arXiv subjects

Alexis Roquefeuil

Publications and source records attributed to Alexis Roquefeuil.

3 recordsLinked to original sources

arXiv2022

Quantum $K$-theory of projective spaces and confluence of $q$-difference equations

Givental's $K$-theoretical $J$-function can be used to reconstruct genus zero $K$-theoretical Gromov--Witten invariants. We view this function as a fundamental solution of a $q$-difference system. In the case of projective spaces, we show that we can use the confluence of $q$-difference systems to obtain the cohomological $J$-function from its $K$-theoretic analogue. This provides another point of view to one of the statements of Givental--Tonita's quantum Hirzebruch--Riemann--Roch theorem. Furthermore, we compute connection numbers in the equivariant setting.

math.AG↗

arXiv2021

Confluence in quantum K-theory of weak Fano manifolds and q-oscillatory integrals for toric manifolds

For a smooth projective variety whose anti-canonical bundle is nef, we prove confluence of the small $K$-theoretic $J$-function, i.e., after rescaling appropriately the Novikov variables, the small $K$-theoretic $J$-function has a limit when $q\to 1$, which coincides with the small cohomological $J$-function. Furthermore, in the case of a Fano toric manifold $X$ of Picard rank 2, we prove the $K$-theoretic version of an identity due to Iritani that compares the $I$-function of the toric manifold and the oscillatory integral of the toric mirror. In particular, our confluence result yields a new proof of Iritani's identity in the case of a toric manifold of Picard rank 2.

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arXiv2019

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

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.

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