Chern–Simons theory

Builds on 3-manifold Principal bundle Connection on a principal bundle Differential form

Definition

Let $G = SU(n)$ , a compact Lie group with Lie algebra $\mathfrak{su}(n)$ , and let $M$ be a closed oriented 3-manifold. Every principal $G$ -bundle over $M$ is trivial, so a connection on it is given by a $\mathfrak{su}(n)$ -valued differential $1$ -form $A$ on $M$ . The Chern–Simons functional is $CS(A) = \frac{1}{8\pi^2} \int_M \tr\left(A \wedge dA + \tfrac{2}{3} A \wedge A \wedge A\right).$ For an integer $k$ , called the level, Chern–Simons theory is the gauge theory on $M$ with gauge group $G$ and action $S_k(A) = 2\pi k \, CS(A)$ . Its quantum partition function is the formal path integral $Z_k(M) = \int e^{i S_k(A)} \, \mathcal{D}A$ over connections modulo gauge transformations.

Remarks

Example: the trivial connection $A = 0$ has $CS(0) = 0$ . The first variation of $CS$ is $\delta CS = \frac{1}{4\pi^2} \int_M \tr(\delta A \wedge F_A)$ , where $F_A = dA + A \wedge A$ is the curvature, so the critical points of $CS$ are exactly the flat connections.

$CS$ is not invariant under gauge transformations: a gauge transformation $g \colon M \to G$ changes $CS(A)$ by an integer, the winding number of $g$ up to sign. So $CS$ is not a function on connections modulo gauge, but $e^{i S_k}$ is, because $k$ is an integer; this is why the level must be an integer.

The path integral is not a rigorous mathematical definition. Its predicted values are realised rigorously by the Witten–Reshetikhin–Turaev invariants.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

3-manifold concept
Connection on a principal bundle concept
Differential form concept
Principal bundle concept
Chern–Simons theory definition

Tag	`E0NXWECXTB`
URI	`m2://E0NXWECXTB`
Kind	definition
Writing	W0 stub
MSC	57K16, 81T45, 58J28
Defines	Chern–Simons theory
Uses	3-manifold, Principal bundle, Connection on a principal bundle, Differential form
Changed	2 November 2026

Chern–Simons theory

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here