Nonlocal brackets and integrable string models

Abstract:
The closed string model in the background gravity field is considered as bihamiltonian system assuming that the string model is an integrable model for the particular kind of the background fields. The dual nonlocal Poisson brackets are obtained on the phase space. The integrability condition is formulated as the compatibility condition of the bihamiltonity condition and the Jacobi identity of the dual Poisson brackets. It is shown, that the Jacobi identity reduced to the nonlocal analogy of the WDVV equations. The local solutions of this equations are reduced to zero curvature and to the constant background fields. There is the difference between the dual nonlocal brackets on the phase space and Ferapontov nonlocal brackets of the hydrodynamical type. It is shown, that dual brackets and dual hamiltonians can be obtained from the canonical Poisson brackets and from the initial hamiltonian by imposing the second kind constraints on the initial dynamical system.