Spectrum of the $\bar{\partial }$-Laplace operator on zero forms for the quantum quadric ${\mathcal{O}}_q({\text{Q}}_N)$

Abstract

We study the Laplacian operator ${\Delta }_{\bar{\partial }}$ associated to a Kähler structure $({\Omega }^{(\bullet ,\bullet )},\kappa )$ for the Heckenberger--Kolb differential calculus of the quantum quadrics ${\mathcal{O}}_q({\text{Q}}_N)$, which is to say, the irreducible quantum flag manifolds of types $B_n$ and $D_n$. We show that the eigenvalues of ${\Delta }_{\bar{\partial }}$ on zero forms tend to infinity and have finite multiplicity.

25 pages