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TMF, 2011, Volume 166, Number 1, Pages 3–27 (Mi tmf6592)  

This article is cited in 58 scientific papers (total in 58 papers)

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

A. D. Mironovab, A. Yu. Morozovb, S. M. Natanzoncd

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

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 $GL$ 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 Hurwitz–Kontsevich $\tau$-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


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Theoretical and Mathematical Physics, 2011, 166:1, 1–22

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Received: 07.06.2010

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
\by A.~D.~Mironov, A.~Yu.~Morozov, S.~M.~Natanzon
\paper Complete set of cut-and-join operators in the~Hurwitz--Kontsevich theory
\jour TMF
\yr 2011
\vol 166
\issue 1
\pages 3--27
\jour Theoret. and Math. Phys.
\yr 2011
\vol 166
\issue 1
\pages 1--22

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    This publication is cited in the following articles:
    1. Brown T.W., “Complex matrix model duality”, Phys. Rev. D, 83:8 (2011), 085002  crossref  adsnasa  isi  scopus
    2. Mironov A., Morozov A., Natanzon S., “Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations”, J. High Energy Phys., 2011, no. 11, 097  crossref  mathscinet  zmath  isi  elib  scopus
    3. Alexandrov A., Mironov A., Morozov A., Natanzon S., “Integrability of Hurwitz partition functions”, J. Phys. A, 45:4 (2012), 045209  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. D. V. Galakhov, A. D. Mironov, A. Yu. Morozov, A. V. Smirnov, “Three-dimensional extensions of the Alday–Gaiotto–Tachikawa relation”, Theoret. and Math. Phys., 172:1 (2012), 939–962  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
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    6. A. Yu. Morozov, “Challenges of $\beta$-deformation”, Theoret. and Math. Phys., 173:1 (2012), 1417–1437  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
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