Contents Online

# Communications in Number Theory and Physics

## Volume 17 (2023)

### Number 1

### On arithmetic Dijkgraaf–Witten theory

Pages: 1 – 61

DOI: https://dx.doi.org/10.4310/CNTP.2023.v17.n1.a1

#### Authors

#### 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.

#### Keywords

arithmetic Chern–Simons theory, arithmetic topology, Dijkgraaf–Witten theory, topological quantum field theory

#### 2010 Mathematics Subject Classification

Primary 11Rxx, 81Txx. Secondary 57Mxx.

Dedicated to the memory of Professor Toshie Takata.

Received 7 June 2021

Accepted 24 September 2022

Published 23 February 2023