Confluence of quantum $K$-theory to quantum cohomology for projective spaces

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 $K$-theoretical Gromov--Witten invariants. These invariants are obtained by replacing cohomological constructions used in the definition of the usual Gromov--Witten invariants by their $K$-theoretical analogues. Then, an essential question is to understand how these two invariants 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 equations verify a property called "confluence", which guarantees that we can take some limit of these functional equations to obtain differential equations. In this thesis, we propose to compare the two Gromov--Witten theories through the confluence of $q$-difference equation. We show that, in the case of complex projective spaces, the confluence of Givental's small $K$-theoretical $J$-function as a solution of a $q$-difference equations outputs its cohomological analogue.