Givental symmetries of Frobenius manifolds and multi-component KP tau-functions

TL;DR

This paper links Givental's and van de Leur's constructions of twisted loop group actions on Frobenius manifolds, showing their equivalence at genus zero and connecting Gromov-Witten theory with multi-component KP hierarchies.

Contribution

It establishes the equivalence between Givental's and van de Leur's descriptions of loop group actions on Frobenius structures at genus zero.

Findings

Givental and van de Leur actions coincide at genus zero.

Explicit formulas for tangent actions are connected.

The work bridges Gromov-Witten theory and integrable hierarchies.

Abstract

We establish a link between two different constructions of the action of the twisted loop group on the space of Frobenius structures. The first construction (due to Givental) describes the action of the twisted loop group on the partition functions of formal Gromov-Witten theories. The explicit formulas for the corresponding tangent action were computed by Y.-P. Lee. The second construction (due to van de Leur) describes the action of the same group on the space of Frobenius structures via the multi-component KP hierarchies. Our main theorem states that the genus zero restriction of the Y.-P. Lee formulas coincides with the tangent van de Leur action.