Spectral gaps for twisted Dolbeault-Dirac operators over the irreducible quantum flag manifolds

TL;DR

This paper demonstrates that twisting Dirac and Laplace operators on quantum flag manifolds with negative line bundles creates spectral gaps, leveraging noncommutative K"ahler geometry and positivity of quantum metrics.

Contribution

It introduces a framework for spectral gaps in noncommutative K"ahler structures and applies it to quantum flag manifolds, establishing spectral gaps for q close to 1.

Findings

Spectral gaps are produced by twisting operators with negative line bundles.

Positivity of the quantum Fubini-Study metric is crucial for the results.

Even degree de Rham cohomology groups do not vanish in this setting.

Abstract

We show that tensoring the Laplace and Dolbeault-Dirac operators of a K\"ahler structure (with closed integral) by a negative Hermitian holomorphic module, produces operators with spectral gaps around zero. The proof is based on the recently established Akizuki-Nakano identity of a noncommutative K\"ahler structure. This general framework is then applied to the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds, and it is shown that twisting their Dirac and Laplace operators by negative line bundles produces a spectral gap, for q sufficiently close to 1. The main technical challenge in applying the framework is to establish positivity of the quantum Fubini-Study metric of the quantum flag manifold. Importantly, combining positivity with the noncommutative hard Lefschetz theorem, it is additionally observed that the even degree de Rham cohomology groups of the Heckenberger-Kolb calculi do not vanish.