$N=1$ formal genus $0$ Gromov-Witten theories and Givental's formalism

TL;DR

This paper connects Givental's formalism with the geometric Lagrangian cone approach in genus zero Gromov-Witten theories, providing an explicit identification of the space of such theories.

Contribution

It demonstrates the equivalence between the Lagrangian cone description and Givental's quantum Hamiltonian formalism in genus zero GW theories, and explicitly characterizes the space of N=1 theories.

Findings

Lagrangian cones description matches Givental's formalism

Explicit identification of N=1 genus zero GW theories

Characterization of the space modulo the string flow

Abstract

In [Gi3] Givental introduced and studied a space of formal genus zero Gromov-Witten theories $GW_0$, 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.