On arithmetic Dijkgraaf–Witten theory

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.