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Brane actions, categorifications of Gromov-Witten theory and quantum \(K\)-theory. (English) Zbl 1423.14320
Summary: Let \(X\) be a smooth projective variety. Using the idea of brane actions discovered by B. Toën [“Operations on derived moduli spaces of branes”, Preprint, arXiv:1307.0405], 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 \(\operatorname{Qcoh}(X)\) which allows us to recover the quantum \(K\)-theory of A. B. Givental [in: Frobenius manifolds. Quantum cohomology and singularities. Proceedings of the workshop, Bonn, Germany, July 8–19, 2002. Wiesbaden: Vieweg. 91–112 (2004; Zbl 1075.53091)] and Y. P. Lee [Duke Math. J. 121, No. 3, 389–424 (2004; Zbl 1051.14064)].

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
55P48 Loop space machines and operads in algebraic topology
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