Formula omitted] formal genus zero Gromov–Witten theories and Givental’s formalism

Abstract

In [A. Givental, Symplectic geometry of Frobenius structures. arxiv: math.AG/0305409] Givental introduced and studied a space of formal genus zero Gromov–Witten theories G W 0 , i.e. functions satisfying string and dilaton equations and topological recursion relations. A central role in the theory plays the geometry of certain Lagrangian cones and a twisted symplectic group of hidden symmetries. In this note we show that the Lagrangian cones description of the action of this group coincides with the genus zero part of Givental’s quantum Hamiltonian formalism. As an application we identify explicitly the space of N = 1 formal genus zero GW theories with lower-triangular twisted symplectic group modulo the string flow.