---
tag: F28G886PWD
kind: concept
name: Chern–Simons theory
msc: [57K16, 81T45, 58J28]
specializes: [DAP92B80J2]
sources:
  - url: https://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theory
same_as:
  wikidata: Q1528019
  wikipedia: Chern–Simons theory
---

## Summary

Chern–Simons theory is a gauge theory on $3$-manifolds whose action is the Chern–Simons functional of a connection.
The functional does not use a metric, so the quantum theory is a [[topological quantum field theory|DAP92B80J2]].
Witten showed in 1989 that its expectation values of Wilson loops give the [[Jones polynomial|4SPC17FPXE]].
Its partition functions give new invariants of [[3-manifolds|R20TB2KEQQ]], the [[quantum invariants|YGPR4VR3GP]].
Its classical solutions are the flat connections, which link it with representations of fundamental groups and with the [[Casson invariant|33GY6P5A4T]].

## Explanations

### L1 human

Chern–Simons theory is a physical theory of fields on a $3$-dimensional space that does not care about lengths or angles, only about shape.
Because of that, the numbers it produces are invariants of the space and of knots inside it.
Its most famous output is the Jones polynomial of a knot, which was first found by quite different methods.
It is a meeting point of knot theory, topology and quantum physics.