发表机构
Institut des Hautes Études Scientifiques; CNRS, Laboratoire Alexander Grothendieck（高等科学研究所; 法国国家科学研究中心，亚历山大·格罗滕迪克实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明实二次域的 Stark 单位是代数数，通过发现模量子 dilogarithm 特殊值满足超定多项式方程组，并利用 Ocneanu 刚性定理，同时得到无限族无理近群融合范畴及 Stark 单位二次关系。
AI 中文摘要
我们证明了与实二次域相关的 Stark-Shintani 射线类不变量（Stark 单位）是代数数。这些不变量由 Faddeev 的模量子 dilogarithm 的特殊值给出，该函数由 Garoufalidis-Kashaev-Zagier 引入。我们的主要发现是，模量子 dilogarithm 的特殊值满足一个显式的超定多项式方程组，该方程组与 Andersen-Kashaev 在两个循环群的乘积上定义的量子 dilogarithm 的定义方程的变体相匹配。该方程组的解可用于对 Izumi 引入的融合环进行分类，而特殊值的代数性则可由 Ocneanu 的刚性定理得出。作为副产品，我们得到了一个显式的无限族无理近群融合范畴。作为进一步的应用，我们证明了 Appleby、Flammia 和 Kopp 最近基于 Zauner 关于 SIC-POVM（复等角线）猜想所提出的 Stark 单位的一族二次关系。
英文摘要
We prove that Stark--Shintani ray class invariants (Stark units) associated to real quadratic fields are algebraic numbers. These invariants are given by special values of Faddeev's modular quantum dilogarithm, introduced by Garoufalidis--Kashaev--Zagier. Our main discovery is that special values of the modular quantum dilogarithm satisfy an explicit overdetermined system of polynomial equations, matching a variation on the defining equations of Andersen--Kashaev's notion of a quantum dilogarithm on a product of two cyclic groups. We give two and a half proofs that this system of equations defines a zero-dimensional variety. The simplest follow from an uncertainty principle for finite Fourier transform and $2$-adic valuation bounds. The last proof is more involved and shows finite quanatum dilogarithms can be used to categorify fusion rings introduced by Izumi, and the algebraicity of the special values then follows by Ocneanu's rigidity theorem. As a byproduct, we obtain an explicit infinite family of irrational near-group fusion categories. As a further application, we prove a family of quadratic relations for Stark units recently conjectured by Appleby, Flammia, and Kopp motivated by Zauner's conjecture about SIC-POVMs (complex equiangular lines).
Comments36 pages, 1 figure
