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arXiv · 2106.02308

On arithmetic Dijkgraaf-Witten theory

Hikaru Hirano·Junhyeong Kim·Masanori Morishita
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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.

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Hikaru Hirano, Junhyeong Kim, Masanori Morishita. 2021-06-04. On arithmetic Dijkgraaf-Witten theory. https://arxiv.org/abs/2106.02308

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