4-dimensional Frobenius manifolds and Painlevé VI. (English) Zbl 1307.53074
In the early 90’s B. Dubrovin observed that there is a bi-Hamiltonian hierarchy associated to any Frobenius manifold. In the paper under review the author extends the construction of Dubrovin to a tri-Hamiltonian hierarchy imposing however additional constraints on the Frobenius manifold.
The author proves that 4-dimensional Frobenius manifolds having a tri-Hamiltonian structure are in one-to-one correspondance with the solutions to the Painlevé VI equation. In his textbook B. Dubrovin has shown that the solution to the Painlevé VI equation defines a 3-dimensional Frobenius manifold. Using these two results, where the same equation plays a completely different role, the author describes explicitly the procedure of constructing all tri-Hamiltonian 4-dimensional Frobenius manifolds from the 3-dimensional Frobenius manifolds.
The author proves that 4-dimensional Frobenius manifolds having a tri-Hamiltonian structure are in one-to-one correspondance with the solutions to the Painlevé VI equation. In his textbook B. Dubrovin has shown that the solution to the Painlevé VI equation defines a 3-dimensional Frobenius manifold. Using these two results, where the same equation plays a completely different role, the author describes explicitly the procedure of constructing all tri-Hamiltonian 4-dimensional Frobenius manifolds from the 3-dimensional Frobenius manifolds.
Reviewer: Alexey Basalaev (Hannover)
MSC:
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
References:
| [1] | Adler, M., van Moerbeke, P.: Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice. Commun. Math. Phys. 237, 397-440 (2003) · Zbl 1090.37051 |
| [2] | Adler, M., Van Moerbeke, P.: Integrals over classical groups, random permutations, Toda and Toeplitz lattices. Commun. Pure Appl. Math. 54, 153-205 (2001) · Zbl 1086.34545 |
| [3] | Appell, P.: Sur les fonctions hypergéometriques de plusiers variables. Mem. des Sciences de Paris, III (1925) · JFM 51.0281.09 |
| [4] | Carlet, G.: The Hamiltonian structures of the two-dimensional Toda lattice and \[R\] R-matrices. Lett. Math. Phys. 71(3), 209-226 (2005) · Zbl 1084.37051 · doi:10.1007/s11005-005-0629-y |
| [5] | Dinar, Y.: On classification and construction of algebraic Frobenius manifolds. J. Geom. Phys. 58, 9 (2008) · Zbl 1149.53322 · doi:10.1016/j.geomphys.2008.04.001 |
| [6] | Dubrovin, B.: Geometry of 2D topological field theories. In: Donagi, R., Dubrovin, B., Frenkel, E., Previato, E., Francavilla, M., Greco, S. (eds.) Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, vol. 1620. Springer, Berlin, pp. 120-348 (1996) · Zbl 0665.32011 |
| [7] | Dubrovin, B.; Conte, R. (ed.), Painlevé transcendents and topological field theory, 287-412 (1999), Berlin · Zbl 1026.34095 · doi:10.1007/978-1-4612-1532-5_6 |
| [8] | Dubrovin, B.: On almost duality for Frobenius manifolds. In: Geometry, Topology and Mathematical Physics. Amer. Math. Soc. Transl. Ser., vol. 2, no. 212, pp. 75-132 (2004) · Zbl 1063.53092 |
| [9] | Dubrovin, B.: Flat pencils of metrics and Frobenius manifolds. In: Proceedings of 1997 Taniguchi Symposium “Integrable Systems and Algebraic Geometry”, pp. 47-72 (1998) · Zbl 0963.53054 |
| [10] | Dubrovin, B.: Differential geometry of moduli spaces and its applications to soliton equations and to topological conformal field theory. Surveys Differ. Geom. 4, 213-238 (1999) · Zbl 0947.35117 · doi:10.4310/SDG.1998.v4.n1.a5 |
| [11] | Dubrovin, B.: Geometry and analytic theory of Frobenius manifolds. In: Proceedings of the International Congress of Mathematicians, Doc. Math. Extra. vol. II, Berlin, pp. 315-326 (1998) · Zbl 0916.32018 |
| [12] | Dubrovin, B., Mazzocco, M.: Monodromy of certain Painlevé VI transcendents and reflection groups. Invent. Math. 141, 55-147 (2000) · Zbl 0960.34075 · doi:10.1007/PL00005790 |
| [13] | Dubrovin, B., Mazzocco, M.: On the reductions and classical solutions of the Schlesinger equations, differential equations and quantum groups. Andrey A. Bolibruch memorial volume, IRMA Lectures in Mathematics and Theoretical Physics, vol. 9, pp. 157-187 (2006) · Zbl 1356.32007 |
| [14] | Dubrovin, B., Novikov, S.: Hyrdodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. Uspekhi Mat. Nauk 44, 29-98 [English translation in Russ. Math. Surveys 44, 35-124 (1989)] · Zbl 0712.58032 |
| [15] | Dubrovin, B., Novikov, S.: On Poisson brackets of hydrodynamic type. Soviet Math. Dokl. 279(2), 294-297 (1984) · Zbl 0591.58012 |
| [16] | Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, G.: Higher transcendental functions, vol. I. McGraw-Hill, New York (1953) · Zbl 0051.30303 |
| [17] | Gambier, B.: Sur les equations differentielles du second ordre et du primier degrè dont l’ integrale est a points critiques fixes. Acta Math. 33 (1910) · Zbl 0999.34079 |
| [18] | Givental, A.: Twisted Picard-Lefschetz formulas. Funktsional. Anal. i Prilozhen. 22, 12-22 (1988) · Zbl 0665.32011 |
| [19] | Guzzetti, D.: Inverse problem and monodromy data for three-dimensional Frobenius manifolds. Math. Phys. Anal. Geom. 4, 245-291 (2001) · Zbl 1001.53061 · doi:10.1023/A:1012933622521 |
| [20] | Ismail, M., Pitman, J.: Algebraic evaluation of some Euler integrals, duplication formulae for Appell’s hypergeometric function \[F_1\] F1, and Brownian variations. Can. J. Math. Phys. 52, 961-981 (2000) · Zbl 0961.33012 |
| [21] | Korotkin, D., Shramchenko, V.: Riemann-Hilbert problem for Hurwitz Frobenius manifolds: regular singularities. J. Reine Angew. Math. 661, 125-187 (2011) · Zbl 1250.34072 |
| [22] | Krichever, I.M.: Integration of nonlinear equations by the methods of algebraic geometry. Funct. Anal. Appl. 11, 12-26 (1977) · Zbl 0368.35022 · doi:10.1007/BF01135528 |
| [23] | Kupershmidt, B.: Discrete Lax equations and differential-difference calculus. Asterisque 123, Societé Mathématique de France, Paris (1985) · Zbl 0565.58024 |
| [24] | Manin, Yu.I.: Sixth Painlevé equation, universal elliptic curve, and mirror of \[{\bf P}^2\] P2. In: Geometry of Differential Equations. Amer. Math. Soc. Transl., vol. 2, no. 186, pp. 131-151 (1996) · Zbl 0948.14025 |
| [25] | Mazzocco, M.: Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321(1), 157-195 (2001) · Zbl 0999.34079 · doi:10.1007/PL00004500 |
| [26] | Okamoto, K.: Studies on the Painlevé equations. I. Sixth Painlevé equation PVI. Ann. Mat. Pura Appl. 146(4), 337-381 (1987) · Zbl 0637.34019 |
| [27] | Olver, P., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E (53) 53(2), 1900-1906 (1996) · doi:10.1103/PhysRevE.53.1900 |
| [28] | Oevel, W., Ragnisco, \[O.: R\] R-matrices and higher Poisson brackets for integrable systems. Phys. A 161(1), 181-220 (1989) · Zbl 0717.35081 |
| [29] | Oevel, W., Ragnisco, O.: An abstract tri-Hamiltonian Lax hierarchy. Nonlinear evolution equations and dynamical systems. Res. Rep. Phys, Springer, Berlin, pp. 144-147 (1989) · Zbl 0368.35022 |
| [30] | Painlevé, P.: Sur les equations differentielles du second ordre et d’ordre superieur, dont linterable generale est uniforme. Acta Math. 25 (1902) · JFM 32.0340.01 |
| [31] | Pavlov, M.V., Tsarev, S.P.: Tri-Hamiltonian structures of the Egorov systems. Funct. Anal. Appl. 37(1), 32-45 (2003) · Zbl 1019.37048 · doi:10.1023/A:1022971910438 |
| [32] | Pavlyk, O.: Solutions to WDVV from generalized Drinfeld-Sokolov hierarchies. math-ph/0003020 (2003, preprint) |
| [33] | Schlesinger, L.: Über eine Klasse von Differentsialsystemen beliebliger Ordnung mit festen kritischer Punkten. J. für Math. 141, 96-145 (1912) · JFM 43.0385.01 |
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