nLab arithmetic Chern-Simons theory

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.

Related concepts

• number theory and physics

• arithmetic gauge theory

• p-adic physics

  • p-adic string theory

  • p-adic AdS/CFT correspondence

References

• Minhyong Kim, Arithmetic Chern-Simons Theory I, in: Galois Covers, Grothendieck-Teichmüller Theory and Dessins d’Enfants, Proceedings in Mathematics & Statistics 330, Spinger (2020) [arXiv:1510.05818, doi:10.1007/978-3-030-51795-3_8]

• 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)

On entanglement entropy in arithmetic Chern-Simons theory:

• Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo, Entanglement entropies in the abelian arithmetic Chern-Simons theory [arXiv:2312.17138]

An introductory talk“

• Minhyong Kim, Arithmetic topological quantum field theory?, (slides)

Introduction in the broader context of arithmetic gauge theory:

• Minhyong Kim, Arithmetic Gauge Theory: A Brief Introduction, Modern Physics Letters A 33 29 (2018) 1830012 [arxiv:1712.07602, doi:10.1142/S0217732318300124]