nLab level (Chern-Simons theory)



\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory






Special and general types

Special notions


Extra structure





The local Lagrangian of Chern-Simons theory gives rise to an action functional which is gauge invariant only for certain discrete choices of the scale (a global prefactor) of the Lagrangian. This is called the level of the theory. The fact that it takes values in a discrete subgroup of the real numbers is called level quantization of the theory.


As a characteristic class

Formally, for gauge group being some compact Lie group GG, the level is identified with a choice of universal characteristic class, hence an element

[c]H 4(BG,) [c] \in H^4(B G, \mathbb{Z})

in the integral cohomology of the classifying space of GG. Under the Chern-Weil homomorphism every such cc corresponds to an invariant polynomial , c\langle -,-\rangle_c on the Lie algebra. (The point being here that while every scalar multiple of an invariant polynomial is itself again an invariant polynomial, only a lattice of these does corespond to a class in integral cohomology.)

If GG is furthermore connected and simply connected, then the local Lagrangian of Chern-Simons theory is the Chern-Simons form CS cCS_c of this specific invariant polynomial , c\langle -,-\rangle_c.

Generally, for c:BGB 3U(1)K(,4)c \;\colon\; B G \to B^3 U(1) \simeq K(\mathbb{Z},4) a universal characteristic map the extended Lagrangian of the corresponding 3d Chern-Simons theory is a differential refinement c^\hat \mathbf{c} of a lift c\mathbf{c} through geometric realization of smooth infinity-groupoids

BG conn c^ B 3U(1) conn H BG c B 3U(1) |Π()| BG c B 3U(1) L wheTop. \array{ \mathbf{B}G_{conn} &\stackrel{\hat \mathbf{c}}{\to}& \mathbf{B}^3 U(1)_{conn} \\ \downarrow && \downarrow &&& \mathbf{H} \\ \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^3 U(1) \\ && &&& \downarrow^{\mathrlap{{\vert \Pi(-)\vert}}} \\ B G &\stackrel{c}{\to}& B^3 U(1) &&& L_{whe} Top } \,.

Hence c^\hat \mathbf{c} is the “differentially refined level” of the theory. Notice that in terms of this the statement that the action functional is gauge invariant for “given level” is the statement that for Σ 3\Sigma_3 a closed manifold of dimension 3, the transgression of the extended Lagrangian c^\hat \mathbf{c} to the moduli stack fo fields on Σ 3\Sigma_3 is a map of smooth stacks

exp(2πi Σ 3[Σ 3,c^]):[Σ 3,BG conn]U(1). \exp\left( 2 \pi i \int_{\Sigma_3} [\Sigma_3, \hat \mathbf{c}] \right) \;\colon\; [\Sigma_3, \mathbf{B}G_{conn}] \to U(1) \,.

Analogous reasoning and termionology applies to higher dimensional Chern-Simons theory and generally to ∞-Chern-Simons theory.

Level renormalization

Upon quantization, the Chern-Simons level receives a renormalization which shifts it by the dual Coxeter number of the gauge group (see the references below).

extended prequantum field theory

0kn0 \leq k \leq n(off-shell) prequantum (n-k)-bundletraditional terminology
00differential universal characteristic maplevel
11prequantum (n-1)-bundleWZW bundle (n-2)-gerbe
kkprequantum (n-k)-bundle
n1n-1prequantum 1-bundle(off-shell) prequantum bundle
nnprequantum 0-bundleaction functional



For traditional accounts see at Chern-Simons theory - References.

Introductory discussion is in the section Physics in Higher Geometry: Motivation and Survey at

Level renormalization

On the renormalization of the Chern-Simons level (shift by the dual Coxeter number):

Last revised on May 30, 2022 at 10:18:08. See the history of this page for a list of all contributions to it.