Spectral gaps for twisted Dolbeault-Dirac operators over the irreducible quantum flag manifolds
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.
Keywords
Cite
@article{arxiv.2206.10719,
title = {Spectral gaps for twisted Dolbeault-Dirac operators over the irreducible quantum flag manifolds},
author = {Biswarup Das and Réamonn Ó Buachalla and Petr Somberg},
journal= {arXiv preprint arXiv:2206.10719},
year = {2022}
}
Comments
The associated preprint arXiv:1910.14007 has now been divided in two. This article forms the second of the two new papers. arXiv admin note: text overlap with arXiv:1910.14007