Affine Kac–Moody Algebras and Tau-Functions for the Drinfeld–Sokolov Hierarchies: the Matrix-Resolvent Method

Boris Dubrovin ; Daniele Valeri ; Di Yang

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

For each affine Kac–Moody algebra $X_n^{(r)}$ of rank $\ell$, $r=1,2$, or $3$, and for every choice of a vertex $c_m$, $m=0,\dots,\ell$, of the corresponding Dynkin diagram, by using the matrix-resolvent method we define a gauge-invariant tau-structure for the associated Drinfeld–Sokolov hierarchy and give explicit formulas for generating series of logarithmic derivatives of the tau-function in terms of matrix resolvents, extending the results of [Mosc. Math. J. 21 (2021), 233–270, arXiv:1610.07534] with $r=1$ and $m=0$. For the case $r=1$ and $m=0$, we verify that the above-defined tau-structure agrees with the axioms of Hamiltonian tau-symmetry in the sense of [Adv. Math. 293 (2016), 382–435, arXiv:1409.4616] and [arXiv:math.DG/0108160].

Détail

Keywords: Kac–Moody algebra, tau-function, Drinfeld–Sokolov hierarchy, matrix resolvent.

Bibliographie

[1] Balog J., Fehér L., O'Raifeartaigh L., Forgács P., Wipf A., “Toda theory and $\mathcal{W}$-algebra from a gauged WZNW point of view”, Ann. Physics, 203 (1990), 76–136 | DOI | MR

[2] Bertola M., Dubrovin B., Yang D., “Correlation functions of the KdV hierarchy and applications to intersection numbers over $\overline{\mathcal{M}}_{g,n}$”, Phys. D, 327 (2016), 30–57, arXiv: 1504.06452 | DOI | MR

[3] Bertola M., Dubrovin B., Yang D., “Simple Lie algebras, Drinfeld–Sokolov hierarchies, and multi-point correlation functions”, Mosc. Math. J., 21 (2021), 233–270, arXiv: 1610.07534 | DOI | MR

[4] Buryak A., Dubrovin B., Guéré J., Rossi P., “Tau-structure for the double ramification hierarchies”, Comm. Math. Phys., 363 (2018), 191–260, arXiv: 1602.05423 | DOI | MR

[5] Cafasso M., Wu C.-Z., “Borodin–Okounkov formula, string equation and topological solutions of Drinfeld–Sokolov hierarchies”, Lett. Math. Phys., 109 (2019), 2681–2722, arXiv: 1505.00556 | DOI | MR

[6] Collingwood D.H., McGovern W.M., Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993 | MR

[7] de Groot M.F., Hollowood T.J., Miramontes J.L., “Generalized Drinfel'd–Sokolov hierarchies”, Comm. Math. Phys., 145 (1992), 57–84 | DOI | MR

[8] De Sole A., Jibladze M., Kac V.G., Valeri D., “Integrability of classical affine $\mathcal W$-algebras”, Transf. Groups, 26 (2021), 479–500, arXiv: 2007.01244 | DOI | MR

[9] De Sole A., Kac V.G., Valeri D., “Classical $\mathcal W$-algebras and generalized Drinfeld–Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras”, Comm. Math. Phys., 323 (2013), 663–711, arXiv: 1207.6286 | DOI | MR

[10] De Sole A., Kac V.G., Valeri D., “Structure of classical (finite and affine) $\mathcal W$-algebras”, J. Eur. Math. Soc., 18 (2016), 1873–1908, arXiv: 1404.0715 | DOI | MR

[11] De Sole A., Kac V.G., Valeri D., “Classical affine $\mathcal W$-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras”, Comm. Math. Phys., 360 (2018), 851–918, arXiv: 1705.10103 | DOI | MR

[12] Dinar Y.I., “Frobenius manifolds from subregular classical $W$-algebras”, Int. Math. Res. Not., 2013 (2013), 2822–2861, arXiv: 1108.5445 | DOI | MR

[13] Dinar Y.I., “$W$-algebras and the equivalence of bihamiltonian, Drinfeld–Sokolov and Dirac reductions”, J. Geom. Phys., 84 (2014), 30–42, arXiv: 0911.2116 | DOI | MR

[14] Drinfel'd V.G., Sokolov V.V., “Lie algebras and equations of Korteweg–de Vries type”, Soviet J. Math., 30 (1985), 1975–2036 | DOI | MR

[15] Dubrovin B., “Geometry of $2$D topological field theories”, Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., 1620, Springer, Berlin, 1996, 120–348, arXiv: hep-th/9407018 | DOI | MR

[16] Dubrovin B., Liu S.-Q., Yang D., Zhang Y., “Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs”, Adv. Math., 293 (2016), 382–435, arXiv: 1409.4616 | DOI | MR

[17] Dubrovin B., Liu S.-Q., Zhang Y., “Frobenius manifolds and central invariants for the Drinfeld–Sokolov biHamiltonian structures”, Adv. Math., 219 (2008), 780–837, arXiv: 0710.3115 | DOI | MR

[18] Dubrovin B., Yang D., Zagier D., “On tau-functions for the KdV hierarchy”, Selecta Math. (N.S.), 27 (2021), 12, 47 pp., arXiv: 1812.08488 | DOI | MR

[19] Dubrovin B., Zhang Y., Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants, arXiv: math.DG/0108160

[20] Frenkel E., Givental A., Milanov T., “Soliton equations, vertex operators, and simple singularities”, Funct. Anal. Other Math., 3 (2010), 47–63, arXiv: 0909.4032 | DOI | MR

[21] Hollowood T., Miramontes J.L., “Tau-functions and generalized integrable hierarchies”, Comm. Math. Phys., 157 (1993), 99–117, arXiv: hep-th/9208058 | DOI | MR

[22] Kac V.G., “Infinite-dimensional algebras, Dedekind's $\eta $-function, classical Möbius function and the very strange formula”, Adv. Math., 30 (1978), 85–136 | DOI | MR

[23] Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990 | DOI | MR

[24] Kac V.G., Wakimoto M., “Exceptional hierarchies of soliton equations”, Theta Functions–Bowdoin 1987 (Brunswick, ME, 1987), v. 1, Proc. Sympos. Pure Math., 49, Amer. Math. Soc., Providence, RI, 1989, 191–237 | DOI | MR

[25] Kaup D.J., “On the inverse scattering problem for cubic eigenvalue problems of the class $\psi_{xxx}+6Q\psi_{x}+6R\psi =\lambda \psi $”, Stud. Appl. Math., 62 (1980), 189–216 | DOI | MR

[26] Kostant B., “The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group”, Amer. J. Math., 81 (1959), 973–1032 | DOI | MR

[27] Liu S.-Q., Ruan Y., Zhang Y., “BCFG Drinfeld–Sokolov hierarchies and FJRW-theory”, Invent. Math., 201 (2015), 711–772, arXiv: 1312.7227 | DOI | MR

[28] Liu S.-Q., Wu C.-Z., Zhang Y., “Virasoro Constraints for Drinfeld–Sokolov hierarchies and equations of Painlevé type”, J. London Math. Soc., 106 (2022), 1443–1500, arXiv: 1908.06707 | DOI | MR

[29] Liu S.-Q., Wu C.-Z., Zhang Y., Zhou X., “Drinfeld–Sokolov hierarchies and diagram automorphisms of affine Kac–Moody algebras”, Comm. Math. Phys., 375 (2020), 785–832, arXiv: 1811.10137 | DOI | MR

[30] Magri F., “A simple model of the integrable Hamiltonian equation”, J. Math. Phys., 19 (1978), 1156–1162 | DOI | MR

[31] Sawada K., Kotera T., “A method for finding $N$-soliton solutions of the Kd.V equation and Kd.V-like equation”, Progr. Theoret. Phys., 51 (1974), 1355–1367 | DOI | MR

[32] Wu C.-Z., “Tau functions and Virasoro symmetries for Drinfeld–Sokolov hierarchies”, Adv. Math., 306 (2017), 603–652, arXiv: 1203.5750 | DOI | MR

[33] Zhou J., On absolute $N$-point function associated with Gelfand–Dickey polynomials, Unpublished, 2015