Explicit classes in Habiro cohomology
Stavros Garoufalidis, Campbell Wheeler

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 over the spectrum of an \'etale -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…
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
