A. D. Mironov, A. Yu. Morozov, S. M. Natanzon, “Complete set of cut-and-join operators in the Hurwitz–Kontsevich theory”, TMF, 166:1 (2011), 3–27; Theoret. and Math. Phys., 166:1 (2011), 1

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Teoreticheskaya i Matematicheskaya Fizika, 2011, Volume 166, Number 1, Pages 3–27
DOI: https://doi.org/10.4213/tmf6592 (Mi tmf6592)

This article is cited in 113 scientific papers (total in 113 papers)

Complete set of cut-and-join operators in the Hurwitz–Kontsevich theory

A. D. Mironov^ab, A. Yu. Morozov^b, S. M. Natanzon^cd

^a Lebedev Physical Institute, RAS, Moscow, Russia
^b Institute for Theoretical and Experimental Physics, Moscow, Russia
^c Higher School of Economics, Moscow, Russia
^d Institute of Physico-Chemical Biology, Lomonosov Moscow State University, Moscow, Russia

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DOI: https://doi.org/10.4213/tmf6592

Abstract: 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

G L

$GL$ characters are their eigenfunctions and the symmetric group characters are their eigenvalues; (2) they can be represented as

W

$W$ -type differential operators (in particular, acting on the time variables in the Hurwitz–Kontsevich

τ

$\tau$ -function). The operators have the simplest form when expressed in terms of the Miwa variables. They form an important commutative associative algebra, a universal Hurwitz algebra, generalizing all group algebra centers of particular symmetric groups used to describe the universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams evaluated on the product of all diagrams characterizing particular ramification points of the branched covering.

Keywords: matrix model, Hurwitz number, symmetric group character.

Received: 07.06.2010

English version:
Theoretical and Mathematical Physics, 2011, Volume 166, Issue 1, Pages 1–22
DOI: https://doi.org/10.1007/s11232-011-0001-6

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: A. D. Mironov, A. Yu. Morozov, S. M. Natanzon, “Complete set of cut-and-join operators in the Hurwitz–Kontsevich theory”, TMF, 166:1 (2011), 3–27; Theoret. and Math. Phys., 166:1 (2011), 1–22

Citation in format AMSBIB

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This publication is cited in the following 113 articles:

Chuanzhong Li, A. Mironov, A.Yu. Orlov, “Hopf link invariants and integrable hierarchies”, Physics Letters B, 860 (2025), 139170
A. Mironov, A. Morozov, A. Popolitov, “Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems”, Nuclear Physics B, 1012 (2025), 116809
A. Mironov, A. Morozov, “On the status of DELL systems”, Nuclear Physics B, 999 (2024), 116448
Ya. Drachov, A. Mironov, A. Popolitov, “W1+∞ and W˜ algebras, and Ward identities”, Physics Letters B, 849 (2024), 138426
Dmitry Galakhov, Alexei Morozov, Nikita Tselousov, “Toward a theory of Yangians”, Phys. Rev. D, 109:6 (2024)
Dmitry Galakhov, Alexei Morozov, Nikita Tselousov, “Simple representations of BPS algebras: the case of $Y(\widehat{\mathfrak {gl}}_2)$”, Eur. Phys. J. C, 84:6 (2024)
Dmitry Galakhov, Alexei Morozov, Nikita Tselousov, “Macdonald polynomials for super-partitions”, Physics Letters B, 856 (2024), 138911
A. Mironov, A. Morozov, A. Popolitov, Sh. Shakirov, “Deformation of superintegrability in the Miwa-deformed Gaussian matrix model”, Phys. Rev. D, 110:4 (2024)
A. Morozov, A. Oreshina, “On character expansion and Gaussian regularization of Itzykson-Zuber measure”, Physics Letters B, 857 (2024), 139006
Yaroslav Drachov, “Generalized $\widetilde{W}$ algebras”, Eur. Phys. J. C, 84:10 (2024)
A. Mironov, A. Morozov, A. Popolitov, “Cherednik-Mehta-Macdonald formula as a superintegrability property of a unitary model”, Phys. Rev. D, 110:12 (2024)
A. Yu. Orlov, “Integrals of tau functions: A round dance tau function and multimatrix integrals”, Theoret. and Math. Phys., 215:3 (2023), 784–792
A. Yu. Orlov, “Polygon gluing and commuting bosonic operators”, Theoret. and Math. Phys., 216:2 (2023), 1110–1118
Yaroslav Drachov, Aleksandr Zhabin, “Genus expansion of matrix models and $\hbar $ expansion of BKP hierarchy”, Eur. Phys. J. C, 83:5 (2023)
Na Wang, Can Zhang, Ke Wu, “3D boson representation of affine Yangian of gl(1) and 3D cut-and-join operators”, Journal of Mathematical Physics, 64:11 (2023)
A. Mironov, A. Morozov, “On combinatorial generalization(s) of Borel transform: Averaging method in combinatorics of symmetric polynomials”, Physics Letters B, 843 (2023), 138037
A. Morozov, N. Tselousov, “3-Schurs from explicit representation of Yangian $ \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) $. Levels 1–5”, J. High Energ. Phys., 2023:11 (2023)
Dmitry Galakhov, Alexei Morozov, Nikita Tselousov, “Super-Schur polynomials for Affine Super Yangian Y($ \hat{\mathfrak{gl}} $1|1)”, J. High Energ. Phys., 2023:8 (2023)
Yannick Mvondo-She, “From Hurwitz numbers to Feynman diagrams: Counting rooted trees in log gravity”, Nuclear Physics B, 995 (2023), 116350
A. Mironov, A. Morozov, “Many-body integrable systems implied by WLZZ models”, Physics Letters B, 842 (2023), 137964

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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