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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

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

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English version:
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
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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
    5. JETP Letters, 95:11 (2012), 586–593  mathnet  crossref  isi  elib  elib
    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
    7. Mironov A. Morozov A., “Equations on Knot Polynomials and 3D/5D Duality”, Sixth International School on Field Theory and Gravitation-2012, AIP Conf. Proc., 1483, ed. Rodrigues W. Kerner R. Pires G. Pinheiro C., Amer. Inst. Physics, 2012, 189–211  crossref  adsnasa  isi  scopus
    8. Mironov A., Morozov A., Shakirov Sh., “Torus HOMFLYPT as the Hall-Littlewood polynomials”, J. Phys. A, 45:35 (2012), 355202  crossref  mathscinet  zmath  isi  elib  scopus
    9. Mironov A., Morozov A., Morozov And., “Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid”, J. High Energy Phys., 2012, no. 3, 034  crossref  mathscinet  zmath  isi  scopus
    10. Mironov A., Morozov A., Shakirov Sh., Sleptsov A., “Interplay between Macdonald and Hall-Littlewood expansions of extended torus superpolynomials”, J. High Energy Phys., 2012, no. 5, 070  crossref  mathscinet  zmath  isi  scopus
    11. A. D. Mironov, A. Yu. Morozov, A. V. Sleptsov, “Genus expansion of HOMFLY polynomials”, Theoret. and Math. Phys., 177:2 (2013), 1435–1470  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. Mironov A., Morozov A., Natanzon S., “Cardy-Frobenius Extension of the Algebra of Cut-and-Join Operators”, J. Geom. Phys., 73 (2013), 243–251  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    13. Mironov A. Morozov A. Sleptsov A., “On Genus Expansion of Knot Polynomials and Hidden Structure of Hurwitz Tau-Functions”, Eur. Phys. J. C, 73:7 (2013), 2492  crossref  adsnasa  isi  scopus
    14. Dunin-Barkowski P. Mironov A. Morozov A. Sleptsov A. Smirnov A., “Superpolynomials for Torus Knots From Evolution Induced by Cut-and-Join Operators”, J. High Energy Phys., 2013, no. 3, 021  crossref  mathscinet  zmath  isi  elib  scopus
    15. Itoyama H. Mironov A. Morozov A. Morozov A.N.D., “Eigenvalue Hypothesis for Racah Matrices and Homfly Polynomials for 3-Strand Knots in Any Symmetric and Antisymmetric Representations”, Int. J. Mod. Phys. A, 28:3-4, SI (2013), 1340009  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    16. Mironov A., Morozov A., Natanzon S., “A Hurwitz Theory Avatar of Open-Closed Strings”, Eur. Phys. J. C, 73:2 (2013), 2324  crossref  adsnasa  isi  scopus
    17. Dolotin V. Morozov A., “Introduction to Khovanov Homologies I. Unreduced Jones Superpolynomial”, J. High Energy Phys., 2013, no. 1, 065  crossref  mathscinet  zmath  isi  elib  scopus
    18. JETP Letters, 99:2 (2014), 109–113  mathnet  crossref  crossref  isi  elib  elib
    19. A. Alexandrov, “From Hurwitz numbers to Kontsevich–Witten tau-function: a connection by Virasoro operators”, Lett. Math. Phys., 104:1 (2014), 75–87  crossref  mathscinet  zmath  adsnasa  isi  scopus
    20. V. Dolotin, A. Morozov, “Introduction to Khovanov homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants”, Nuclear Phys. B, 878 (2014), 12–81  crossref  mathscinet  zmath  adsnasa  isi
    21. JETP Letters, 100:4 (2014), 271–278  mathnet  crossref  crossref  isi  elib  elib
    22. Mironov A., Morozov A., Sleptsov A., Smirnov A., “On Genus Expansion of Superpolynomials”, Nucl. Phys. B, 889 (2014), 757–777  crossref  mathscinet  zmath  adsnasa  isi  scopus
    23. Sleptsov A., “Hidden Structures of Knot Invariants”, Int. J. Mod. Phys. A, 29:29 (2014), 1430063  crossref  mathscinet  zmath  adsnasa  isi  scopus
    24. Alexandrov A. Mironov A. Morozov A. Natanzon S., “On KP-Integrable Hurwitz Functions”, J. High Energy Phys., 2014, no. 11, 080  crossref  mathscinet  zmath  isi  scopus
    25. Mironov A. Morozov A. Morozov A., “On Colored Homfly Polynomials For Twist Knots”, Mod. Phys. Lett. A, 29:34 (2014), 1450183  crossref  zmath  adsnasa  isi  scopus
    26. Mironov A., Morozov A., Natanzon S., “Infinite-Dimensional Topological Field Theories From Hurwitz Numbers”, J. Knot Theory Ramifications, 23:6 (2014), 1450033  crossref  mathscinet  zmath  isi  scopus
    27. Anokhina A. Mironov A. Morozov A. Morozov A., “Knot Polynomials in the First Non-Symmetric Representation”, Nucl. Phys. B, 882 (2014), 171–194  crossref  mathscinet  zmath  adsnasa  isi  scopus
    28. Morozov A. Smirnov A., “Towards the Proof of AGT Relations With the Help of the Generalized Jack Polynomials”, Lett. Math. Phys., 104:5 (2014), 585–612  crossref  mathscinet  zmath  adsnasa  isi  scopus
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    30. Galakhov D., Melnikov D., Mironov A., Morozov A., “Knot Invariants From Virasoro Related Representation and Pretzel Knots”, Nucl. Phys. B, 899 (2015), 194–228  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    31. Mironov A., Morozov A., “Towards Effective Topological Field Theory For Knots”, Nucl. Phys. B, 899 (2015), 395–413  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    32. Harnad J., Orlov A.Yu., “Hypergeometric Tau-Functions, Hurwitz Numbers and Enumeration of Paths”, Commun. Math. Phys., 338:1 (2015), 267–284  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    33. Mironov A. Morozov A. Morozov A. Ramadevi P. Singh V.K., “Colored Homfly Polynomials of Knots Presented as Double Fat Diagrams”, J. High Energy Phys., 2015, no. 7, 109  crossref  mathscinet  zmath  isi  scopus
    34. Mironov A. Morozov A. Sleptsov A., “Colored Homfly Polynomials For the Pretzel Knots and Links”, J. High Energy Phys., 2015, no. 7, 069  crossref  mathscinet  isi  scopus
    35. Kononov Ya., Morozov A., “On factorization of generalized Macdonald polynomials”, Eur. Phys. J. C, 76:8 (2016), 424  crossref  isi  scopus
    36. Prochazka T., “$ \mathcal{W} $-symmetry, topological vertex and affine Yangian”, J. High Energy Phys., 2016, no. 10, 077  crossref  mathscinet  isi  elib  scopus
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    39. Zheng Q., “Genus expanded cut-and-join operators and generalized Hurwtiz numbers”, Acta. Math. Sin.-English Ser., 32:9 (2016), 1089–1098  crossref  mathscinet  zmath  isi  scopus
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    41. Smirnov A., “On the Instanton R-matrix”, Commun. Math. Phys., 345:3 (2016), 703–740  crossref  mathscinet  zmath  isi  elib  scopus
    42. Zheng Q., “Shifted genus expanded W algebra and shifted Hurwitz numbers”, J. Math. Phys., 57:5 (2016), 051705  crossref  mathscinet  zmath  isi  elib  scopus
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    46. A. Yu. Orlov, “Hurwitz numbers and products of random matrices”, Theoret. and Math. Phys., 192:3 (2017), 1282–1323  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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