Frobenius manifold

A Frobenius manifold is a manifold $M$ equipped with a smoothly (or analytically) varying structure of a Frobenius algebra on each tangent space which satisfies a series of integrability conditions (e.g. the induced metric is required to be flat). The most relevant example arises in Gromov-Witten theory where the genus 0 theory of a variety $X$ imposes a Frobenius manifold structure on $H^{\ast}(X,\mathbb{C})$ (more precisely, the structure of a formal, graded-commutative Frobenius manifold). The integrability condition in this case is equivalent to the associativity of the quantum product. Other examples arise in the deformation theory of isolated singularities and in Hodge theory.