On KP-integrable skew Hurwitz \(\tau\)-functions and their \(\beta\)-deformations. (English) Zbl 1520.81094
Summary: We extend the old formalism of cut-and-join operators in the theory of Hurwitz \(\tau\)-functions to description of a wide family of KP-integrable skew Hurwitz \(\tau\)-functions, which include, in particular, the newly discovered interpolating WLZZ models. Recently, the simplest of them was related to a superintegrable two-matrix model with two potentials and one external matrix field. Now we provide detailed proofs, and a generalization to a multi-matrix representation, and propose the \(\beta\)-deformation of the matrix model as well. The general interpolating WLZZ model is generated by a \(W\)-representation given by a sum of operators from a one-parametric commutative sub-family (a commutative subalgebra of \(w_\infty\)). Different commutative families are related by cut-and-join rotations. Two of these sub-families (‘vertical’ and ‘45-degree’) turn out to be nothing but the trigonometric and rational Calogero-Sutherland Hamiltonians, the ‘horizontal’ family is represented by simple derivatives. Other families require an additional analysis.
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 11M35 | Hurwitz and Lerch zeta functions |
| 81T32 | Matrix models and tensor models for quantum field theory |
| 14D15 | Formal methods and deformations in algebraic geometry |
| 70H05 | Hamilton’s equations |
References:
| [1] | Wang, R.; Liu, F.; Zhang, C. H.; Zhao, W. Z. |
| [2] | Mironov, A.; Mishnyakov, V.; Morozov, A.; Popolitov, A.; Wang, R.; Zhao, W. Z. |
| [3] | Alexandrov, A.; Mironov, A.; Morozov, A.; Natanzon, S., J. High Energy Phys., 11, Article 080 pp. (2014) · Zbl 1333.81192 |
| [4] | Macdonald, I. G., Symmetric Functions and Hall Polynomials (1995), Oxford University Press · Zbl 0899.05068 |
| [5] | Mironov, A.; Morozov, A.; Natanzon, S., Theor. Math. Phys., 166, 1-22 (2011); Mironov, A.; Morozov, A.; Natanzon, S., J. Geom. Phys., 62, 148-155 (2012) |
| [6] | Fulton, W., Young Tableaux: With Applications to Representation Theory and Geometry, LMS (1997) · Zbl 0878.14034 |
| [7] | Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Transformation groups for soliton equations, (RIMS Symp. Non-Linear Integrable Systems - Classical Theory and Quantum Theory (1983), World Scientific: World Scientific Singapore) · Zbl 0571.35105 |
| [8] | Ueno, K.; Takasaki, K., Adv. Stud. Pure Math., 4, 1 (1984) |
| [9] | Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A., Int. J. Mod. Phys. A, 10, 2015 (1995) · Zbl 0985.81639 |
| [10] | Okounkov, A., Math. Res. Lett., 7, 447-453 (2000) · Zbl 0969.37033 |
| [11] | Alexandrov, A.; Mironov, A.; Morozov, A.; Natanzon, S., J. Phys. A, 45, Article 045209 pp. (2012) · Zbl 1259.37040 |
| [12] | Lando, S., (Koolen; Kwak; Xu, Applications of Group Theory to Combinatorics (2008), Taylor & Francis Group: Taylor & Francis Group London), 109-132 |
| [13] | Orlov, A.; Shcherbin, D. M., Theor. Math. Phys., 128, 906-926 (2001) · Zbl 0992.37063 |
| [14] | Belyi, G., Math. USSR, Izv., 14, 2, 247-256 (1980); Grothendieck, A., Sketch of a programme, (Lond. Math. Soc. Lect. Note Ser., vol. 242 (1997)), 243-283; Grothendieck, A., Esquisse d’un programme, (Lochak, P.; Schneps, L., Geometric Galois Action (1997), Cambridge University Press: Cambridge University Press Cambridge), 5-48; Shabat, G. B.; Voevodsky, V. A., (The Grothendieck Festschrift, vol. III (1990), Birkhäuser), 199-227; Lando, S. K.; Zvonkin, A. K., Graphs on Surfaces and Their Applications, Encycl. of Math. Sciences, vol. 141 (2004), Springer; Zograf, P., Int. Math. Res. Not., 24, 13533-13544 (2015) |
| [15] | Balantekin, A. B., Phys. Rev. D, 62, Article 085017 pp. (2000); Morozov, A., Theor. Math. Phys., 162, 1-33 (2010), See also a review in: |
| [16] | Bakas, I., Phys. Lett. B, 228, 57-63 (1989); Awata, H.; Fukuma, M.; Matsuo, Y.; Odake, S., Prog. Theor. Phys. Suppl., 118, 343-374 (1995) |
| [17] | Mironov, A.; Morozov, A.; Natanzon, S., J. High Energy Phys., 11, Article 097 pp. (2011) · Zbl 1306.81291 |
| [18] | Takasaki, K., Adv. Stud. Pure Math., 4, 139-163 (1984) |
| [19] | Mironov, A.; Morozov, A., Eur. Phys. J. C, 83, 71 (2023) |
| [20] | Wang, L. Y.; Mishnyakov, V.; Popolitov, A.; Liu, F.; Wang, R. |
| [21] | Mironov, A.; Morozov, A.; Shakirov, Sh., J. High Energy Phys., 03, Article 102 pp. (2011) · Zbl 1301.81151 |
| [22] | Goulden, D.; Jackson, D. M.; Vainshtein, A., Ann. Comb., 4, 27-46 (2000) · Zbl 0957.58011 |
| [23] | Sergeev, A. N.; Veselov, A. P. |
| [24] | Marshakov, A.; Mironov, A.; Morozov, A., Phys. Lett. B, 274, 280-288 (1992) |
| [25] | Morozov, A.; Shakirov, S., J. High Energy Phys., 04, Article 064 pp. (2009) |
| [26] | Wang, R.; Zhang, C. H.; Zhang, F. H.; Zhao, W. Z., Nucl. Phys. B, 985, Article 115989 pp. (2022) · Zbl 1516.81159 |
| [27] | Mironov, A.; Mishnyakov, V.; Morozov, A.; Rashkov, R., Eur. Phys. J. C, 81, 1140 (2021); Mironov, A.; Mishnyakov, V.; Morozov, A.; Rashkov, R., JETP Lett., 113, 728-732 (2021) |
| [28] | Mishnyakov, V.; Oreshina, A., Eur. Phys. J. C, 82, 548 (2022) |
| [29] | Mironov, A.; Morozov, A.; Semenoff, G. W., Int. J. Mod. Phys. A, 11, 5031 (1996) · Zbl 1044.81723 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
