arXiv 2509.020022025-09math.DGmath.AGmath.RA

对合代数上群的对称空间及其在Higgs丛中的应用

Symmetric spaces for groups over involutive algebras and applications to Higgs bundles

Pengfei Huang, Georgios Kydonakis, Eugen Rogozinnikov, Anna Wienhard

备注：: 95 pages

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AI总结：

研究对合代数上的辛群和不定正交群，识别其极大紧子群并构造黎曼对称空间的几何模型，进而应用于Higgs丛模空间的计数和Hitchin态射的分解。

摘要：

我们研究对合（可能非交换）代数$(A, \sigma)$上的辛群和不定正交群。当代数$(A, \sigma)$是Hermite的，或是Hermite对合代数的复化$(A_{\mathbb C}, \sigma_{\mathbb C})$时，可以识别这些群的极大紧子群，并考虑其相关的黎曼对称空间。这一新视角使得对称空间的各种几何模型得以实现。我们显式描述了每个模型的复化切空间，以及它们之间的微分同胚及其微分。在文章的第二部分，我们给出了该理论的一些应用。第一部分描述的黎曼对称空间的几何实现为Higgs丛数据提供了新的几何解释，这些数据可用于研究Hermite对合代数上辛群或不定正交群的基本群表示。我们利用相应极大紧子群的拓扑而非Morse-Bott理论技术，给出了$\rm{Sp}_2(A_{\mathbb C}, \sigma_{\mathbb C})$-Higgs丛和$\rm O(A_{\mathbb C}, \sigma_{\mathbb C})$-Higgs丛模空间的精确分量计数。此外，我们利用非交换对称空间模型构造了$\rm{Sp}_2(A_{\mathbb C},\sigma_{\mathbb C})$-Higgs丛的Hitchin态射的分解，以及实群$\rm{Sp}_2(A,\sigma)$和$\rm O_{(1,1)}(A,\sigma)$的类似分解。这些分解由从相应切模型到Jordan代数目标的二次范数映射诱导，并通过中间仿射GIT商。因此，它们降低了显式刻画Hitchin基所需的代数复杂度。

英文摘要：

We study symplectic groups and indefinite orthogonal groups over involutive, possibly noncommutative, algebras $(A, σ)$. In the case when the algebra $(A, σ)$ is Hermitian, or the complexification $(A_{\mathbb C}, σ_{\mathbb C})$ of a Hermitian involutive algebra, one can identify maximal compact subgroups of such groups, and consider their associated Riemannian symmetric spaces. This new perspective allows for the realization of various geometric models for the symmetric space. We describe explicitly the complexified tangent space for each of the models, as well as the diffeomorphisms between them and their differentials. In the second part of the article, we give a number of applications of this theory. The geometric realizations of the Riemannian symmetric spaces described in the first part provide new geometric interpretations of Higgs bundle data that can be used for the study of fundamental group representations into symplectic or into indefinite orthogonal groups over Hermitian involutive algebras. We give an exact component count for the moduli spaces of $\rm{Sp}_2(A_{\mathbb C}, σ_{\mathbb C})$-Higgs bundles and of $\rm O(A_{\mathbb C}, σ_{\mathbb C})$-Higgs bundles, using the topology of the corresponding maximal compact subgroups rather than Morse-Bott theory techniques. Furthermore, we use the noncommutative symmetric-space models to construct a factorization of the Hitchin morphism for $\rm{Sp}_2(A_{\mathbb C},σ_{\mathbb C})$-Higgs bundles, together with analogous factorizations for the real groups $\rm{Sp}_2(A,σ)$ and $\rm O_{(1,1)}(A,σ)$. These factorizations are induced by quadratic norm maps from the corresponding tangent models to Jordan-algebraic targets and pass through intermediate affine GIT quotients. As a consequence, they reduce the algebraic complexity required in order to characterize the Hitchin base explicitly.

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