Flat coordinates on orbit spaces: from Novikov algebras to cyclic quotient singularities

Abstract

The Gauss-Manin equations are solved for a class of flat-metrics defined by Novikov algebras, this generalizing a result of Balinskii and Novikov who solved this problem in the case of commutative Novikov algebras (where the algebraic conditions reduce to those of a Frobenius algebra). The problem stems from the theory of first-order Hamiltonian operators and their reduction to a constant, or Darboux, form. The monodromy group associated with the Novikov algebra gives rise to an orbit space, which is, for a wide range of Novikov algebras, a cyclic quotient singularity.