Arithmetic Chern-Simons Theory II

TL;DR

This paper extends the concepts of Chern-Simons theory to arithmetic curves, defining actions on Galois representations and providing computational formulas with potential applications in number theory.

Contribution

It introduces a novel arithmetic analogue of Chern-Simons theory, connecting topological quantum field theory with algebraic number theory.

Findings

Defined classical Chern-Simons actions on Galois representation spaces

Provided formulas for computations in specific cases

Suggested potential applications in arithmetic geometry

Abstract

We apply ideas of Dijkgraaf and Witten on three-dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of cases and point towards some arithmetic applications.