Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit classes in Habiro cohomology

Stavros Garoufalidis, Campbell Wheeler

TL;DR

This work develops explicit cycle descriptions of Habiro cohomology $\,\mathcal{H}(X/B)\,$ for smooth varieties over étale $\,\mathbb{Z}[\lambda]\,$-algebras, revealing deformations of de Rham/Hodge theory around roots of unity. It introduces two complementary construction paths: (i) from hypergeometric motives via Gauss–Manin connections and $q$-deformations that yield $q$-holonomic modules and $q$-Picard–Fuchs equations, and (ii) push-forwards of Habiro rings along étale maps to produce cohomology classes. The authors demonstrate the framework with the Legendre family, the $A$-polynomial curve of the figure-eight knot, and the quintic Calabi–Yau, illustrating how $q$-hypergeometric origins tie quantum $K$-theory and complex Chern–Simons theory into a unified Habiro-cohomological picture. They develop $p$-adic convergence/gluing techniques, provide residue-based constructions, and discuss applications to perturbative invariants, $q$-deformations of PF equations, and the structure of Habiro cohomology as a module over Habiro rings. These results offer a path toward systematic, computable encodings of geometric and topological data in Habiro cohomology with potential links to TQFTs and CY/kit invariants.

Abstract

We propose a cycle description of the Habiro cohomology of a smooth variety $X$ over the spectrum $B$ of an étale $Z[λ]$-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.

Explicit classes in Habiro cohomology

TL;DR

This work develops explicit cycle descriptions of Habiro cohomology for smooth varieties over étale -algebras, revealing deformations of de Rham/Hodge theory around roots of unity. It introduces two complementary construction paths: (i) from hypergeometric motives via Gauss–Manin connections and -deformations that yield -holonomic modules and -Picard–Fuchs equations, and (ii) push-forwards of Habiro rings along étale maps to produce cohomology classes. The authors demonstrate the framework with the Legendre family, the -polynomial curve of the figure-eight knot, and the quintic Calabi–Yau, illustrating how -hypergeometric origins tie quantum -theory and complex Chern–Simons theory into a unified Habiro-cohomological picture. They develop -adic convergence/gluing techniques, provide residue-based constructions, and discuss applications to perturbative invariants, -deformations of PF equations, and the structure of Habiro cohomology as a module over Habiro rings. These results offer a path toward systematic, computable encodings of geometric and topological data in Habiro cohomology with potential links to TQFTs and CY/kit invariants.

Abstract

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

Paper Structure

This paper contains 32 sections, 23 theorems, 250 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A dream
  3. An experiment
  4. A definition
  5. Habiro cohomology classes from hypergeometric motives
  6. Elements in the Habiro ring of an étale $\mathbb Z[t]$-algebra
  7. Habiro cohomology classes from push-forward
  8. Habiro cohomology classes of Calabi--Yau manifolds
  9. Future directions
  10. From hypergeometric motives to Habiro cohomology
  11. $q$-Pochhammer symbols near roots of unity
  12. The Fermat curve
  13. Basics on balanced hypergeometric series
  14. From $q$-deformations to $D$-modules
  15. Convergence and gluing
  16. ...and 17 more sections

Key Result

Theorem 1.5

The collection $\omega_q=(\omega_{m,q-\zeta_m})_{m\geq1,(m,N)=1} \in \mathcal{H}_\mathrm{naiv}^{n}(X/B)$.

Figures (3)

  • Figure 1: The projection $Y\to\mathbb{G}_m$.
  • Figure 2: Henkel contours on the curve $X/\mathbb Z[1/30]$ over $u=\log(x)$-plane. The red curves cut of the principal branch of $\sqrt{\Delta(\exp(u))}$. When integrating this function one must change the sign of the integrand when crossing the red lines in order to perform the correct analytic continuation.
  • Figure 3: The Newton polygon of the 6th order operator \ref{['order6']}.

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 39 more