ON THE HAMILTONIAN REDUCTION OF THE ASSOCIATIVITY EQUATIONS IN THE CASE OF FOUR PRIMARY FIELDS

  • N. A. Strizhova L.D.Landau Institute for Theoretical Physics of RAS
DOI 10.29006/1564-2291.JOR-2019.47(1).37
Keywords associativity equations, reduction on the set of stationary points of its nondegenerate integral, canonical Hamiltonian system

Abstract

The associativity equations arose in the papers of Witten (Witten, 1990) and Dijkgraaf, Verlinde, (Dijkgraaf et al., 1991) on two-dimensional topological field theories and subsequently they became to play a key role in many other important domains of mathematics and mathematical physics: in quantum cohomology, Gromov–Witten invariants, enumerative geometry, theory of submanifolds and so on. In Mokhov’s papers (Mokhov, 1984), (Mokhov, 1987) a general fundamental principle stating a canonical Hamiltonian property for the restriction of an arbitrary flow on the set of stationary points of its nondegenerate integral was proposed and proved. In this paper the Hamiltonians of the reductions of the associativity equations with antidiagonal matrix ηij in the case of four primary fields according to Mokhov`s construction is found in an explicit form.

References


  1. Mokhov O.I. The Hamiltonian property of an evolutionary flow on the set of stationary points of its integral. Russian Mathematical Surveys, 1984, Vol. 39, No. 4. pp. 133–134.

  2. Mokhov O.I. On the Hamiltonian property of an arbitrary evolution system on the set of stationary points of its integral. Mathematics of the USSR-Izvestiya, 1988, Vol. 31, No. 3, pp. 657–664.

  3. Mokhov O.I. Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems. Russian Mathematical Surveys, 1998, Vol. 53, No. 3, pp. 515–622.

  4. Dijkgraaf R., Verlinde H., and Verlinde E. Topological strings in d < 1. Nucl. Phys. B, 1991, Vol. 352, No. 1, pp. 59–86.

  5. Dubrovin B.A. Geometry of 2D topological field theories. Preprint SISSA-89/94/FM, SISSA, Trieste: Italy, 1994, Lecture Notes in Math, 1996, Vol. 1620, pp. 120–348; arXiv: hepth/9407018 (1994).

  6. Ferapontov E.V. and Mokhov O.I. On the Hamiltonian representation of the associativity equations. Algebraic aspects of integrable systems: In memory of Irene Dorfman. Eds. I.M. Gelfand, A.S. Fokas, Boston: Birkhäuser, 1996, pp. 75–91.

  7. Pavlov M.V. and Vitolo R.F. On the Bi-Hamiltonian Geometry of WDVV Equations. Lett. Math. Phys, 2015, Vol. 105, No. 8, pp. 1135–1163; arXiv:1409.7647[math-ph] (2014).

  8. Witten E. On the structure of topological phase of two-dimensional gravity. Nucl. Phys. B, 1990, Vol. 340, No. 2–3, pp. 281–332.
Published
2019-05-29
Issue
Vol 47 No 1 (2019): Journal of Oceanological Research
Section
The XXII workshop of the Council of nonlinear dynamics of the Russian Academy of

Transfer of copyrights occurs on the basis of a license agreement between the Author and Shirshov Institute of Oceanology, RAS