et al., 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

Gavrylenko P., Journal of High Energy Physics 2015 No. 09 P. 167

We study the solution of the Schlesinger system for the 4-point $\mathfrak{sl}_N$ isomonodromy problem and conjecture an expression for the isomonodromic τ-function in terms of 2d conformal field theory beyond the known N = 2 Painlevé VI case. We show that this relation can be used as an alternative definition of conformal blocks for the ...

Added: October 9, 2015

Bychkov B., Dunin-Barkowski P., Shadrin S., 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

Mironov A., Morozov A., Natanzon S. M., 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

Yurii Burman, Shapiro B., 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

Dunin-Barkowski P., Kramer R., Popolitov A. et al., 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

Alexandrov A., Mironov A., Morozov A. et al., Journal of High Energy Physics 2014 Vol. 11 No. 80 P. 1-31

There is now a renewed interest to a Hurwitz tau-function, counting the
isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and
Grothiendicks’s dessins d’enfant. It is distinguished by belonging to a particular family
of Hurwitz tau-functions, possessing conventional Toda/KP integrability properties. We
explain how the variety of recent observations about this function fits into the ...

Added: December 2, 2014

Bychkov B., 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

Alexeevski A., Natanzon S. M., 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

Gavrylenko P., Marshakov A., Journal of High Energy Physics 2014 No. 5 P. 97

We study the extended prepotentials for the S-duality class of quiver gauge theories, considering them as quasiclassical tau-functions, depending on gauge theory condensates and bare couplings. The residue formulas for the third derivatives of extended prepotentials are proven, which lead to effective way of their computation, as expansion in the weak-coupling regime. We discuss also ...

Added: October 20, 2014

Costa A., Sergey Natanzon, Shapiro B., 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

Mironov A., Morozov A., Natanzon S. M., Journal of High Energy Physics 2011 No. 11(097) P. 1-31

Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized WDVV equations of. We consider the Hurwitz partition functions, associated in this way with the CAA of cut-and-join operators. ...

Added: October 12, 2012

Natanzon S. M., Zabrodin A., 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 ...

Added: April 2, 2014

Zabrodin A., Alexandrov A., Journal of Geometry and Physics 2013 Vol. 67 P. 37-80

We review the formalism of free fermions used for construction of tau-functions of classical integrable hierarchies and give a detailed derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear relations, the generalized Wick theorem and the bosonization rules. We also consider various examples of tau-functions and give their ...

Added: February 16, 2013

Shapiro B., Yurii Burman, 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

Dunin-Barkowski Petr, Kazarian Maxim, Orantin N. et al., 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

Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., 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

Feigin E., European Journal of Combinatorics 2012 Vol. 33 No. 1 P. 1913-1918

It has been shown recently that the normalized median Genocchi numbers are equal to the Euler characteristics of the degenerate flag varieties. The q-analogues of the Genocchi numbers can be naturally defined as the Poincare polynomials of the degenerate flag varieties. We prove that the generating function of the Poincare polynomials can be written as ...

Added: July 18, 2012