Communications in Number Theory and Physics

Volume 17 (2023)

Number 1

On arithmetic Dijkgraaf–Witten theory

Pages: 1 – 61



Hikaru Hirano (Graduate School of Mathematics, Kyushu University, Nishi-ku, Fukuoka, Japan)

Junhyeong Kim (Tokyo, Japan)

Masanori Morishita (Graduate School of Mathematics, Kyushu University, Nishi-ku, Fukuoka, Japan)


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.


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