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

TL;DR

This paper proves the convergence of the small K-theoretic J-function to the cohomological J-function for certain Fano manifolds and establishes a new proof of Iritani's identity for toric manifolds of Picard rank 2 using q-oscillatory integrals.

Contribution

It demonstrates confluence of K-theoretic J-functions for weak Fano manifolds and extends Iritani's identity to K-theory for specific toric manifolds, providing new insights and proofs.

Findings

K-theoretic J-function converges to cohomological J-function as q approaches 1.

Established K-theoretic version of Iritani's identity for Picard rank 2 toric manifolds.

Provided a new proof of Iritani's identity using q-oscillatory integrals.

Abstract

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.