2024/12/05 by Garoufalidis, Stavros, Scholze, Peter, Wheeler, Campbell +1 · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2412.04241
We introduce the Habiro ring of a number field \mathbbK and modules over it graded by K3(\mathbbK). 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 étale), 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.