Duality for arithmetic Dijkgraaf-Witten theory

TL;DR

This paper extends the classification of equivalences in Dijkgraaf-Witten theories from finite groups to arithmetic settings over number fields, identifying conditions for when these theories are equivalent or not.

Contribution

It establishes an analogy for arithmetic Dijkgraaf-Witten theory over number fields and provides examples and conditions for equivalence or failure of equivalence.

Findings

Identifies conditions under which arithmetic Dijkgraaf-Witten theories are equivalent.

Provides examples with quadratic fields and quaternion groups where equivalences fail.

Proposes criteria for when these theories remain equivalent in arithmetic contexts.

Abstract

Naidu classified pairs of finite groups and 3-cocycles that lead to equivalent Dijkgraaf-Witten theories for 3-manifolds. We establish analogous equivalences for arithmetic Dijkgraaf-Witten theory over totally imaginary number fields F containing n-th roots of unity, where n is invertible on X subset spec O_F. For the full ring of integers X = spec O_F, we give examples with quadratic fields and the quaternion group Q_8 where these equivalences fail, but also identify sufficient conditions under which they still hold.