Gamma conjecture I for flag varieties

TL;DR

This work establishes Gamma conjecture I for all flag varieties by constructing and comparing A- and B-model data through the Rietsch mirror. Central to the approach is identifying the Gamma class $\widehat{\Gamma}_{G^{\vee}/P^{\vee}}$ with the mirror of a totally positive cycle, and then using oscillatory integrals and stationary phase to extract the required asymptotics. The authors prove a mirror theorem connecting the quantum cohomology of $G^{\vee}/P^{\vee}$ with the Brieskorn–Jacobi theory of the Rietsch mirror, and show that the flat sections defined from the Gamma-class coincide with those coming from the mirror. The result broadens Gamma conjecture I to all flag varieties, illuminating the deep link between quantum cohomology, geometric crystals, and positivity phenomena in Lie theory, with potential implications for derived-category statements and integrable systems.

Abstract

We prove Gamma conjecture I for any flag varieties by following a strategy proposed by Galkin and Iritani. The main ingredient is to prove that the $\widehatΓ$-class of a flag variety is mirror to the totally positive part of the corresponding Rietsch mirror.

Theorems & Definitions (160)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Definition 2.1