算术拓扑中的量子统计力学
Quantum statistical mechanics in arithmetic topology
摘要：
本文给出了一个与3-球面中的结点和3-球面的循环分支覆盖相关联的量子统计力学系统的构造，它在算术拓扑意义上是Bost-Connes系统的一个模拟，结点代替了素数，3-球面的循环分支覆盖代替了有理数域的阿贝尔扩张.该系统的算子代数性质显着不同的Bost-Connes的情况下，由于性质的行动的半群的结结群的直接限制。由此产生的代数的可观是一个非交换伯努利交叉产品。我们描述了相关的量子统计力学系统和相关的配分函数，这是从简单的结不变量，如属和交叉数的主要性质。
英文摘要：
This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost-Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of observables is a noncommutative Bernoulli crossed product. We describe the main properties of the associated quantum statistical mechanical system and of the relevant partition functions, which are obtained from simple knot invariants like genus and crossing number.
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