论文标题
Siegel模块化品种的BOST-CONNES-MARCOLLI系统
Bost-Connes-Marcolli system for the Siegel modular variety
论文作者
论文摘要
我们构建了一个量子的满足机械系统，该系统概括了Connes-Marcolli $ GL_2 $系统。特别是，我们介绍了与Siegel模块化程度$ 2 $相关的Connes-Marcolli系统。我们将其$ \ text {kmms}_β$ - 属于反温度$β> 0 $的状态分类，并表明自发相变发生在$β= 3 $。更准确地说，我们证明该系统不承认$ \ text {kms}_β$状态$β<3 $，$β<3 $带有$β\ neq 1 $，构建明显的极gibbs，以$β> 4 $表示$β> 4 $，并显示出一个唯一的$ \文本{kmms}_β$ com $ $ $β> $ $ $ 3 <4.
We construct a quantum satisitical mechanical system which generalizes the Connes-Marcolli $GL_2$ system. In particular we introduce the Connes-Marcolli system associated to the Siegel modular variety of degree $2$. We classify its $\text{KMS}_β$-states for inverse temperatures $β>0$ and show that a spontaneous phase transition occurs at $β=3$. More precisely, we prove that the system does not admit a $\text{KMS}_β$ state for $β< 3$ with $β\neq 1$, construct the explicit extremal Gibbs states for $β>4$ and show that a unique $\text{KMS}_β$ state exists for every $β>0$ with $3 < β\leq 4$.
