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

TL;DR

This paper explores the relationship between quantum K-theory and cohomology for projective spaces by analyzing q-difference systems and their confluence, offering new insights into Givental's K-theoretic J-function and related invariants.

Contribution

It demonstrates how to derive the cohomological J-function from the K-theoretic one using confluence of q-difference systems, connecting to the quantum Hirzebruch--Riemann--Roch theorem.

Findings

Derived cohomological J-function from K-theoretic J-function via confluence.

Computed connection numbers in the equivariant setting.

Provided a new perspective on Givental--Tonita's theorem.

Abstract

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.