nLab Chern-Simons form

Context

Differential cohomology

differential cohomology

Contents

Idea

A Chern-Simons form $\mathrm{CS}\left(A\right)$ is a differential form naturally associated to a differential form $A\in {\Omega }^{1}\left(P,𝔤\right)$ with values in a Lie algebra $𝔤$: it is the form trivializing (locally) a curvature characteristic form $⟨{F}_{A}\wedge \cdots \wedge {F}_{A}⟩$ of $A$, for $⟨\cdots ⟩$ an invariant polynomial:

${d}_{\mathrm{dR}}\mathrm{CS}\left(A\right)=⟨{F}_{A}\wedge \cdots \wedge {F}_{A}⟩\phantom{\rule{thinmathspace}{0ex}},$d_{dR} CS(A) = \langle F_A \wedge \cdots \wedge F_A \rangle \,,

where ${F}_{A}\in {\Omega }^{2}\left(X,𝔤\right)$ is the curvature 2-form of $A$.

Therefore it is often also called a secondary characteristic form.

More generally, for $A,A\prime \in {\Omega }^{1}\left(P,𝔤\right)$ two $𝔤$-valued 1-forms and for $\stackrel{^}{A}\in {\Omega }^{1}\left(P×\left[0,1\right],𝔤\right)$ a “path of connections”, the Chern-Simons form relative to $A$ and $A\prime$ is a form that trivializes the difference between the two curvature characteristic forms

${d}_{\mathrm{dR}}\mathrm{CS}\left(A,A\prime \right)=⟨\left({F}_{A}{\right)}^{k}⟩-⟨\left({F}_{A\prime }{\right)}^{k}⟩\phantom{\rule{thinmathspace}{0ex}}.$d_{dR}CS(A,A') = \langle (F_A)^k \rangle - \langle (F_{A'})^k \rangle \,.

Chern-Simons forms are of interest notably when the differential forms $A,A\prime$ are (local representatives of) connections on a $G$-principal bundle $P\to X$, for instance if $A\in {\Omega }^{1}\left(P,𝔤\right)$ is an Ehresmann connection 1-form.

Often the term Chern-Simons form is taken to refer to the case where $𝔤$ is a semisimple Lie algebra with binary invariant polynomial $⟨-,-⟩$ (e.g. the Killing form) in which case $\mathrm{CS}\left(A\right)$ is the 3-form

$⟨A\wedge {d}_{\mathrm{dR}}A⟩+c⟨A\wedge \left[A\wedge A\right]⟩\phantom{\rule{thinmathspace}{0ex}}.$\langle A \wedge d_{dR} A\rangle + c \langle A \wedge [A \wedge A] \rangle \,.

Even more specifically, often the term is understood to refer to the case where $𝔤\subset \mathrm{𝔤𝔩}\left(n\right)$ is a matrix Lie algebra, for instance $𝔬\left(n\right)$ (for the orthogonal group) or notably $𝔲\left(n\right)$ (for the unitary group). In that case the invariant polynomials may be taken to be given by matrix traces: $⟨\cdots ⟩=\mathrm{tr}\left(\cdots \right)$.

Details

It is sufficient to discuss properties of Chern-Simons forms for $𝔤$-valued 1-forms. The corresponding statements for connections on a $G$-bundle follow straightforwardly.

Paths of connections

Let $U$ be a smooth manifold.

Definition

A smooth path of $𝔤$-valued 1-forms on $U$ is a smooth 1-form $\stackrel{^}{A}\in {\Omega }^{1}\left(U×\left[0,1\right],𝔤\right)$

Call this path pure shift if ${\iota }_{{\partial }_{t}}\stackrel{^}{A}=0$, where $t:U×\left[0,1\right]\to \left[0,1\right]↪ℝ$ is the canonical coordinate along the interval.

We say this path goes from ${A}_{0}:={\psi }_{0}^{*}\stackrel{^}{A}$ to ${A}_{1}:={\psi }_{1}^{*}\stackrel{^}{A}$, where

${\psi }_{t}:U\simeq U×*\stackrel{\mathrm{Id}×t}{\to }U×\left[0,1\right]$\psi_t : U \simeq U \times * \stackrel{Id \times t}{\to} U \times [0,1]

picks the copy of $U$ at parameter $t$.

So a smooth path is a smooth 1-form on the cylinder $U×\left[0,1\right]$ and it is pure shift if it has no “leg” along the $\left[0,1\right]$-direction. We will see that ${\iota }_{{\partial }_{t}}\stackrel{^}{A}$ encodes infinitesimal gauge transformations, while ${\partial }_{t}\stackrel{^}{A}$ is the change by infinitesimal shifts minus infinitesimal gauge transformations of the connection.

Definition/Observation

Let $P$ be an invariant polynomial on $𝔤$ of arity $n$.

Consider the fiber integration

${\mathrm{CS}}_{P}\left({A}_{0},{A}_{1}\right):={\int }_{\left[0,1\right]}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)\phantom{\rule{thinmathspace}{0ex}}.$CS_P(A_0,A_1) := \int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \,.

This defines a $\left(2n-1\right)$-form ${\mathrm{CS}}_{P}\left({A}_{0},{A}_{1}\right)\in {\Omega }^{2n-1}\left(U\right)$.

We have that the exterior differential of this form is the difference of the curvature characteristic forms of ${A}_{0}$ and ${A}_{1}$:

${d}_{\mathrm{dR}}{\mathrm{CS}}_{P}\left({A}_{0},{A}_{1}\right)=P\left({F}_{{A}_{1}}\wedge \cdots \wedge {F}_{{A}_{1}}\right)-P\left({F}_{{A}_{0}}\wedge \cdots \wedge {F}_{{A}_{0}}\right)$d_{dR} CS_P(A_0,A_1) = P(F_{A_1} \wedge \cdots \wedge F_{A_1}) - P(F_{A_0} \wedge \cdots \wedge F_{A_0})
Proof

Write the fiber integration more explicitly as an integral

${\mathrm{CS}}_{P}\left({A}_{0},{A}_{1}\right)={\int }_{\left[0,1\right]}{\psi }_{t}^{*}{\iota }_{{\partial }_{t}}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)dt\phantom{\rule{thinmathspace}{0ex}}.$CS_P(A_0,A_1) = \int_{[0,1]} \psi_t^* \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \,.

Then use that ${d}_{\mathrm{dR}}$ is linear and commutes with pullback, use Cartan's magic formula ${d}_{\mathrm{dR}}\circ {\iota }_{{\partial }_{t}}+{\iota }_{\partial }\circ {d}_{\mathrm{dR}}={ℒ}_{{\partial }_{t}}$ in view of the fact that $P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)$ is a closed form and then finally apply the Stokes theorem:

$\begin{array}{rl}{d}_{\mathrm{dR}}{\int }_{0}^{1}{\psi }_{t}^{*}{\iota }_{{\partial }_{t}}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)dt& ={\int }_{0}^{1}{d}_{\mathrm{dR}}{\psi }_{t}^{*}{\iota }_{{\partial }_{t}}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)dt\\ & ={\int }_{0}^{1}{\psi }_{t}^{*}{d}_{\mathrm{dR}}{\iota }_{{\partial }_{t}}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)dt\\ & ={\int }_{0}^{1}{\psi }_{t}^{*}\frac{d}{dt}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)dt\\ & ={\int }_{0}^{1}\left(\frac{d}{dt}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)\right)\left(t\right)dt\\ & ={\psi }_{1}^{*}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)-{\psi }_{0}^{*}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)\\ & =P\left({F}_{{A}_{1}}\wedge \cdots \wedge {F}_{{A}_{1}}\right)-P\left({F}_{{A}_{0}}\wedge \cdots \wedge {F}_{{A}_{0}}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} d_{dR} \int_0^1 \psi^*_t \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t & = \int_0^1 d_{dR} \psi^*_t \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \\ & = \int_0^1 \psi^*_t d_{dR} \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \\ & = \int_0^1 \psi^*_t \frac{d}{d t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \\ & = \int_0^1 \left(\frac{d}{d t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})\right)(t) d t \\ & = \psi^*_1 P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) - \psi^*_0 P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \\ &= P(F_{A_1} \wedge \cdots \wedge F_{A_1}) - P(F_{A_0} \wedge \cdots \wedge F_{A_0}) \end{aligned} \,.

Explicit formulas

Above we saw that a general expression for the Chern-Simons ${\mathrm{CS}}_{P}\left({A}_{0},{A}_{1}\right)$ obtained from a path of connections $\stackrel{^}{A}$ between ${A}_{0}$ and ${A}_{1}$ is

${\mathrm{CS}}_{P}\left({A}_{0},{A}_{1}\right)={\int }_{\left[0,1\right]}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)\phantom{\rule{thinmathspace}{0ex}}.$CS_P(A_0, A_1) = \int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \,.

We now unwind this to get explicit formulas for the Chern-Simons form in terms of wedge products of connection forms and their curvatures.

For $\stackrel{^}{A}$ a pure shift path, $\stackrel{^}{A}:t↦{A}_{t}$ notice that the curvature 2-form of $\stackrel{^}{A}$ is

${F}_{\stackrel{^}{A}}\left(t\right)={F}_{{A}_{t}}+\left({\partial }_{t}{A}_{t}\right)\wedge dt\phantom{\rule{thinmathspace}{0ex}}.$F_{\hat A}(t) = F_{A_t} + (\partial_t A_t) \wedge d t \,.

Inserting this into the above expression yields

${\mathrm{CS}}_{P}\left({A}_{0},{A}_{1}\right)={\int }_{0}^{1}P\left({\partial }_{t}A\wedge {F}_{{A}_{t}}\wedge \cdots \wedge {F}_{{A}_{t}}\right)\phantom{\rule{thinmathspace}{0ex}}.$CS_P(A_0,A_1) = \int_0^1 P(\partial_t A \wedge F_{A_t} \wedge \cdots \wedge F_{A_t}) \,.

Notably if ${A}_{0}=0$ and $\stackrel{^}{A}$ is the constant path $\stackrel{^}{A}:t↦tA$ to ${A}_{1}:=A$ such that

$\begin{array}{rl}{F}_{\stackrel{^}{A}}& =t{d}_{\mathrm{dR}}A+{t}^{2}\left[A\wedge A\right]\\ & =t{F}_{A}+\left({t}^{2}-t\right)\left[A\wedge A\right]\end{array}$\begin{aligned} F_{\hat A} &= t d_{dR} A + t^2 [A \wedge A] \\ &= t F_A + (t^2 - t) [A \wedge A] \end{aligned}

this yields

${\mathrm{CS}}_{P}\left(A\right):={\int }_{0}^{1}P\left(A\wedge \left(t{F}_{A}+\left({t}^{2}-t\right)\left[A\wedge A\right]\right)\right)\wedge \cdots \left(t{F}_{A}+\left({t}^{2}-t\right)\left[A\wedge A\right]\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$CS_P(A) := \int_0^1 P(A \wedge (t F_{A} + (t^2 - t) [A \wedge A])) \wedge \cdots (t F_{A} + (t^2 - t) [A \wedge A]))) \,.

This is just an integral over a polynomial in $t$ with constant coefficients in forms. Peforming the integral yields a bunch of coefficients ${c}_{i}$ and with these the Chern-Simons form achieves the form

$\mathrm{CS}\left(A\right)={c}_{1}⟨A\wedge {F}_{A}\wedge \cdots \wedge {F}_{A}⟩+{c}_{2}⟨A\wedge \left[A\wedge A\right]\wedge {F}_{A}\wedge \cdots {F}_{A}⟩+\cdots \phantom{\rule{thinmathspace}{0ex}}.$CS(A) = c_1 \langle A \wedge F_A \wedge \cdots \wedge F_A \rangle + c_2 \langle A \wedge [A \wedge A] \wedge F_A \wedge \cdots F_A \rangle + \cdots \,.

Particularly for $n=2$ and using the definition of the curvature 2-form ${F}_{A}={d}_{\mathrm{dR}}A+\left[AwdgeA\right]$ we get

$\mathrm{CS}\left(A\right)=⟨A\wedge dA⟩+c⟨A\wedge \left[A\wedge A\right]⟩\phantom{\rule{thinmathspace}{0ex}}.$CS(A) = \langle A \wedge d A\rangle + c \langle A \wedge [A \wedge A]\rangle \,.

Gauged paths of connections

Above we defined $\mathrm{CS}\left({A}_{0},{A}_{1}\right)$ for every path of connections form ${A}_{0}$ to ${A}_{1}$ which is pure shift . This is a possibly convenient but unnecessary restriction:

Notice that a general (gauged) path is a general 1-form $\stackrel{^}{A}\in {\Omega }^{1}\left(U×\left[0,1\right],𝔤\right)$ which we can decompose in the form

$\stackrel{^}{A}:t↦{A}_{t}+\lambda dt\phantom{\rule{thinmathspace}{0ex}},$\hat A : t \mapsto A_t + \lambda d t \,,

where $\lambda$ is a $𝔤$-valued function. The parallel transport of $\lambda dt$ along $\left[0,1\right]$ defines an element in $G$ and shift $\left({\partial }_{t}A{\right)}_{t}$ of the connection along $\left[0,1\right]$ is now relative to the gauge transformation on $A$ induced by this function: the curvature 2-form now is

${F}_{\stackrel{^}{A}}:t↦{F}_{{A}_{t}}+\left(\left({\partial }_{t}A{\right)}_{t}+{d}_{U}\lambda \left(t\right)+\left[\lambda ,{A}_{t}\right]\right)\wedge dt$F_{\hat A} : t \mapsto F_{A_t} + ((\partial_t A)_t + d_U \lambda(t) + [\lambda,A_t]) \wedge d t
Proposition

The Chern-Simons form ${\int }_{\left[0,1\right]}P\left({F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}\right)$ defined with respect to any gauged lift of a pure shift path of connections differs from that of the pure shift path by an exact term.

In $\infty$-Chern-Weil theory

We discuss now a more encompassing perspective on Chern-Simons forms the way it occurs in ∞-Chern-Weil theory.

Prerequisites

We need to collect a few notions described elsewhere, on which the following discussion is based.

For $𝔤$ a Lie algebra or more generally an ∞-Lie algebra we have the following dg-algebras naturally associated with it:

Given $n\in ℕ$, the Lie integration of $𝔤$ to degree $n$ is the ∞-Lie groupoid which is the $n$-truncation of the simplicial presheaf

$\mathrm{exp}\left(𝔤\right):U,\left[n\right]↦\mathrm{dgAlg}\left(\mathrm{CE}\left(𝔤\right),{C}^{\infty }\left(U\right)\otimes {\Omega }^{•}\left({\Delta }^{n}\right)\right)\phantom{\rule{thinmathspace}{0ex}},$\exp(\mathfrak{g}) : U,[n] \mapsto dgAlg( CE(\mathfrak{g}), C^\infty(U)\otimes \Omega^\bullet(\Delta^n) ) \,,

where here and in the following ${\Omega }^{•}\left({\Delta }^{n}\right)$ denotes the de Rham complex dg-algebra of those smooth differential forms $\omega$ on the standard smooth $n$-simplex that have sitting instants in that for each $k\in ℕ$ every $k$-face of ${\Delta }^{n}$ has an open neighbourhood such that restricted to that neighbourhood $\omega$ is constant in the direction perpendicular to the face.

This is a one-object ∞-Lie groupoid which we may write

$BG={\mathrm{cosk}}_{n+1}\mathrm{exp}\left(𝔤\right)\phantom{\rule{thinmathspace}{0ex}},$\mathbf{B}G = \mathbf{cosk}_{n+1} \exp(\mathfrak{g}) \,,

thus defining the ∞-Lie group $G$ that integrates $𝔤$ in degree $n$.

At ∞-Chern-Weil theory is explained that a resolution of $BG$ that serves to compute curvature characteristic forms in that it encodes pseudo-connections on $G$-principal ∞-bundles is given by the simplicial presheaf

$B{G}_{\mathrm{diff}}:={\mathrm{cosk}}_{n+1}\left(U,\left[n\right]↦\left\{\begin{array}{ccc}{C}^{\infty }\left(U\right)\otimes {\Omega }^{•}\left({\Delta }^{n}\right)& ←& \mathrm{CE}\left(𝔤\right)\\ ↑& & ↑\\ {\Omega }^{•}\left(U\right)\otimes {\Omega }^{•}\left({\Delta }^{n}\right)& ←& W\left(𝔤\right)\end{array}\right\}\right)\phantom{\rule{thinmathspace}{0ex}},$\mathbf{B}G_{diff} := \mathbf{cosk}_{n+1} \left( U,[n] \mapsto \left\{ \array{ C^\infty(U) \otimes \Omega^\bullet(\Delta^n) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(\mathfrak{g}) } \right\} \right) \,,

where the vertical morphisms are the canonical ones.

Much of the subtlety of the full theory of connections of $\infty$-bundles comes from the finite coskeleton-truncation here. For the following discussion of Chern-Simons forms it is helpful to first ignore this issue by taking $n=\infty$, hence ignoring the truncation for the moment. This is sufficient for understand everything about Chern-Simons forms locally.

A cocycle in ∞-Lie algebra cohomology in degree $k$ is a morphism

$\mathrm{CE}\left(𝔤\right)←\mathrm{CE}\left({b}^{k-1}ℝ\right):\mu \phantom{\rule{thinmathspace}{0ex}}.$CE(\mathfrak{g}) \leftarrow CE(b^{k-1} \mathbb{R}) : \mu \,.

Simply by composition (since we ignore the truncation for the moment), this integrates to a cocycle of the corresponding $\infty$-Lie groupoids

$\mathrm{exp}\left(\mu \right):\mathrm{exp}\left(𝔤\right)\to \mathrm{exp}\left({b}^{k-1}ℝ\right)\phantom{\rule{thinmathspace}{0ex}},$\exp(\mu) : \exp(\mathfrak{g}) \to \exp(b^{k-1}\mathbb{R}) \,,

At ∞-Chern-Weil theory it is discussed how the proper lift of this through the extension $B{G}_{\mathrm{diff}}$ that computes the abstractly defined curvature characteristic classes is given by finding the invariant polynomial $⟨-,-⟩\in W\left(𝔤\right)$ that is in transgression with $\mu$ in that we have a commuting diagram

$\begin{array}{ccc}\mathrm{CE}\left(𝔤\right)& \stackrel{\mu }{←}& \mathrm{CE}\left({b}^{k-1}ℝ\right)\\ ↑& & ↑\\ W\left(𝔤\right)& \stackrel{\left(\mathrm{cs},⟨-,-⟩\right)\right)}{←}& W\left({b}^{k-1}ℝ\right)\\ ↑& & ↑\\ \mathrm{inv}\left(𝔤\right)& \stackrel{⟨-,-⟩}{←}& \mathrm{CE}\left({b}^{k}ℝ\right)\end{array}$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(cs, \langle-,-\rangle))}{\leftarrow}& W(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle -,-\rangle}{\leftarrow}& CE(b^k \mathbb{R}) }

with a choice of interpolating Chern-Simons element $\mathrm{cs}\in W\left(𝔤\right)$, which induces by precomposition with its upper part the morphism

$\mathrm{exp}\left(\left(\mathrm{cs},⟨-,-⟩\right)\right):B{G}_{\mathrm{diff}}\to \mathrm{exp}\left({b}^{k-1}ℝ{\right)}_{\mathrm{diff}}\phantom{\rule{thinmathspace}{0ex}}.$\exp((cs,\langle-,-\rangle)) : \mathbf{B}G_{diff} \to \exp(b^{k-1}\mathbb{R})_{diff} \,.

By further projection to its lower part we get furthermore a morphism

$\mathrm{exp}\left({b}^{k-1}ℝ{\right)}_{\mathrm{diff}}\to {♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{simp}}:=\left(U,\left[n\right]↦\left\{{\Omega }^{•}\left(U\right)\otimes {\Omega }^{•}\left({\Delta }^{n}\right)←W\left({b}^{k}ℝ\right)\right\}\right)\phantom{\rule{thinmathspace}{0ex}}.$\exp(b^{k-1}\mathbb{R})_{diff} \to \mathbf{\flat}_{dR}\mathbf{B}^k \mathbb{R}_{simp} := (U,[n] \mapsto \{ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^n) \leftarrow W(b^k \mathbb{R}) \}) \,.

Finally – and this is crucial now for obtaining the incarnation of Chern-Simons forms at integrals of curvature forms as in the above discussion – at ∞-Lie groupoid in the section on simplicial differential forms (see also circle n-bundles with connection the section Models from ∞-Lie integration) it is discussed that the operation that takes the $n$-cells on the right and integrates the corresponding forms over the $n$-simplex yields an equivalence

${\int }_{{\Delta }^{•}}:\left(U,\left[n\right]↦\left\{{\Omega }^{•}\left(U\right)\otimes {\Omega }^{•}\left({\Delta }^{n}\right)←W\left({b}^{k}ℝ\right)\right\}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\to \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\Xi \left(\stackrel{{d}_{\mathrm{dR}}}{\to }{\Omega }^{k-1}\left(-\right)\stackrel{{d}_{\mathrm{dR}}}{\to }{\Omega }_{\mathrm{closed}}^{k}\left(-\right)\right)$\int_{\Delta^\bullet} : (U,[n] \mapsto \{ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^n) \leftarrow W(b^k \mathbb{R}) \}) \;\;\;\;\to \;\;\;\; \Xi( \stackrel{d_{dR}}{\to}\Omega^{k-1}(-)\stackrel{d_{dR}}{\to}\Omega^k_{closed}(-))

to the image of the $ℝ$-Deligne complex of sheaves under the Dold-Kan correspondence.

Higher order Chern-Simons form

With all of the above in hand, we can make now the following observations:

For $X$ a smooth manifold and $𝔤$ an ∞-Lie algebra with coefficient for pseudo-connections being $B{G}_{\mathrm{diff}}$ as above, a morphism

$A:X\to B{G}_{\mathrm{diff}}$A : X \to \mathbf{B}G_{diff}

of simplicial presheaves (no resolution on the left, since we are concentrating on globally defined forms for the present purpose) is effectively a $𝔤$-values differential form on $X$

For $\mu$ a cocycle on $𝔤$ and $⟨-,-⟩$ a corresponding invariant polynomial the composite

$X\to B{G}_{\mathrm{diff}}\to {♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}$X \to \mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}

discussed above produces the corresponding curvature characteristic form.

$\left(\nabla \to \nabla \prime \right):X\cdot \Delta \left[1\right]\to B{G}_{\mathrm{diff}}$(\nabla \to \nabla') : X \cdot \Delta[1] \to \mathbf{B}G_{diff}

is a smooth path in the space of $𝔤$-valued forms on $X$. Under the adjunction

$\left[X\cdot \Delta \left[1\right],{♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}\right]\simeq \left[X,{♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}^{\Delta \left[1\right]}\right]$[X \cdot \Delta[1], \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}] \simeq [X, \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}^{\Delta[1]}]

this corresponds to a $\left(k-1\right)$-form on $X$ this is the Chern-Simons form

$\mathrm{CS}\left(\nabla \to \nabla \prime \right):X\to {♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}^{\Delta \left[1\right]}\phantom{\rule{thinmathspace}{0ex}}.$CS(\nabla \to \nabla') : X \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}^{\Delta[1]} \,.

The higher homotopies are higher order Chern-Simons forms.

The following proposition says this in a more precise way for ordinary Chern-Simons forms.

Ordinary Chern-Simons forms revisited

We now show how the traditional definition of Chern-Simons forms is reproduced by the general abstract mechanism.

Proposition

(ordinary Chern-Simons form)

Let $𝔤$ be a Lie algebra, and $⟨-,-⟩\in W\left(𝔤\right)$ an invariant polynomial.

Then morphisms (of simplicial presheaves)

$A:X\to B{G}_{\mathrm{diff}}$A : X \to \mathbf{B}G_{diff}

are in canonical bijection with Lie-algebra valued 1-forms $A\in {\Omega }^{1}\left(X,𝔤\right)$. Morphisms

$X\to {♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}$X \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}

are in canonical bijection with closed $k$-forms on $X$ and composition with the morphism

$B{G}_{\mathrm{diff}}\to {♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}$\mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}

discussed above and under this canonical identification the composite

$⟨{F}_{A}⟩:X\stackrel{A}{\to }B{G}_{\mathrm{diff}}\stackrel{}{\to }{♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}$\langle F_A \rangle : X \stackrel{A}{\to} \mathbf{B}G_{diff} \stackrel{}{\to} \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}

is the corresponding curvature characteristic form.

Homotopies

$\left({A}_{1}\stackrel{\gamma }{\to }{A}_{2}\right):X×\Delta \left[1\right]\to B{G}_{\mathrm{diff}}$(A_1 \stackrel{\gamma}{\to} A_2) : X\times \Delta[1] \to \mathbf{B}G_{diff}

are in canonical bijection with smooth paths in the space of $𝔤$-valued 1-forms on $X$ and under composition with $B{G}_{\mathrm{diff}}\to {♭}_{\mathrm{dR}}{B}^{k}ℝ$ these identify with the corresponding Chern-Simons form

$⟨{F}_{A}⟩\stackrel{\mathrm{CS}\left(A\stackrel{\gamma }{\to }A\prime \right)}{\to }⟨{F}_{A\prime }⟩:X\to \left({♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}{\right)}^{\Delta \left[1\right]}\phantom{\rule{thinmathspace}{0ex}}.$\langle F_A\rangle \stackrel{CS(A \stackrel{\gamma}{\to} A')}{\to} \langle F_{A'}\rangle : X \to (\mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn})^{\Delta[1]} \,.
Proof

This is a straightforward unwinding of the definitions. We spell it out in the following in order to highlight the way the mechanism works.

By the Yoneda lemma and the definition of $B{G}_{\mathrm{diff}}$, a morphism $X\to B{G}_{\mathrm{diff}}$ is equivalently a diagram

$\begin{array}{ccc}{C}^{\infty }\left(X\right)\otimes {\Omega }^{•}\left({\Delta }^{0}\right)& ←& \mathrm{CE}\left(𝔤\right)\\ ↑& & ↑\\ {\Omega }^{•}\left(X\right)\otimes {\Omega }^{•}\left({\Delta }^{0}\right)& \stackrel{\left(A,{F}_{A}\right)}{←}& W\left(𝔤\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ C^\infty(X) \otimes \Omega^\bullet(\Delta^0) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) \otimes \Omega^\bullet(\Delta^0) &\stackrel{(A,F_A)}{\leftarrow}& W(\mathfrak{g}) } \,.

Since $\mathrm{CE}\left(𝔤\right)$ is trivial in degree 0 and since ${C}^{\infty }\left(X\right)otimess{\Omega }^{•}\left({\Delta }^{0}\right)$ is trivial above degree 0, the top morphism is necessarily 0 and the commutativity of the diagram is an empty condition.

The bottom morphism on the other hand enccodes precisely a $𝔤$-valued form, as discussed in some detail at Weil algebra.

Composition with the morphism $B{G}_{\mathrm{diff}}\to {♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}$ is composition of the bottom morphism of the above digram with $W\left(𝔤\right)←\mathrm{CE}\left({b}^{k-1}ℝ\right):⟨-⟩$ followed by fiber integration of the resulting $k$-form

${\Omega }^{•}\left(X\right)\otimes {\Omega }^{•}\left({\Delta }^{0}\right)\stackrel{\left(A,{F}_{A}\right)}{←}W\left(𝔤\right)\stackrel{⟨-⟩}{←}\mathrm{CE}\left({b}^{k-1}ℝ\right):⟨{F}_{A}⟩$\Omega^\bullet(X)\otimes \Omega^\bullet(\Delta^0) \stackrel{(A,F_A)}{\leftarrow} W(\mathfrak{g}) \stackrel{\langle - \rangle}{\leftarrow} CE(b^{k-1}\mathbb{R}) : \langle F_A \rangle

over the point. This fiber integration is of course trivial, so that we find that indeed $X\stackrel{\left(A,{F}_{A}\right)}{\to }B{G}_{\mathrm{diff}}\stackrel{langlw-⟩}{\to }{♭}_{\mathrm{dR}}{B}^{k}{ℝ}_{\mathrm{chn}}$ is the curvature characteristic form defined on $⟨-⟩$ on $A$.

Next, a homotopy $\left(A\stackrel{\gamma }{\to }A\prime \right):X\cdot \Delta \left[1\right]\to B{G}_{\mathrm{diff}}$ is (again by the Yoneda lemma) a diagram

$\begin{array}{ccc}{C}^{\infty }\left(X\right)\otimes {\Omega }^{•}\left({\Delta }^{1}\right)& \stackrel{\lambda }{←}& \mathrm{CE}\left(𝔤\right)\\ ↑& & ↑\\ {\Omega }^{•}\left(X\right)\otimes {\Omega }^{•}\left({\Delta }^{0}\right)& \stackrel{\left(A,{F}_{A}\right)\stackrel{\gamma }{\to }\left(A\prime ,{F}_{A\prime }\right)}{←}& W\left(𝔤\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ C^\infty(X) \otimes \Omega^\bullet(\Delta^1) &\stackrel{\lambda}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) \otimes \Omega^\bullet(\Delta^0) &\stackrel{(A,F_A) \stackrel{\gamma}{\to} (A',F_{A'})}{\leftarrow}& W(\mathfrak{g}) } \,.

The top morphism defines an $X$-parameterized family of $𝔤$-valued 1-form on the interval $\left[0,1\right]$, which is canonically identified with a smooth function $g:X×\left[0,1\right]\to G$ into the simply connected Lie group integrating] $𝔤$ based at the identity, $g\left(x,0\right)=e$, by the formula

$\lambda ={g}^{*}\theta$\lambda = g^* \theta

where $\theta \in {\Omega }^{1}\left(G,𝔤\right)$ is the Maurer-Cartan form on $G$,

or conversely by parallel transport

$f\left(x,s\right)=P\mathrm{exp}\left({\int }_{\left[0,s\right]}\lambda \left(x,s\right)ds\right)$f(x,s) = P \exp(\int_{[0,s]} \lambda(x,s) d s)

We may think of this as a smooth path of gauge transformations .

The bottom morphism encodes a $𝔤$-valued form

$\stackrel{^}{A}+\lambda \in {\Omega }^{1}\left(X×\left[0,1\right],𝔤\right)$\hat A + \lambda \in \Omega^1(X \times [0,1] , \mathfrak{g})

with $\stackrel{^}{A}\in {\Omega }^{1}\left(X,𝔤\right)\otimes {C}^{\infty }\left(\left[0,1\right]\right)$ and $\lambda$ as before, such that $\stackrel{^}{A}\left(s=0\right)=A$ and $\stackrel{^}{A}\left(s=1\right)=A\prime$.

This is a smooth path in the space of 1-forms . In the case that $\lambda =0$ this is a pure shift path in the terminology above. we look at this case in the following, for ease of notation.

Under composition with $W\left(𝔤\right)←\mathrm{CE}\left({b}^{k-1}ℝ\right):⟨-⟩$ this becomes a $k$-form

$⟨{F}_{\stackrel{^}{A}}⟩\in {\Omega }^{k}\left(X\right)\otimes {C}^{\infty }\left({\Delta }^{1}\right)\oplus {\Omega }^{k-1}\left(X\right)\otimes {\Omega }^{1}\left({\Delta }^{1}\right)\phantom{\rule{thinmathspace}{0ex}}.$\langle F_{\hat A } \rangle \in \Omega^{k}(X)\otimes C^\infty(\Delta^1)\oplus \Omega^{k-1}(X)\otimes \Omega^1(\Delta^1) \,.

The fiber integration of this over ${\Delta }^{1}$ is manifestly the same operation as that in the definition of the Chern-Simons form above.

As secondary characteristic forms

If a curvature characteristic form vanishes (for instance if the connection is flat or the degree of the curvature characteristic form is simply greater than the dimension of $X$) the corresponding Chern-Simons form is a closed form. So in this case the de Rham cohomology class of the curvature characteristic form becomes trivial, but the Chern-Simons form provides another de Rham class. This is therefore called a secondary characteristic class.

Chern-Simons theory

In particular on a 3-dimensional smooth manifold $X$ necessarily the Chern-Simons 3-form is closed. The functional

$\left(A\in {\Omega }^{1}\left(X,𝔤\right)\right)↦{\int }_{X}\mathrm{CS}\left(A\right)$(A \in \Omega^1(X,\mathfrak{g})) \mapsto \int_X CS(A)

is the action functional of the quantum field theory called Chern-Simons theory.

More generally, for $X$ a $\left(2n-1\right)$-dimensional smooth manifold and $⟨-,\cdots ,-⟩$ an invariant polynomial of arity $n$, the analous formula defines the action functional of $\left(2n+1\right)$-dimensional Chern-Simons theory.

In terms of $\infty$-Lie algebroids

As discussed at invariant polynomial, Chern-Simons elements int the Weil algebra $W\left(𝔤\right)$ of a Lie algebra $𝔤$ induce the transgression between invariant polynomials and cocycles in Lie algebra cohomology.

For

$\begin{array}{ccc}{T}_{\mathrm{vert}}P& \stackrel{{A}_{\mathrm{vert}}}{\to }& 𝔤\\ ↓& & ↓\\ TP& \stackrel{\left(A,{F}_{A}\right)}{\to }& \mathrm{inn}\left(𝔤\right)\\ ↓& & ↓\\ TX& \stackrel{\left({P}_{i}\left({F}_{A}\right)\right)}{\to }& \prod _{i}{b}^{{n}_{i}}𝔲\left(1\right)\end{array}$\array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} \\ \downarrow && \downarrow \\ T P &\stackrel{(A,F_A)}{\to}& inn(\mathfrak{g}) \\ \downarrow && \downarrow \\ T X &\stackrel{(P_i(F_A))}{\to}& \prod_i b^{n_i} \mathfrak{u}(1) }

the data of an Ehresmann connection on a $G$-principal bundle expressed as a diagram of ∞-Lie algebroids with the curvature characteristic forms on the bottom, a choice of transgression element ${\mathrm{cs}}_{P}$ for an invariant polynomial $P$ in transgression with a Lie algebra cocycle $\mu$ induces a diagram

$\begin{array}{ccc}𝔤& \stackrel{\mu }{\to }& {b}^{n}𝔲\left(1\right)\\ ↓& & ↓\\ \mathrm{inn}\left(𝔤\right)& \stackrel{\left({\mathrm{cs}}_{P},P\right)}{\to }& e{b}^{n}𝔲\left(1\right)\\ ↓& & ↓\\ \prod _{i}{b}^{{n}_{i}}𝔲\left(1\right)& \stackrel{{p}_{i}}{\to }& {b}^{n+1}𝔲\left(1\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathfrak{g} &\stackrel{\mu}{\to}& b^n \mathfrak{u}(1) \\ \downarrow && \downarrow \\ inn(\mathfrak{g}) &\stackrel{(cs_P,P)}{\to}& e b^{n} \mathfrak{u}(1) \\ \downarrow && \downarrow \\ \prod_i b^{n_i}\mathfrak{u}(1) &\stackrel{p_i}{\to}& b^{n+1} \mathfrak{u}(1) } \,.

The pasting of this to the above Ehresmann connection expresses in the middle horizontal morphism the Chern-Simons form ${\mathrm{cs}}_{P}\left(A\right)$ and its curvature characteristic form $P\left({F}_{A}\right)$

$\begin{array}{ccccc}{T}_{\mathrm{vert}}P& \stackrel{{A}_{\mathrm{vert}}}{\to }& 𝔤& \stackrel{\mu }{\to }& {b}^{n}𝔲\left(1\right)\\ ↓& & ↓& & ↓\\ TP& \stackrel{A}{\to }& \mathrm{inn}\left(𝔤\right)& \stackrel{\left({\mathrm{cs}}_{P},P\right)}{\to }& e{b}^{n}𝔲\left(1\right)\\ ↓& & ↓& & ↓\\ TX& \stackrel{\left({P}_{i}\right)}{\to }& \prod _{i}{b}^{{n}_{i}}𝔲\left(1\right)& \stackrel{{p}_{i}}{\to }& {b}^{n+1}𝔲\left(1\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} &\stackrel{\mu}{\to}& b^n \mathfrak{u}(1) \\ \downarrow && \downarrow && \downarrow \\ T P &\stackrel{A}{\to}& inn(\mathfrak{g}) &\stackrel{(cs_P,P)}{\to}& e b^n \mathfrak{u}(1) \\ \downarrow && \downarrow && \downarrow \\ T X &\stackrel{(P_i)}{\to}& \prod_i b^{n_i} \mathfrak{u}(1) &\stackrel{p_i}{\to}& b^{n+1} \mathfrak{u}(1) } \,.

References

The article introducing the concept is

As it says in the introduction of this article, it was motivated by an attempt to find a combinatorial formula for the Pontrjagin class of a Riemannian manifold (i.e. that associated to the O(n)-principal bundle to which the tangent bundle is associated) and the Chern-Simons form appeared as a boundary term that obstructed to original attempt to derive the Pontrjagin class by integrating curvature classes simplex-by-simplex. But A combinatorial formula of the kind these authors were looking for was however (nevertheless) given later in

• Jean-Luc Brylinski, Dennis McLaughlin? Čech cocycles for characteristic classes , Comm. Math. Phys. 178 (1996) (pdf)

The statements about “pure shift” paths are reviewed on the first few pages of

which discusses the relevance of Chern-Simons forms in differential K-theory.

The L-∞-algebra-formulation is discussed in SSS08.

An abstract algebraic model of the algebra of Chern’s characteristic classes and Chern-Simons secondary characteristic classes and of the gauge group action on this algebra (which also enables some noncommutative generalizations) is pioneered in 2 articles

• Israel M. Gelfand, Mikhail M. Smirnov, The algebra of Chern-Simons classes, the Poisson bracket on it, and the action of the gauge group, Lie theory and geometry, 261–288, Progr. Math. 123, Birkhäuser 1994; Chern-Simons classes and cocycles on the Lie algebra of the gauge group, The Gelfand Mathematical Seminars, 1993–1995, 101–122, Birkhäuser 1996.

Revised on January 13, 2013 20:29:45 by Urs Schreiber (203.116.137.162)