Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies and the equations of associativity

In this paper we solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics and establish that this nontrivial special class of Hamiltonian operators is closely connected with the associativity equations of twodimensional topological quantum field theories and the theory of Frobenius manifolds. It is shown that the Hamiltonian operators of this class are of special interest for many other reasons too. In particular, we prove in this paper that any such Hamiltonian operator always defines integrable structural flows (systems of hydrodynamic type), always gives a nontrivial pencil of compatible Hamiltonian operators and generates integrable hierarchies of hydrodynamic type. It is proved that the affinors of any such Hamiltonian operator generate some special integrals in involution. The nonlinear systems describing integrals in involution are of independent great interest. The equations of associativity of two-dimensional topological quantum field theories (the Witten–Dijkgraaf–Verlinde–Verlinde and Dubrovin equations) describe an important special class of integrals in involution and a special class of nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. It is shown that anyN -dimensional Frobenius manifold can be locally presented by a certain special flat N -dimensional submanifold with flat normal bundle in a 2N -dimensional pseudo-Euclidean space and this submanifold is defined uniquely up to motions. We will devote a separate paper to the properties of this construction and to the properties of this special class of flat submanifolds with flat normal bundle (we mean the class corresponding to Frobenius manifolds).