Brane actions, categorifications of Gromov–Witten theory and quantum K–theory

Abstract

Let $X$ be a smooth projective variety. Using the idea of brane actions discovered by Toën, 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 $Qcoh\left(X\right)$ which allows us to recover the quantum K–theory of Givental and Lee.