nLab
arithmetic Chern-Simons theory

Contents

Context

Arithmetic geometry

Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

Arithmetic Chern-Simons theory names the attempt to apply constructions from Chern-Simons theory to the field of arithmetic, in view of patterns in the function field analogy. In particular, the papers (Kim1, Kim2) apply ideas of Dijkgraaf-Witten theory on 2+1 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields.

This theory pursues the surprising analogies between 3-dimensional topology and number theory, where knots embedded in a 3-manifold behave like prime ideals in a ring of algebraic integers, known as arithmetic topology.

References

  • Minhyong Kim, Arithmetic Chern-Simons Theory I, (arXiv:1510.05818).

  • Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo, Arithmetic Chern-Simons Theory II, (arXiv:1609.03012)

  • Frauke M. Bleher, Ted Chinburg, Ralph Greenberg, Mahesh Kakde, George Pappas, Martin J. Taylor, Unramified arithmetic Chern-Simons invariants, (arXiv:1705.07110)

  • Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, George Pappas, Jeehoon Park, Hwajong Yoo, Abelian arithmetic Chern-Simons theory and arithmetic linking numbers, (arXiv:1706.03336)

  • Hikaru Hirano, On mod 2 arithmetic Dijkgraaf-Witten invariants for certain real quadratic number fields, (arXiv:1911.12964)

  • Jungin Lee, Jeehoon Park, Arithmetic Chern-Simons theory with real places, (arXiv:1905.13610)

For an introductory talk see

Last revised on January 8, 2020 at 08:00:34. See the history of this page for a list of all contributions to it.