arXiv 2401.037052024-01math.RThep-lathep-thmath-phmath.MPmath.OA

布拉特利网络与箭图上的谱作用

Bratteli networks and the Spectral Action on quivers

Carlos I. Perez-Sanchez

备注：: 36 pp, many quivers; v3 several significant amends, new results

AI总结：

本文在非交换几何背景下研究有限维预谱三元组上的箭图表示，引入布拉特利网络刻画表示空间等结构，基于表示论数据构建谱三元组并计算谱作用，应用于格箭图得到杨-米尔斯格规范理论等结果，还推导出自环涌现的希格斯场及光滑极限下的杨-米尔斯-希格斯理论。

摘要：

在非交换几何背景下，我们研究箭图表示——并非传统的向量空间上的表示，而是有限维预谱三元组（‘离散拓扑非交换空间’）上的表示。类似的想法曾出现在Marcolli-van Suijlekom关于谱三元组中箭图表示（‘离散非交换几何’）的原始工作中，该工作为我们在独立方向上的部分结果奠定了基础。我们引入布拉特利网络这一结构，它为箭图Q的预谱三元组表示空间Rep Q、规范群及其商群提供了简洁的组合刻画。我们认为，这些使‘在Rep Q上积分’成为可能的主张不仅符合随机非交换几何的精神——即建立狄拉克算子上的路径积分，还包含一个物理相关的案例。具体而言，本文为新范畴建立的箭图表示与路代数模之间的等价性启发了如下构造：仅从表示论数据出发，我们为箭图构建一个谱三元组，并通过闭合路径上的一般公式计算谱作用泛函。当将此构造应用于格箭图时，我们不仅得到威尔逊型杨-米尔斯格规范理论，还在Symanzik改进规范理论的背景下得到Weisz-Wohlert胞。我们证明，厄米（‘希格斯’）矩阵场从箭图的自环中涌现，并推导出平坦空间上的杨-米尔斯-希格斯理论作为光滑极限。

英文摘要：

In the context of noncommutative geometry, we consider quiver representations -- not on vector spaces, as traditional, but on finite-dimensional prespectral triples (`discrete topological noncommutative spaces'). A similar idea appeared in the original work of Marcolli-van Suijlekom on quiver representations in spectral triples (`discrete noncommutative geometries'), which paved the way for some of our results in independent directions. We introduce Bratteli networks, a structure that yields a neat combinatorial characterisation of the space $\mathrm{Rep}~Q$ of prespectral-triple-representations of a quiver $\mathrm{Rep}~Q$, as well as of the gauge group and of their quotient. Not only these claims that make it possible to `integrate over $\mathrm{Rep}~Q$' are, as we now argue, in line with the spirit of random noncommutative geometry -- formulating path integrals over Dirac operators -- but they also contain a physically relevant case. Namely, the equivalence between quiver representations and path algebra modules, established here for the new category, inspired the following construction: Only from representation theory data, we build a spectral triple for the quiver and evaluate the spectral action functional from a general formula over closed paths. When we apply this construction to lattice-quivers, we obtain not only Wilsonian Yang-Mills lattice gauge theory, but also the Weisz-Wohlert-cells in the context of Symanzik's improved gauge theory. We show that a hermitian (`Higgs') matrix field emerges from the self-loops of the quiver and derive the Yang-Mills--Higgs theory on flat space as a smooth limit.

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