×

zbMATH — the first resource for mathematics

Complete set of cut-and-join operators in the Hurwitz-Kontsevich theory. (English. Russian original) Zbl 1312.81125
Theor. Math. Phys. 166, No. 1, 1-22 (2011); translation from Teor. Mat. Fiz. 166, No. 1, 3-27 (2011).
Summary: 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.

MSC:
81T45 Topological field theories in quantum mechanics
20C35 Applications of group representations to physics and other areas of science
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
17B69 Vertex operators; vertex operator algebras and related structures
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] A. Hurwitz, Math. Ann., 39, 1–60 (1891); 55, 53–66 (1902). · JFM 23.0429.01 · doi:10.1007/BF01199469
[2] G. Frobenius, Berl. Ber., 985–1021 (1896).
[3] R. Dijkgraaf, ”Mirror symmetry and elliptic curves,” in: The Moduli Spaces of Curves (Progr. Math., Vol. 129, D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, and M. Thaddeus, eds.), Birkhäuser, Boston, Mass. (1995), pp. 149–163. · Zbl 0913.14007
[4] R. Vakil, ”Enumerative geometry of curves via degeneration methods,” Doctoral dissertation, Harvard University, Cambridge, Mass. (1997).
[5] I. P. Goulden and D. M. Jackson, Proc. Amer. Math. Soc., 125, 51–60 (1997); arXiv:math.CO/9903094v1 (1999). · Zbl 0861.05006 · doi:10.1090/S0002-9939-97-03880-X
[6] D. Zvonkine and S. K. Lando, Funct. Anal. Appl., 33, No. 3, 178–188 (1999); ”Counting ramified coverings and intersection theory on spaces of rational functions I (Cohomology of Hurwitz spaces),” arXiv: math.AG/0303218v1 (2003). · Zbl 0953.32007 · doi:10.1007/BF02465202
[7] S. M. Natanzon and V. Turaev, Topology, 38, 889–914 (1999). · Zbl 0929.57001 · doi:10.1016/S0040-9383(98)00036-6
[8] I. P. Goulden, D. M. Jackson, and A. Vainshtein, Ann. Comb., 4, 27–46 (2000); arXiv:math.AG/9902125v1 (1999). · Zbl 0957.58011 · doi:10.1007/PL00001274
[9] A. Okounkov, Math. Res. Lett., 7, 447–453 (2000); arXiv:math.AG/0004128v1 (2000). · Zbl 0969.37033 · doi:10.4310/MRL.2000.v7.n4.a10
[10] A. Givental, Moscow Math. J., 1, 551–568 (2001); arXiv:math.AG/0108100v2 (2001).
[11] T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, Invent. Math., 146, 297–327 (2001); arXiv: math.AG/0004096v3 (2000). · Zbl 1073.14041 · doi:10.1007/s002220100164
[12] S. K. Lando, Russ. Math. Surveys, 57, 463–533 (2002). · Zbl 1054.14037 · doi:10.1070/RM2002v057n03ABEH000511
[13] A. V. Alexeevski and S. M. Natanzon, Selecta Math., 12, 307–377 (2006); arXiv:math.GT/0202164v2 (2002). · Zbl 1158.57304 · doi:10.1007/s00029-006-0028-y
[14] A. V. Alekseevskii and S. M. Natanzon, Russ. Math. Surveys, 61, 767–769 (2006); S. M. Natanzon, ”Disk single Hurwitz numbers,” arXiv:0804.0242v2 [math.GT] (2008); A. Alexeevski and S. Natanzon, ”Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebras of finite groups,” in: Geometry, Topology, and Mathematical Physics (Amer. Math. Soc. Transl. Ser. 2, Vol. 224, V. M. Buchstaber and I. M. Krichever, eds.), Amer. Math. Soc., Providence, R. I. (2008), pp. 1–25; A. V. Alekseevskii, Izv. Math., 72, 627–646 (2008). · Zbl 1131.30018 · doi:10.1070/RM2006v061n04ABEH004345
[15] J. Zhou, ”Hodge integrals, Hurwitz numbers, and symmetric groups,” arXiv:math.AG/0308024v1 (2003).
[16] A. Okounkov and R. Pandharipande, Ann. Math., 163, 517–560 (2006); arXiv:math.AG/0204305v1 (2002). · Zbl 1105.14076 · doi:10.4007/annals.2006.163.517
[17] T. Graber and R. Vakil, Compos. Math., 135, 25–36 (2003); arXiv:math.AG/0003028v1 (2000). · Zbl 1063.14032 · doi:10.1023/A:1021791611677
[18] M. E. Kazarian and S. K. Lando, Izv. Math., 68, 82–113 (2004); arXiv:math.AG/0410388v1 (2004); M. E. Kazarian and S. K. Lando, J. Amer. Math. Soc., 20, 1079–1089 (2007); arXiv:math.AG/0601760v1 (2006).
[19] M. Kazarian, Adv. Math., 221, 1–21 (2009); arXiv:0809.3263v1 [math.AG] (2008). · Zbl 1168.14006 · doi:10.1016/j.aim.2008.10.017
[20] S. Lando, ”Combinatorial facets of Hurwitz numbers,” in: Applications of Group Theory to Combinatorics (J. Koolen, J. H. Kwak, and M.-Y. Xu, eds.), CRC, Boca Raton, Fla. (2008), pp. 109–131. · Zbl 1178.30053
[21] V. Bouchard and M. Mariño, ”Hurwitz numbers, matrix models, and enumerative geometry,” in: From Hodge Theory to Integrability and TQFT: tt*-Geometry (Proc. Sympos. Pure Math., Vol. 78, R. Y. Donagi and K. Wendland, eds.), Amer. Math. Soc., Providence, R. I. (2008), pp. 263–283; arXiv:0709.1458v2 [math.AG] (2007). · Zbl 1151.14335
[22] A. Mironov and A. Morozov, JHEP, 0902, 024 (2009); arXiv:0807.2843v3 [hep-th] (2008). · Zbl 1245.14059 · doi:10.1088/1126-6708/2009/02/024
[23] A. Mironov, A. Morozov, and S. Natanzon, ”Integrability and N-point Hurwitz numbers” (to appear). · Zbl 1306.81291
[24] A. Morozov and Sh. Shakirov, JHEP, 0904, 064 (2009); arXiv:0902.2627v3 [hep-th] (2009). · doi:10.1088/1126-6708/2009/04/064
[25] D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Clarendon, Oxford (1950); M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley, Reading, Mass. (1962); I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, Oxford (1995); W. Fulton, Young Tableaux: With Applications to Representation Theory and Geometry (London Math. Soc. Stud. Texts, Vol. 35), Cambridge Univ. Press, Cambridge (1997).
[26] N. Nekrasov and A. Okounkov, ”Seiberg-Witten theory and random partitions,” in: The Unity of Mathematics (Progr. Math., Vol. 244, P. Etingof, V. Retakh, and I. M. Singer, eds.), Birkhäuser, Boston, Mass. (2006), pp. 525–596; arXiv:hep-th/0306238v2 (2003); A. Marshakov and N. Nekrasov, JHEP, 0701, 104 (2007); arXiv:hep-th/0612019v2 (2006); B. Eynard, J. Stat. Mech., 0807, P07023 (2008); arXiv:0804.0381v2 [math-ph] (2008); A. Klemm and P. Sułkowski, Nucl. Phys. B, 819, 400–430 (2009); arXiv:0810.4944v2 [hep-th] (2008).
[27] S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 10, 2015–2051 (1995); arXiv:hep-th/9312210v1 (1993). · Zbl 0985.81639 · doi:10.1142/S0217751X9500098X
[28] S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Modern Phys. Lett. A, 8, 1047–1061 (1993); arXiv:hep-th/9208046v2 (1992); S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, Theor. Math. Phys., 95, 571–582 (1993). · Zbl 1021.81833 · doi:10.1142/S0217732393002531
[29] M. L. Kontsevich, Funct. Anal. Appl., 25, No. 2, 123–129 (1991); M. L. Kontsevich, Comm. Math. Phys., 147, 1–23 (1992); S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and A. Zabrodin, Phys. Lett. B, 275, 311–314 (1992); arXiv:hep-th/9111037v1 (1991); S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and A. Zabrodin, Nucl. Phys. B, 380, 181–240 (1992); arXiv:hep-th/9201013v1 (1992); A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 274, 280–288 (1992); arXiv:hep-th/9201011v1 (1992); S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Nucl. Phys. B, 397, 339–378 (1993); arXiv:hep-th/9203043v1 (1992); P. Di Francesco, C. Itzykson, and J.-B. Zuber, Comm. Math. Phys., 151, 193–219 (1993); arXiv:hep-th/9206090v1 (1992). · Zbl 0742.14021 · doi:10.1007/BF01079591
[30] A. Yu. Morozov, Sov. Phys. Uspekhi, 37, 1–55 (1994); arXiv:hep-th/9303139v2 (1993); A. Yu. Morozov, ”Matrix models as integrable systems,” arXiv:hep-th/9502091v1 (1995); ”Challenges of matrix models,” in: String Theory: From Gauge Interactions to Cosmology (NATO Sci. Ser. II Math. Phys. Chem., Vol. 208, L. Baulieu, J. de Boer, B. Pioline, and E. Rabinovici, eds.), Springer, Dordrecht (2006), pp. 129-162; arXiv:hepth/0502010v2 (2005); A. Mironov, Internat. J. Mod. Phys. A, 9, 4355–4405 (1994); arXiv:hep-th/9312212v1 (1993); A. D. Mironov, Phys. Part. Nucl., 33, 537–582 (2002).
[31] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, and A. Orlov, Nucl. Phys. B, 357, 565–618 (1991). · doi:10.1016/0550-3213(91)90482-D
[32] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and S. Pakuliak, Nucl. Phys. B, 404, 717–750 (1993); arXiv:hep-th/9208044v1 (1992). · Zbl 1009.81550 · doi:10.1016/0550-3213(93)90595-G
[33] A. Mironov and A. Morozov, Phys. Lett. B, 252, 47–52 (1990); F. David, Modern Phys. Lett. A, 5, 1019–1029 (1990); J. Ambjørn and Yu. M. Makeenko, Modern Phys. Lett. A, 5, 1753–1763 (1990); H. Itoyama and Y. Matsuo, Phys. Lett. B, 255, 202–208 (1991); Yu. Makeenko, A. Marshakov, A. Mironov, and A. Morozov, Nucl. Phys. B, 356, 574–628 (1991). · doi:10.1016/0370-2693(90)91078-P
[34] A. Alexandrov, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 19, 4127–4163 (2004); arXiv:hep-th/0310113v1 (2003); A. S. Alexandrov, A. D. Mironov, and A. Yu. Morozov, Theor. Math. Phys., 142, 349–411 (2005); A. Alexandrov, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 21, 2481–2517 (2006); arXiv:hep-th/0412099v1 (2004); Fortschr. Phys., 53, 512–521 (2005); arXiv:hep-th/0412205v1 (2004); B. Eynard, JHEP, 0411, 031 (2004); arXiv:hep-th/0407261v1 (2004); B. Eynard and N. Orantin, Commun. Number Theory Phys., 1, 347–452 (2007); arXiv:math-ph/0702045v4 (2007); N. Orantin, ”From matrix models’ topological expansion to topological string theories: Counting surfaces with algebraic geometry,” arXiv:0709.2992v1 [hep-th] (2007); A. Alexandrov, A. Mironov, A. Morozov, and P. Putrov, Internat. J. Mod. Phys. A, 24, 4939–4998 (2009); arXiv:0811.2825v2 [hep-th] (2008). · Zbl 1087.81051 · doi:10.1142/S0217751X04018245
[35] A. S. Alexandrov, A. D. Mironov, and A. Yu. Morozov, Theor. Math. Phys., 150, 153–164 (2007); arXiv:hep-th/0605171v1 (2006); A. Alexandrov, A. Mironov, and A. Morozov, Phys. D, 235, 126–167 (2007); arXiv:hep-th/0608228v1 (2006); N. Orantin, ”Symplectic invariants, Virasoro constraints, and Givental decomposition,” arXiv:0808.0635v2 [math-ph] (2008). · Zbl 1118.81057 · doi:10.1007/s11232-007-0011-6
[36] A. B. Zamolodchikov, Theor. Math. Phys., 65, 1205–1213 (1985); V. A. Fateev and A. B. Zamolodchikov, Nucl. Phys. B, 280, 644–660 (1987); A. Gerasimov, A. Marshakov, and A. Morozov, Phys. Lett. B, 236, 269–272 (1990); Nucl. Phys. B, 328, 664–676 (1989); A. Marshakov and A. Morozov, Nucl. Phys. B, 339, 79–94 (1990); A. Morozov, Nucl. Phys. B, 357, 619–631 (1991). · doi:10.1007/BF01036128
[37] M. Sato, ”Soliton equations as dynamical systems on an infinite dimensional Grassmann manifolds,” in: Random Systems and Dynamical Systems (RIMS Kokyuroku, Vol. 439, H. Totoki, ed.), Kyoto Univ., Kyoto (1981), pp. 30–46. · Zbl 0507.58029
[38] G. Segal and G. Wilson, Publ. Math. Publ. IHES, 61, 5–65 (1985); D. Friedan and S. Shenker, Phys. Lett. B, 175, 287–296 (1986); Nucl. Phys. B, 281, 509–545 (1987); N. Ishibashi, Y. Matsuo, and H. Ooguri, Modern Phys. Lett. A, 2, 119–132 (1987); L. Alvarez-Gaumé, C. Gomez, and C. Reina, Phys. Lett. B, 190, 55–62 (1987); A. Morozov, Phys. Lett. B, 196, 325–328 (1987); A. S. Schwarz, Nucl. Phys. B, 317, 323–343 (1989). · Zbl 0592.35112 · doi:10.1007/BF02698802
[39] M. Jimbo and T. Miwa, Publ. RIMS Kyoto Univ., 19, 943–1001 (1983). · Zbl 0557.35091 · doi:10.2977/prims/1195182017
[40] K. Ueno and K. Takasaki, ”Toda lattice hierarchy,” in: Group Representations and Systems of Differential Equations (Adv. Stud. Pure Math., Vol. 4, K. Okamoto, ed.), North-Holland, Amsterdam (1984), pp. 1–95. · Zbl 0577.58020
[41] S. Helgason, Differential Geometry and Symmetric Spaces (Pure Appl. Math., Vol. 12), Acad. Press, New York (1962); D. P. Zelobenko, Compact Lie Groups and their Representations (Transl. Math. Monogr., Vol. 40), Amer. Math. Soc., Providence, R. I. (1973). · Zbl 0111.18101
[42] A. Alexandrov, A. Mironov, and A. Morozov, ”Cut-and-join operators, matrix models, and characters” (to appear).
[43] A. Grothendieck, ”Esquisse d’un programme,” in: Geometric Galois Actions (London Math. Soc. Lect. Note Ser., Vol. 242, L. Schneps and P. Lochak, eds.), Vol. 1, Cambridge Univ. Press, Cambridge (1997), pp. 5–48; G. V. Belyi, Math. USSR-Izv., 14, 247–256 (1980); G. B. Shabat and V. A. Voevodsky, ”Drawing curves over number fields,” in: The Grothendieck Festschrift (Progr. Math., Vol. 88, P. Cartier, L. Illusie, N. M. Katz, G. Laumon, Y. Manin, and K. A. Ribet, eds.), Vol. 3, Birkhäuser, Boston, Mass. (1990), pp. 199–227; A. Levin and A. Morozov, Phys. Lett. B, 243, 207–214 (1990); S. K. Lando and A. K. Zvonkine, Graphs on Surfaces and Their Applications (Encycl. Math. Sci., Vol. 141), Springer, Berlin (2004); N. M. Adrianov, N. Ya. Amburg, V. A. Dremov, Yu. A. Levitskaya, E. M. Kreines, Yu. Yu. Kochetkov, V. F. Nasretdinova, and G. B. Shabat, ”Catalog of dessins d’enfants with 4 edges,” arXiv:0710.2658v1 [math.AG] (2007).
[44] A. Mironov, A. Morozov, and S. Natanzon, ”Universal algebras of Hurwitz numbers,” arXiv:0909.1164v2 [math.GT] (2009).
[45] M. Atiyah, Publ. Math. IHES, 68, 175–186 (1988). · Zbl 0692.53053 · doi:10.1007/BF02698547
[46] R. Dijkgraaf and E. Witten, Comm. Math. Phys., 129, 393–429 (1990). · Zbl 0703.58011 · doi:10.1007/BF02096988
[47] A. Morozov, Sov. Phys. Usp., 35, 671–714 (1992); A. Mironov, A. Morozov, and L. Vinet, Theor. Math. Phys., 100, 890–899 (1994); arXiv:hep-th/9312213v2 (1993); A. Gerasimov, S. Khoroshkin, D. Lebedev, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 10, 2589–2614 (1995); arXiv:hep-th/9405011v1 (1994); S. Kharchev, A. Mironov and A. Morozov, Theor. Math. Phys., 104, 866–878 (1995); arXiv:q-alg/9501013v1 (1995); A. Mironov, ”Quantum deformations of \(\tau\) -functions, bilinear identities, and representation theory,” in: Symmetries and Integrability of Difference Equations (CRM Proc. Lect. Notes, Vol. 9, D. Levi, L. Vinet, and P. Winternitz, eds.), Amer. Math. Soc., Providence, R. I. (1996), pp. 219–237; arXiv:hep-th/9409190v2 (1994); A. D. Mironov, Theor. Math. Phys., 114, 127–183 (1998); arXiv:q-alg/9711006v2 (1997). · doi:10.1070/PU1992v035n08ABEH002255
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.