nLab level (Chern-Simons theory)

Idea

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.

Properties

As a characteristic class

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

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

in the integral cohomology of the classifying space of $G$. Under the Chern-Weil homomorphism every such $c$ corresponds to an invariant polynomial $\langle -,-\rangle_c$ 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 $G$ is furthermore connected and simply connected, then the local Lagrangian of Chern-Simons theory is the Chern-Simons form $CS_c$ of this specific invariant polynomial $\langle -,-\rangle_c$.

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

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

Hence $\hat \mathbf{c}$ 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 $\Sigma_3$ a closed manifold of dimension 3, the transgression of the extended Lagrangian $\hat \mathbf{c}$ to the moduli stack fo fields on $\Sigma_3$ is a map of smooth stacks

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

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).

Related concepts

• su(2)-anyon

• quantization of 3d Chern-Simons theory

extended prequantum field theory

$0 \leq k \leq n$	(off-shell) prequantum (n-k)-bundle	traditional terminology
$0$	differential universal characteristic map	level
$1$	prequantum (n-1)-bundle	WZW bundle (n-2)-gerbe
$k$	prequantum (n-k)-bundle
$n-1$	prequantum 1-bundle	(off-shell) prequantum bundle
$n$	prequantum 0-bundle	action functional

References

General

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

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

• geometry of physics.

Level renormalization

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

• Luis Alvarez-Gaumé, J. M. F. Labastida, A. V. Ramallo, A note on perturbative Chern-Simons theory, Nuclear Physics B 334 1 (1990) 103-124 (doi:10.1016/0550-3213(90)90658-Z)

• Mikhail A. Shifman, Four-dimension aspect of the perturbative renormalization in three-dimensional Chern-Simons theory, Nuclear Physics B 352 1 (1991) 87-112 (doi:10.1016/0550-3213(91)90130-P)

• M. Asorey, F. Falceto, J. L. Lopez, G. Luzon, Universalty and Ultraviolet Regularizations of Chern-Simons Theory, Nucl.Phys. B 429 (1994) 344-374 (arXiv:hep-th/9403117, doi:10.1016/0550-3213(94)00331-9)