AI 中文总结
研究克莱因曲面上向量丛和希格斯丛模空间实轨迹连通分支数，利用规范理论方法，确定一般\(r\)和\(d\)时的连通分支数，推广结果到希格斯丛并应用于相关膜拓扑研究，揭示不同条件下连通分支数的变化。
AI 中文摘要
设\(X\)为亏格\(g\geqslant2\)的黎曼曲面，\(\sigma:X\to X\)为\(X\)上的反全纯对合。设\(\mathcal{N}(r,d)\)为\(X\)上秩为\(r\)、次数为\(d\)的半稳定向量丛的模空间，并带有诱导实结构。利用规范理论方法，我们确定了一般\(r\)和\(d\)时\(\mathcal{N}(r,d)\)实轨迹的连通分支数。特别地，当基曲线有实点时，偶数秩和次数的四元数向量丛可存在，但\(\mathbb{R}\mathcal{N}(r,d)\)的连通分支数仍等于\(\mathbb{R}\mathrm{Pic}_d\)的连通分支数。相反，当基曲线实轨迹为空且\(r\)和\(d\)不互质时，\(\mathbb{R}\mathcal{N}(r,d)\)的连通分支数可能小于\(\mathbb{R}\mathrm{Pic}_d\)的连通分支数。然后我们将这些结果推广到希格斯丛模空间的实轨迹，并将其应用于相关超凯勒商中某些\((A,A,A)\)和\((A,B,A)\)膜的拓扑研究。
英文摘要
Let $X$ be a Riemann surface of genus $g \geqslant 2$ and let $σ: X \to X$ be an antiholomorphic involution on $X$. Let $\mathcal{N}(r,d)$ be the moduli space of semistable vector bundles of rank $r$ and degree $d$ on $X$, with the induced real structure. Using a gauge-theoretic approach, we determine the number of connected components of the real locus of $\mathcal{N}(r,d)$ for general $r$ and $d$. We show in particular that, when the base curve has real points, quaternionic vector bundles can exist for even rank and degree but that the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ is still equal to that of $\mathbb{R}\mathrm{Pic}_d$. In contrast, when the base curve has empty real locus and $r$ and $d$ are not coprime, the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ can be smaller than that of $\mathbb{R}\mathrm{Pic}_d$. We then generalize these results to real loci of moduli spaces of Higgs bundles and apply them to the study of the topology of certain $(A,A,A)$ and $(A,B,A)$ branes in the associated hyperkähler quotient.
Comments21 pages, 1 figure, All comments and questions are welcome!
