#### Volume 22, issue 3 (2018)

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

### Etienne Mann and Marco Robalo

Geometry & Topology 22 (2018) 1759–1836
##### 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.

##### Keywords
Gromov–Witten theory, higher category, derived algebraic geometry
Primary: 14N35
##### Publication
Received: 7 December 2016
Revised: 4 April 2017
Accepted: 13 June 2017
Published: 16 March 2018
Proposed: Richard Thomas
Seconded: Jim Bryan, Peter Teichner
##### Authors
 Etienne Mann LAREMA UMR - 6093 Département de Mathématiques Bâtiment I Faculté des Sciences 2 Université d’Angers Angers France Marco Robalo Sorbonne Université Faculté des Sciences et Ingénierie Institut de Mathématiques de Jussieu-PRG Paris France