Nonlocal Hamiltonian Operators of Hydrodynamic Type with Flat Metrics, Integrable Hierarchies, and the Associativity Equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten–Dijkgraaf–Verlinde–Verlinde and Dubrovin equations). We show that each $N$-dimensional Frobenius manifold can locally be represented by a special flat $N$-dimensional submanifold with flat normal bundle in a $2N$-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.