On arithmetic Dijkgraaf-Witten theory

TL;DR

This paper develops an arithmetic analogue of Dijkgraaf-Witten topological quantum field theory, constructing key elements like the Chern-Simons cocycle, prequantization bundle, and partition functions within number theory.

Contribution

It introduces a novel arithmetic framework for Chern-Simons theory and Dijkgraaf-Witten invariants, extending topological quantum field concepts to number fields.

Findings

Constructed arithmetic Chern-Simons 1-cocycle and prequantization bundle.

Defined arithmetic quantum Hilbert space and partition function.

Proved decomposition and gluing formulas for arithmetic invariants.

Abstract

We present basic constructions and properties in arithmetic Chern-Simons theory with finite gauge group along the line of topological quantum field theory. For a finite set $S$ of finite primes of a number field $k$, we construct arithmetic analogues of the Chern-Simons 1-cocycle, the prequantization bundle for a surface and the Chern-Simons functional for a $3$-manifold. We then construct arithmetic analogues for $k$ and $S$ of the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf-Witten partition function in (2+1)-dimensional Chern-Simons TQFT. We show some basic and functorial properties of those arithmetic analogues. Finally we show decomposition and gluing formulas for arithmetic Chern-Simons invariants and arithmetic Dijkgraaf-Witten partition functions.