Explicit classes in Habiro cohomology

TL;DR

This paper introduces explicit cycle descriptions of Habiro cohomology for smooth varieties, linking hypergeometric motives, $q$-deformations, and quantum $K$-theory with concrete examples like elliptic curves and Calabi-Yau families.

Contribution

It provides explicit cycle constructions in Habiro cohomology using hypergeometric motives and $q$-hypergeometric functions, unifying quantum $K$-theory and Chern-Simons theory approaches.

Findings

Explicit classes for Calabi-Yau families are constructed.

Cycles generate $q$-holonomic modules for $q$-deformations.

Applications demonstrated on elliptic curves, knot theory, and quintic three-folds.

Abstract

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.