标题： 显式类在Habiro上同调中 显示英文标题

标题： Explicit classes in Habiro cohomology

摘要： 我们提出了一种关于Habiro上同调的循环描述，该描述适用于光滑簇$X$上，该簇的谱为 étale$Z[\lambda]$-代数的$B$，并构造了显式的非平凡循环。这些循环要么通过超几何动机$X/B$上的Picard-Fuchs方程获得，要么通过Habiro环$X/B$中元素的推进得到。特别是，我们给出了1参数Calabi-Yau族的显式类。 $q$-超几何起源的我们的循环表明它们生成定义经典Picard-Fuchs方程的$q$-惠更斯模的$q$-变形。 我们用三个例子来说明我们的定理：勒让德椭圆曲线族，图八纽结的$A$-多项式曲线，以及五次三维流形，在其亏格$0$-量子$K$-理论中出现了其$q$-Picard Fuchs方程。 我们的方法对高维临界点附近的量子$K$-理论和复Chern-Simons理论给出了统一处理。 显示英文摘要

摘要： We propose a cycle description of the Habiro cohomology of a smooth variety $X$ over the spectrum $B$ of an \'etale $Z[\lambda]$-algebra and construct explicit nontrivial cycles using either the Picard-Fuchs equation on $X/B$ of a hypergeometric motive, or a push-forward of elements of the Habiro ring of $X/B$. In particular, we give explicit classes for 1-parameter Calabi--Yau families. The $q$-hypergeometric origin of our cycles imply that they generate $q$-holonomic modules that define $q$-deformations of the classical Picard-Fuchs equation. We illustrate our theorems with three examples: the Legendre family of elliptic curves, the $A$-polynomial curve of the figure eight knot, and for the quintic three-fold, whose $q$-Picard Fuchs equation appeared in its genus $0$-quantum $K$-theory. Our methods give a unified treatment of quantum $K$-theory and complex Chern-Simons theory around higher dimensional critical loci.