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

Etienne Mann (1, 2) , Marco Robalo (3)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
3 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

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(X) which allows us to recover the formulas for Quantum K-theory of Givental-Lee. This paper is the first step of a larger project. We believe that this action in correspondences encodes the full classical cohomological Gromov-Witten invariants of X. This will appear in a second paper.

Mots clés: algebraic topology; Gromov-Witten invariants; algebraic geometry; homotopy theory

Domaines: Mathématiques [math]

Etienne Mann, Marco Robalo. Brane actions, categorifications of Gromov–Witten theory and quantum K–theory. Geometry and Topology, 2018, 22 (3), pp.1759-1836. ⟨10.2140/gt.2018.22.1759⟩. ⟨hal-01151061v2⟩