nLab level (Chern-Simons theory)

Context

$\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Contents

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.

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.

extended prequantum field theory

$0 \leq k \leq n$(off-shell) prequantum (n-k)-bundletraditional terminology
$0$differential universal characteristic maplevel
$1$prequantum (n-1)-bundleWZW bundle (n-2)-gerbe
$k$prequantum (n-k)-bundle
$n-1$prequantum 1-bundle(off-shell) prequantum bundle
$n$prequantum 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