Quantum spectrum and Gamma structure for standard flips

TL;DR

This paper explores the quantum cohomology and Gamma structures of standard flips, revealing a decomposition compatible with semi-orthogonal decompositions through asymptotic analysis of special functions.

Contribution

It introduces a novel decomposition of quantum cohomology for standard flips that aligns with existing semi-orthogonal decompositions, using asymptotic analysis of Meijer G-functions.

Findings

Decomposition of quantum cohomology into asymptotic Gamma classes.

Compatibility with semi-orthogonal decompositions.

Use of asymptotic behavior of special functions in proofs.

Abstract

We investigate the quantum spectrum and Gamma structure for projective bundles, blow-ups, and standard flips. After restricting the quantum multiplication to the exceptional curve direction, we obtain a decomposition of the quantum cohomology of standard flips into asymptotic Gamma classes. We then show that this decomposition is compatible with the semi-orthogonal decompositions for these spaces constructed in work of Orlov and Belmans-Fu-Raedschelders. The proof involves a sequence of reductions to a local model and the asymptotic behavior of Meijer G-functions.