The symmetry of Spin^c Dirac spectrums on Riemannian product manifolds

TL;DR

This paper investigates the symmetry of the spectrum of spin^c Dirac operators on odd-dimensional product manifolds, establishing conditions under which the spectrum remains symmetric based on the factors' properties.

Contribution

It provides a necessary and sufficient condition for the symmetry of the Dirac spectrum on product manifolds with spin^c structures, extending known results to odd-dimensional cases.

Findings

Spectrum symmetry depends on the factors' spectra or topological invariants.

Symmetry holds if the second factor's spectrum is symmetric or a specific topological condition is satisfied.

Results apply to product manifolds with spin^c structures, broadening understanding of Dirac operator spectra.

Abstract

It is well-known that the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator on a closed Riemannian $\text{spin}^{\mathbb{C}}$ manifold $M^{2k}$ of dimension $2k$ for $k \in \mathbb{N}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M_{1}^{2p} \times M_{2}^{2q+1}$ with a product $\text{spin}^{\mathbb{C}}$ structure for $p \geq 1, q \geq 0$, the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the $\text{spin}^{\mathbb{C}}$ Dirac spectrum of $M_{2}^{2q+1}$ is symmetric or $(e^{\frac{1}{2}c_{1}(L_{1})} \hat{A}(M_1))[M_{1}]=0$, where $L_1$ is the associated line bundle for the given $\text{spin}^{\mathbb{C}}$ structure of $M_1$.