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

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→1, which coincides with the small cohomological J-function. Furthermore, in the case of a Fano toric manifold 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.