The Habiro ring of a number field

TL;DR

This paper introduces the Habiro ring for number fields, connecting algebraic K-theory, power series at roots of unity, and quantum invariants, with implications for arithmetic and enumerative geometry.

Contribution

It defines the Habiro ring for number fields and explores modules over it, linking algebraic K-theory, power series, and quantum invariants in a novel framework.

Findings

Defined the Habiro ring for number fields.

Connected power series to algebraic K-theory and regulators.

Suggested arithmetic and enumerative interpretations of quantum invariants.

Abstract

We introduce the Habiro ring of a number field $\mathbb{K}$ and modules over it graded by $K_3(\mathbb{K})$. Elements of these modules are collections of power series at each complex root of unity that arithmetically glue with each other after applying a Frobenius endomorphism, and after dividing at each prime by a collection of series that depends solely on an element of the Bloch group. The main theorems of this paper concern number fields, their algebraic $K$-theory and its regulator maps (Borel, $p$-adic and \'etale), whereas the explicit collections of series are defined by a careful algebraic analysis of the infinite Pochhammer symbol at roots of unity. The origin of the above mentioned power series comes from perturbative Chern--Simons theory and by expansions of the admissible series of Kontsevich--Soibelman, both ultimately related to the infinite Pochhammer symbol. This link suggests that some Donaldson-Thomas invariants have arithmetic meaning and that some elements of the Habiro ring of a number field have enumerative meaning. Added subsection 1.1 explaining what the paper is about and subsection 1.8 explaining the relation to perturbative complex Chern-Simons theory.