具有磁索结的Dirac算子的谱流
Spectral flow of Dirac operators with magnetic cable knot
摘要：
研究了$\mathbb{S}^3$上带磁链Dirac算子的谱流.这些是Aharonov-Bohm磁链的概括，其中磁场包含与链路的分量重合的许多场线，每个场线的通量表现出与A-B磁链相同的$2π$周期性。我们研究了在只有一条场线的情况下，将磁通从$0$调谐到$2π$所得到的环的谱流：我们将给定的一个纽结和它的索纽结所得到的谱流联系起来，得到环面纽结具有平凡的谱流。在$\mathbb{R}^3$（通过球极投影可看作是$\mathbb{S}^3$的图）中研究了算符的库仑规范，它由Biot和Savart公式简单地给出。
英文摘要：
We study the spectral flow of Dirac operators with magnetic links on $\mathbb{S}^3$. These are generalisations of Aharonov-Bohm solenoids where the magnetic fields contain finitely many field lines coinciding with the components of a link, the flux of each exhibiting the same $2π$-periodicity as A-B solenoids. We study the spectral flow of the loop obtained as tuning the flux from $0$ to $2π$ in the case of only one field line: we relate the spectral flows obtained for one given knot and its cable knots, and obtain that torus knots have trivial spectral flow. The operators are studied in their Coulomb gauge in $\mathbb{R}^3$ (seen as a chart of $\mathbb{S}^3$ through the stereographic projection), which is simply given by the Biot and Savart formula.
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