Explicit classes in Habiro cohomology
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918034835243008 |
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| author | Garoufalidis, Stavros Wheeler, Campbell |
| author_facet | Garoufalidis, Stavros Wheeler, Campbell |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit classes in Habiro cohomology Garoufalidis, Stavros Wheeler, Campbell Algebraic Geometry High Energy Physics - Theory Geometric Topology 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. |
| title | Explicit classes in Habiro cohomology |
| topic | Algebraic Geometry High Energy Physics - Theory Geometric Topology |
| url | https://arxiv.org/abs/2505.19885 |