Brane actions, Categorification of Gromov-Witten theory and Quantum K-theory

TL;DR

This paper constructs a geometric framework using brane actions to encode Gromov-Witten theory and quantum K-theory of a smooth projective variety, linking operad actions to derived categories.

Contribution

It introduces a novel geometric approach to Gromov-Witten and quantum K-theory via operad actions on derived stacks, extending Toën's brane actions.

Findings

Operad of stable curves acts on X in derived correspondences

Gromov-Witten theory is encoded geometrically

Quantum K-theory is recovered from the action

Abstract

Let X be a smooth projective variety. Using the idea of brane actions discovered by To\"en, we construct a lax associative action of the operad of stable curves of genus zero on the variety X seen as an object in correspondences in derived stacks. This action encodes the Gromov-Witten theory of X in purely geometrical terms and induces an action on the derived category Qcoh(X) which allows us to recover the Quantum K-theory of Givental-Lee.