论文标题
在4功能球上的非亚伯式霍奇对应的循环组方法
Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere
论文作者
论文摘要
非亚伯式杂物的对应关系是稳定希格斯束的模量空间与不可减至的平面连接的derham模量空间之间的真实分析图，该空间是由解决方案介导的对自偶性方程的介导的。在本文中，我们在强烈的抛物线$ \ mathfrak {sl}（2，\ mathbb c）$ 4 $ higgs字段上构建了自duality解决方案。 We identify the rescaled limit hyper-Kähler moduli space $\mathcal M_t$ at $t=0$ to be the completion of the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ modulo a $\mathbb Z_2\times\mathbb Z_2$ action, equipped with the Eguchi-Hanson metric.我们的方法和计算基于使用Deligne和Simpson的$λ$ -Connections解释的扭曲器方法进行自偶性方程。通过施工，我们可以计算$ \ Mathcal m_t $ t = 0 $上的全体形态符号形式的泰勒扩展，$ \ varpi_t $，这些$ \ mathcal m_t $ at $ t = 0 $，这些$ t = 0 $ t = 0 $，根据多个多聚类（MPLS）的封闭表达式。 $ \ Mathcal M_T $的几何特性导致了某些MPL的某些身份，我们认为这应该得到进一步的研究。
The non-abelian Hodge correspondence is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions to the self-duality equations. In this paper we construct self-duality solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a $4$-punctured sphere with parabolic weights $t \sim 0$ using complex analytic methods. We identify the rescaled limit hyper-Kähler moduli space $\mathcal M_t$ at $t=0$ to be the completion of the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ modulo a $\mathbb Z_2\times\mathbb Z_2$ action, equipped with the Eguchi-Hanson metric. Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson's $λ$-connections interpretation. By construction we can compute the Taylor expansions of the holomorphic symplectic form $\varpi_t$ on $\mathcal M_t$ at $t=0$ which turn out to have closed form expressions in terms of multiple polylogarithms (MPLs). The geometric properties of $\mathcal M_t$ lead to some identities of certain MPLs which we believe deserve further investigations.
