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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

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 quantum K–theory of Givental and Lee.

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Gromov–Witten theory, higher category, derived algebraic geometry
Mathematical Subject Classification 2010
Primary: 14N35
Received: 7 December 2016
Revised: 4 April 2017
Accepted: 13 June 2017
Published: 16 March 2018
Proposed: Richard Thomas
Seconded: Jim Bryan, Peter Teichner
Etienne Mann
Département de Mathématiques Bâtiment I
Faculté des Sciences 2
Université d’Angers
Marco Robalo
Sorbonne Université
Faculté des Sciences et Ingénierie
Institut de Mathématiques de Jussieu-PRG