3-环面上无质量Dirac算子的谱不对称性
Spectral asymmetry of the massless Dirac operator on a 3-torus
摘要：
考虑一个具有欧氏度规和标准自旋结构的3-环面上的无质量狄拉克算符。已知本征值可以显式计算：谱关于零对称，零本身是双本征值。本文的目的是发展一个关于度量扰动的模最小本征值的扰动理论。在这里，微扰技术的应用受到以下事实的阻碍，即无质量狄拉克算子的本征值具有偶数重，这是该算子与电荷共轭的反线性算子（3维的一个特殊特征）交换的结果。我们推导出一个渐近公式的特征值与最小模的度量的任意扰动，并提出了两个特殊的家庭的黎曼度量的特征值与最小模可以明确评估。我们还建立了我们的渐近公式和eta不变量之间的关系。
英文摘要：
Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
