论文标题
扭曲的Dolbeault-Dirac操作员在不可还原量子旗歧管上的光谱差距
Spectral gaps for twisted Dolbeault-Dirac operators over the irreducible quantum flag manifolds
论文作者
论文摘要
我们表明，张开了由负遗产式Holomorphic模块的Kähler结构（具有封闭积分）的Laplace和Dolbeault-Dirac操作员，可产生左右光谱差距的操作员。该证明是基于最近建立的非共同kähler结构的Akizuki-Nakano身份。然后将此一般框架应用于不可约束的量子标志歧管的heckenberger-kolb计算，并显示出负线捆绑包的狄拉克和拉普拉斯操作员会产生频谱差距，q会产生q足够接近量子的量子量的主要技术挑战。重要的是，将阳性与非共同硬左列施兹定理相结合，还观察到，heckenberger-kolb calculi的均匀程度de rham同胞组不会消失。
We show that tensoring the Laplace and Dolbeault-Dirac operators of a Kähler 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 Kähler 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.
