Around spin Hurwitz numbers. (English) Zbl 1477.05006
Summary: We present a review of the spin Hurwitz numbers, which count the ramified coverings with spin structures. It is known that they are related to the characters of the Sergeev group and to the \(Q\) Schur functions. This allows one to put the whole story into the context of matrix models and integrable hierarchies. The generating functions of the spin Hurwitz numbers \(\tau^{\pm}\) are hypergeometric \(\tau\)-functions of the BKP integrable hierarchy; we present their fermionic realization. The cut-and-join equation in the form of a heat equation is written down. We explain, how a special \(d\)-soliton \(\tau\)-functions of KdV and Veselov-Novikov hierarchies generate the spin Hurwitz numbers \(H^{\pm} \left( \Gamma^b_d \right)\) and \(H^{\pm} \left( \Gamma^b_d,\Delta \right)\). We present the well-known Kontsevich matrix integral as the BKP \(\tau\)-function in the form of special neutral fermion vacuum expectation values (few different ones). We also explain how to rewrite certain BKP \(\tau\)-functions (including the Kontsevich one) as the hypergeometric BKP \(\tau\)-functions using certain relations between the projective Schur functions.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 05E05 | Symmetric functions and generalizations |
| 14H81 | Relationships between algebraic curves and physics |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
References:
| [1] | Alexandrov, A.: Intersection numbers on \(\overline{\cal{M}}_{g,n}\) and BKP hierarchy. arXiv:2012.07573 |
| [2] | Alexandrov, A.: KdV solves BKP. arXiv:2012.10448 |
| [3] | Alexandrov, A.; Mironov, A.; Morozov, A., Physica D, 235, 126-167 (2007) · Zbl 1183.81082 · doi:10.1016/j.physd.2007.04.018 |
| [4] | Alexandrov, A.; Mironov, A.; Morozov, A., BGWM as second constituent of complex matrix model, JHEP, 12, 053 (2009) · doi:10.1088/1126-6708/2009/12/053 |
| [5] | Alexandrov, A.; Mironov, A.; Morozov, A.; Natanzon, S., Integrability of Hurwitz partition functions. I. Summary, J. Phys. A, 045209, 045 (2012) · Zbl 1259.37040 |
| [6] | A. Alexandrov, A. Mironov, A. Morozov and S. Natanzon, “On KP-integrable Hurwitz functions” JHEP 11 (2014) 080 · Zbl 1333.81192 |
| [7] | Aokage, K.; Shinkawa, E.; Yamada, HF, Pfaffian identities and Virasoro operators, Lett. Math. Phys., 110, 1381-1389 (2020) · Zbl 1487.17049 · doi:10.1007/s11005-020-01265-1 |
| [8] | Atiyah, M., Riemann surfaces and spin structures, Ann. Sci. Ecole Norm. Sup., 4, 47-56 (1971) · Zbl 0212.56402 · doi:10.24033/asens.1205 |
| [9] | Burnside, W., Theory of Groups of Finite Order (1911), Cambridge: Cambridge University Press, Cambridge · JFM 42.0151.02 |
| [10] | Chekhov, L.; Makeenko, Y., A hint on the external field problem for matrix models, Phys. Lett. B, 278, 271-278 (1992) · doi:10.1016/0370-2693(92)90192-7 |
| [11] | Chekhov, L.; Makeenko, Y., The Multicritical Kontsevich-Penner Model, Mod. Phys. Lett. A, 7, 1223-1236 (1992) · Zbl 1021.81704 · doi:10.1142/S0217732392003700 |
| [12] | Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Jimbo, M., Miwa, T. (eds.). Nonlinear integrable systems—classical theory and quantum theory, pp. 39-120. World Scientific (Singapore) (1983) · Zbl 0536.00011 |
| [13] | Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Publ. RIMS, Kyoto Univ. 18, 1077-1110 (1982) · Zbl 0571.35103 |
| [14] | Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Transformation groups for soliton equations IV. A new hierarchy of soliton equations of KP type, Physica, 4D, 343-365 (1982) · Zbl 0571.35100 |
| [15] | Dijkgraaf, R., Mirror symmetry and elliptic curves, The moduli spaces of curves, Prog. Math., 129, 149-163 (1995) · Zbl 0913.14007 |
| [16] | Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry-Methods and Applications: Part I. The Geometry of Surfaces, Transformation Groups, and Fields (Springer-Verlag New York, 1984) · Zbl 0529.53002 |
| [17] | Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry-Methods and Applications: Part II: The Geometry and Topology of Manifolds (Springer Science and Business Media, 2012) |
| [18] | Dubrovin, B.A.: Symplectic field theory of a disk, quantum integrable systems, and Schur polynomials. arxiv:1407.5824 · Zbl 1342.81626 |
| [19] | Eskin, A.; Okounkov, A.; Pandharipande, R., The theta characteristic of a branched covering, Adv. Math., 217, 873-888 (2008) · Zbl 1157.14014 · doi:10.1016/j.aim.2006.08.001 |
| [20] | Eynard, B.; Orantin, N., Invariants of algebraic curves and topological expansion, Commun. Numer. Theor. Phys., 1, 347-452 (2007) · Zbl 1161.14026 · doi:10.4310/CNTP.2007.v1.n2.a4 |
| [21] | Frobenius, G., Ueber Gruppencharaktere, Berl. Ber., 1896, 985-1021 (1896) · JFM 27.0092.01 |
| [22] | Fukuma, M.; Kawai, H.; Nakayama, R., Continuum Schwinger-Dyson equations and universal structures in two-dimensional quantum gravity, Int. J. Mod. Phys. A, 6, 1385-1406 (1991) · doi:10.1142/S0217751X91000733 |
| [23] | Goulden, D.; Jackson, DM; Vainshtein, A., The number of ramified coverings of the Sphereby torus and surfaces of higher genera, Ann. Comb., 4, 27-46 (2000) · Zbl 0957.58011 · doi:10.1007/PL00001274 |
| [24] | Gunningham, S., Spin Hurwitz numbers and topological quantum field theory, Geom. Topol., 20, 1859-1907 (2016) · Zbl 1347.81070 · doi:10.2140/gt.2016.20.1859 |
| [25] | Harnad, J.; van de Leur, JW; Orlov, AY, Multiple sums and integrals as neutral BKP \(\tau \)-functions, Theor. Math. Phys., 168, 951-962 (2011) · doi:10.1007/s11232-011-0077-z |
| [26] | Hurwitz, A., Ueber die Nullstellen der hypergeometrischen Reihe, Math. Ann., 38, 452-45 (1891) · JFM 23.0454.01 · doi:10.1007/BF01199430 |
| [27] | Jimbo, M.; Miwa, T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci., 19, 943-1001 (1983) · Zbl 0557.35091 · doi:10.2977/prims/1195182017 |
| [28] | Kac, V.; van de Leur, J., The geometry of spinors and the multicomponent BKP and DKP hierarchies, CRM Proc. Lect. Notes, 14, 159-202 (1998) · Zbl 0924.35114 · doi:10.1090/crmp/014/13 |
| [29] | Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A.; Zabrodin, A., Unification of all string models with \(C < 1\), Phys. Lett. B, 275, 311 (1992) · doi:10.1016/0370-2693(92)91595-Z |
| [30] | Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A.; Zabrodin, A., Towards unified theory of 2-d gravity, Nucl. Phys. B, 380, 181 (1992) · doi:10.1016/0550-3213(92)90521-C |
| [31] | Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A., Generalized Kontsevich model versus Toda hierarchy and discrete matrix models, Nucl. Phys. B, 397, 339-378 (1993) · doi:10.1016/0550-3213(93)90347-R |
| [32] | Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A., Generalized Kazakov-Migdal-Kontsevich model: group theory aspects, Int. J. Mod. Phys. A, 10, 2015 (1995) · Zbl 0985.81639 · doi:10.1142/S0217751X9500098X |
| [33] | Knizhnik, V.: Multiloop amplitudes in the theory of quantum strings and complex geometry. Sov. Phys. Uspekhi 32 (1989) 945 (in Russian Edition: vol.159, p. 451) |
| [34] | Kodama, Y., Young diagrams and N-soliton solutions of the KP equation, J. Phys. A, 37, 11169-11190 (2004) · Zbl 1086.35093 · doi:10.1088/0305-4470/37/46/006 |
| [35] | Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys., 147, 1 (1992) · Zbl 0756.35081 · doi:10.1007/BF02099526 |
| [36] | Lee, J.; Parker, TH, A structure theorem for Gromov-Witten invariants of Kahler surfaces, J. Differ. Geom., 77, 3, 483-513 (2007) · Zbl 1130.53059 · doi:10.4310/jdg/1193074902 |
| [37] | Lee, J.: A note on Gunningham’s formula. arXiv:1407.0055 · Zbl 1471.14115 |
| [38] | Lee, J., A square root of Hurwitz numbers, Manuscr. Math., 162, 1-2, 99-113 (2020) · Zbl 1439.14117 · doi:10.1007/s00229-019-01113-0 |
| [39] | Macdonald, IG, Symmetric Functions and Hall Polynomials (1995), Oxford: Oxford University Press, Oxford · Zbl 0824.05059 |
| [40] | Maulik, D., Pandharihande, R.: Pure and Applied Mathematics Quarterly 4 no 2. Special Issue: in honor of Fedor Bogomolov I, 469-500 (2008) · Zbl 1156.14042 |
| [41] | Manakov, S.V., Novikov, S.P., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons, ed. Nauka, S. P. Novikov (1979) · Zbl 0598.35003 |
| [42] | Mironov, A., Morozov, A.: Superintegrability of Kontsevich matrix model. arXiv:2011.12917 |
| [43] | Mironov, A., Morozov, A., Semenoff, G.: Unitary matrix integrals in the framework of Generalized Kontsevich Model 1. Brezin-Gross-Witten Model. Int. J. Mod. Phys. A10, 2015 (1995) |
| [44] | Mironov, AD; Morozov, AY; Natanzon, S., Cut-and-join structure and integrability for spin Hurwitz numbers, Eur. Phys. J. C, 80, 97 (2020) · doi:10.1140/epjc/s10052-020-7650-2 |
| [45] | Mironov, A.; Morosov, A.; Natanzon, S., Algebra of differential operators associated with Young diagrams, J. Geom. Phys., 62, 148-155 (2012) · Zbl 1242.22008 · doi:10.1016/j.geomphys.2011.09.001 |
| [46] | Mironov, A.; Morozov, A.; Natanzon, S., Complete set of cut-and-join operators in the Hurwitz-Kontsevich theory, Theor. Math. Phys., 166, 1-22 (2011) · Zbl 1312.81125 · doi:10.1007/s11232-011-0001-6 |
| [47] | Mironov, A.; Morozov, A.; Sleptsov, A., On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions, Eur. Phys. J. C, 73, 2492 (2013) · doi:10.1140/epjc/s10052-013-2492-9 |
| [48] | Mironov, A.; Morozov, A., On the complete perturbative solution of one-matrix models, Phys. Lett. B, 771, 503 (2017) · Zbl 1372.81111 · doi:10.1016/j.physletb.2017.05.094 |
| [49] | Mironov, A.; Morozov, A., Sum rules for characters from character-preservation property of matrix models, JHEP, 1808, 163 (2018) · Zbl 1396.81203 · doi:10.1007/JHEP08(2018)163 |
| [50] | Mironov, A., Morozov, A.: To appear |
| [51] | Morozov, A., Phys. Usp. (UFN), 37, 1 (1994) · Zbl 1217.81130 · doi:10.1070/PU1994v037n01ABEH000001 |
| [52] | Mironov, A., Int. J. Mod. Phys. A, 9, 4355 (1994) · Zbl 0988.81538 · doi:10.1142/S0217751X94001746 |
| [53] | Mironov, A., Phys. Part. Nucl., 33, 537 (2002) |
| [54] | Mumford, D.: Theta characteristics of an algebraic curve. Ann. Sci. Ecole Norm. Sup. (4) 4 (1971) 181-192 · Zbl 0216.05904 |
| [55] | Nimmo, J.J.C.: Hall-Littlewood symmetric functions and the BKP equation. J. Phys. A 23, 751-760 · Zbl 0721.35069 |
| [56] | Nimmo, J.J.C., Orlov, A.Y.: A relationship between rational and multi-soliton solutions of the BKP hierarchy. Glasgow Math. J. 47 (A), 149-168 (2005) · Zbl 1076.37063 |
| [57] | Okounkov, A., Toda equations for Hurwitz numbers, Math. Res. Lett., 7, 447-45 (2000) · Zbl 0969.37033 · doi:10.4310/MRL.2000.v7.n4.a10 |
| [58] | Orlov, AY, Hypergeometric Functions Related to Schur Q-polynomials and the BKP Equation, Theor. Math. Phys., 137, 1574-1589 (2003) · Zbl 1178.33015 · doi:10.1023/A:1027370004436 |
| [59] | Orlov, A.; Shcherbin, DM, Hypergeometric solutions of soliton equations, Theor. Math. Phys., 128, 906-926 (2001) · Zbl 0992.37063 · doi:10.1023/A:1010402200567 |
| [60] | Orlov, A., Shcherbin, D.M.: arXiv:nlin/0001001 |
| [61] | Orlov, A.Y.: Vertex operator, \({\bar{\partial }} \)-problem, symmetries, variational identities and Hamiltonian formalism for 2+1 D integrable systems. Nonlinear and Turbulent Processes in Physics 1987. In: V. Baryakhtar, V., Zakharov, V.E. (eds.) Singapore (1988) |
| [62] | Orlov, A.Y.: Soliton collapse in integrable models. Preprint EAiE N 221 Novosibirsk (1983) |
| [63] | Pogrebkov, A.K., Sushko, V.N.: Quantization of the \((sin\psi )_2\) interaction in terms of fermion variables. Translated from Teoreticheskaya i Mathematicheskaya Fizika, 24 (1975) 425-429 (September, 1975). The original paper was submitted on May 15, 1975 |
| [64] | Sergeev, A.: The tensor algebra of the identity representation as a module over the Lie superalgebras Gl.n;m/ and Q(n), Math. Sb. USSR, 51, 419-427 (1985) · Zbl 0573.17002 |
| [65] | van de Leur, JW, The Adler-Shiota-van Moerbeke formula for the BKP hierarchy, J. Math. Phys., 36, 4940-4951 (1995) · Zbl 0844.35109 · doi:10.1063/1.531352 |
| [66] | van de Leur, JW, The n-th reduced BKP hierarchy, the string equation and BW \(1 \infty \)-constraints, Acta Appl. Math., 44, 185-206 (1996) · Zbl 0866.17017 · doi:10.1007/BF00116521 |
| [67] | You, Y.: Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups. In: Infinite-Dimensional Lie Algebras and Groups, Advanced Series in Mathematical Physics 7 (1989). World Scientific Publications, Teaneck, NJ · Zbl 0744.35052 |
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