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 τ-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.
Similar content being viewed by others
References
A. Hurwitz, Math. Ann., 39, 1–60 (1891); 55, 53–66 (1902).
G. Frobenius, Berl. Ber., 985–1021 (1896).
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.
R. Vakil, “Enumerative geometry of curves via degeneration methods,” Doctoral dissertation, Harvard University, Cambridge, Mass. (1997).
I. P. Goulden and D. M. Jackson, Proc. Amer. Math. Soc., 125, 51–60 (1997); arXiv:math.CO/9903094v1 (1999).
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).
S. M. Natanzon and V. Turaev, Topology, 38, 889–914 (1999).
I. P. Goulden, D. M. Jackson, and A. Vainshtein, Ann. Comb., 4, 27–46 (2000); arXiv:math.AG/9902125v1 (1999).
A. Okounkov, Math. Res. Lett., 7, 447–453 (2000); arXiv:math.AG/0004128v1 (2000).
A. Givental, Moscow Math. J., 1, 551–568 (2001); arXiv:math.AG/0108100v2 (2001).
T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, Invent. Math., 146, 297–327 (2001); arXiv: math.AG/0004096v3 (2000).
S. K. Lando, Russ. Math. Surveys, 57, 463–533 (2002).
A. V. Alexeevski and S. M. Natanzon, Selecta Math., 12, 307–377 (2006); arXiv:math.GT/0202164v2 (2002).
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).
J. Zhou, “Hodge integrals, Hurwitz numbers, and symmetric groups,” arXiv:math.AG/0308024v1 (2003).
A. Okounkov and R. Pandharipande, Ann. Math., 163, 517–560 (2006); arXiv:math.AG/0204305v1 (2002).
T. Graber and R. Vakil, Compos. Math., 135, 25–36 (2003); arXiv:math.AG/0003028v1 (2000).
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).
M. Kazarian, Adv. Math., 221, 1–21 (2009); arXiv:0809.3263v1 [math.AG] (2008).
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.
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).
A. Mironov and A. Morozov, JHEP, 0902, 024 (2009); arXiv:0807.2843v3 [hep-th] (2008).
A. Mironov, A. Morozov, and S. Natanzon, “Integrability and N-point Hurwitz numbers” (to appear).
A. Morozov and Sh. Shakirov, JHEP, 0904, 064 (2009); arXiv:0902.2627v3 [hep-th] (2009).
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).
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).
S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 10, 2015–2051 (1995); arXiv:hep-th/9312210v1 (1993).
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).
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).
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).
A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, and A. Orlov, Nucl. Phys. B, 357, 565–618 (1991).
S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and S. Pakuliak, Nucl. Phys. B, 404, 717–750 (1993); arXiv:hep-th/9208044v1 (1992).
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).
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).
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).
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).
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.
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).
M. Jimbo and T. Miwa, Publ. RIMS Kyoto Univ., 19, 943–1001 (1983).
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.
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).
A. Alexandrov, A. Mironov, and A. Morozov, “Cut-and-join operators, matrix models, and characters” (to appear).
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).
A. Mironov, A. Morozov, and S. Natanzon, “Universal algebras of Hurwitz numbers,” arXiv:0909.1164v2 [math.GT] (2009).
M. Atiyah, Publ. Math. IHES, 68, 175–186 (1988).
R. Dijkgraaf and E. Witten, Comm. Math. Phys., 129, 393–429 (1990).
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 τ -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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 166, No. 1, pp. 3–27, January, 2011.
Rights and permissions
About this article
Cite this article
Mironov, A.D., Morozov, A.Y. & Natanzon, S.M. Complete set of cut-and-join operators in the Hurwitz-Kontsevich theory. Theor Math Phys 166, 1–22 (2011). https://doi.org/10.1007/s11232-011-0001-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-011-0001-6