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Confluence de la K-théorique quantique vers la cohomologie quantique pour les espaces projectifs

Abstract : In algebraic geometry, Gromov— Witten invariants are enumerative invariants that count the number of complex curves in a smooth projective variety satisfying some incidence conditions. In 2001, A. Givental and Y.P. Lee defined new invariants, called Ktheoretical Gromov—Witten invariants. These invariants are obtained by replacing cohomological objects used in the definition of the usual Gromov—Witten invariants by their Ktheoretical analogues. Then, an essential question is to understand how these two theories are related. In 2013, Iritani-Givental- Milanov-Tonita show that K-theoretical Gromov—Witten invariants can be embedded in a function which satisfies a q-difference equation. In general, these functional equations verify a property called “confluence”, which guarantees that we can degenerate these equations to obtain a differential equation. In this thesis, we propose to compare our two Gromov—Witten theories through the confluence of q-difference equations. We show that, in the case of complex projective spaces, this property can be adapted to degenerate Ktheoretical invariants into their cohomological analogues. More precisely, we show that theconfluence of Givental’s small K-theoretical Jfunction produces its cohomological analogue after applying the Chern character.
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Alexis Roquefeuil. Confluence de la K-théorique quantique vers la cohomologie quantique pour les espaces projectifs. General Mathematics [math.GM]. Université d'Angers, 2019. English. ⟨NNT : 2019ANGE0019⟩. ⟨tel-02447681⟩

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