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A Hurwitz theory avatar of open-closed strings
The European Physical Journal C - Particles and Fields. 2013. Vol. 73. No. 2. P. 1-10.
We review and explain an infinite-dimensional counterpart of the Hurwitz theory realization (Alexeevski and Natanzon, Math. Russ. Izv. 72:3-24, 2008) of algebraic open-closed-string model à la Moore and Lazaroiu, where the closed and open sectors are represented by conjugation classes of permutations and the pairs of permutations, i.e. by the algebra of Young diagrams and bipartite graphs, respectively. An intriguing feature of this Hurwitz string model is the coexistence of two different multiplications, reflecting the deep interrelation between the theory of symmetric and linear groups, S∞ and GL(∞).
Keywords: числа ГурвицаHurwitz numbersтопологическая теория полятеория струнtopological field theorystring theory
Publication based on the results of:
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., / 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
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
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
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
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., 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
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
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
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
Alexandrov A., Mironov A., Morozov A. et al., 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
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
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
Natanzon S. M., 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
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
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
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
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
Belavin V., Geiko R., Journal of High Energy Physics 2018 No. 112 P. 1-23
We develop a recursive approach to computing Neveu-Schwarz conformal blocks associated with n-punctured Riemann surfaces. This work generalizes the results of [1] obtained recently for the Virasoro algebra. The method is based on the analysis of the analytic properties of the superconformal blocks considered as functions of the central charge c. It consists of two main ...
Added: August 24, 2018
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
Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., / 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
Lvov B. G., Vetrov V. A., Вакуумная техника и технология 2007 Т. 17 № 1 С. 3-6
The method of the multichamber vacuum systems mathematical models forming on basis of the element-node structures method is considered in this paper. Offered method allows to simulate vacuum pumping under free vacuum system structure variations. ...
Added: September 4, 2012
М. : НИУ ВШЭ, 2014
Сборник трудов участников Студенческой научно-практической конференции 11-13 марта 2014 г. факультета бизнес-информатики НИУ ВШЭ. ...
Added: September 28, 2014
Gafarov E., Lazarev A. A., Werner F., Annals of Operations Research 2012 Vol. 196 No. 1 P. 247-261
We consider the problem of maximizing total tardiness on a single machine, where the first job starts at time zero and idle times between the processing of jobs are not allowed.We present a modification of an exact pseudo-polynomial algorithm based on a graphical approach, which has a polynomial running time. This result settles the complexity ...
Added: November 24, 2012