# Schreiber infinity-Chern-Simons theory -- action functionals

This is a sub-entry of infinity-Chern-Simons theory. See there for context

# Contents

## 3.) The action functionals

We discuss the realization of the general abstract notion of the Chern-Simons functional in a cohesive (∞,1)-topos in the concrete context Smooth∞Grpd over a manifold with coefficients given by the circle group $U\left(1\right):=ℝ/ℤ$.

### Integration of the Lagrangian

Recall (here) the general abstract definition of the extended action functional.

###### Definition

As an (∞,1)-functor on discrete ∞-groupoids, the action functional defined by a Lagrangian $\mathrm{exp}\left(iL\left(-\right)\right):{A}_{\mathrm{conn}}\to {B}^{n}U\left(1{\right)}_{\mathrm{conn}}$ over an object $\Sigma$ is the composite of the induced morphism on cocycles with the 0-truncation morphism

$\mathrm{exp}\left(i{S}_{c}\left(-\right){\right)}_{\mathrm{disc}}:H\left(\Sigma ,{A}_{\mathrm{conn}}\right)\stackrel{\mathrm{exp}\left(i{L}_{c}\right)\left(-\right)}{\to }H\left(\Sigma ,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right)\to {\tau }_{\le n-\mathrm{dim}\Sigma }H\left(\Sigma ,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right)\phantom{\rule{thinmathspace}{0ex}}.$\exp(i S_{\mathbf{c}}(-))_{disc} : \mathbf{H}(\Sigma, A_{conn}) \stackrel{\exp(i L_{\mathbf{c}})(-)}{\to} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \to \tau_{\leq n-dim \Sigma} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \,.

Refined to a morphism of smooth ∞-groupoids the action functional is the corresponding composite on internal hom-objects

$\mathrm{exp}\left(i{S}_{c}\left(-\right){\right)}_{\mathrm{smooth}}:\left[\Sigma ,{A}_{\mathrm{conn}}\right]\stackrel{\mathrm{exp}\left(i{L}_{c}\right)\left(-\right)}{\to }\left[\Sigma ,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right]\to {\tau }_{\le n-\mathrm{dim}\Sigma }\left[\Sigma ,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right]\to \mathrm{Conc}{\tau }_{\le n-\mathrm{dim}\Sigma }\left[\Sigma ,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right]\phantom{\rule{thinmathspace}{0ex}},$\exp(i S_{\mathbf{c}}(-))_{smooth} : [\Sigma, A_{conn}] \stackrel{\exp(i L_{\mathbf{c}})(-)}{\to} [\Sigma, \mathbf{B}^n U(1)_{conn}] \to \tau_{\leq n-dim \Sigma} [\Sigma, \mathbf{B}^n U(1)_{conn}] \to Conc\tau_{\leq n-dim \Sigma} [\Sigma, \mathbf{B}^n U(1)_{conn}] \,,

where the last morphism is concretification.

###### Theorem

Let $n\ge 1$. If $\Sigma \in$ SmoothMfd $↪$ Smooth∞Grpd $=:H$ is a closed smooth manifold of dimension $\mathrm{dim}\Sigma \le n$ then

${\tau }_{\le n-\mathrm{dim}\Sigma }H\left(\Sigma ,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right)\simeq {B}^{n-\mathrm{dim}\Sigma }U\left(1{\right)}_{\mathrm{disc}}\simeq K\left(U\left(1\right),n-\mathrm{dim}\Sigma \right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in \infty \mathrm{Grpd}$\tau_{\leq n-dim \Sigma} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \simeq B^{n - dim \Sigma} U(1)_{disc} \simeq K(U(1), n-dim \Sigma) \;\;\;\;\; \in \infty Grpd

and

${\tau }_{\le n-\mathrm{dim}\Sigma }\left[\Sigma ,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right]\simeq {B}^{n-\mathrm{dim}\Sigma }U\left(1{\right)}_{\mathrm{disc}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in \mathrm{Smooth}\infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}},$\tau_{\leq n-dim \Sigma} [\Sigma, \mathbf{B}^n U(1)_{conn}] \simeq \mathbf{B}^{n - dim \Sigma} U(1)_{disc} \;\;\;\;\; \in Smooth \infty Grpd \,,

where in the first – discrete – version ${B}^{n-\mathrm{dim}\Sigma }U\left(1{\right)}_{\mathrm{disc}}$ is the discrete ∞-groupoid corresponding to the Eilenberg-MacLane space $K\left(U\left(1\right),n-\mathrm{dim}\Sigma \right)$, whereas in the second – smooth – version ${B}^{n-\mathrm{dim}\Sigma }U\left(1\right)$ is the smooth circle n-group.

###### Proof

We give the argument for the discrete case. The statement for the smooth case follows from this with a proposition at concrete smooth ∞-groupoid .

Since $\mathrm{dim}\Sigma \le n$ we have by this proposition that $H\left(\Sigma ,{♭}_{\mathrm{dR}}{B}^{n+1}ℝ\right)\simeq {H}_{\mathrm{dR}}^{n+1}\left(\Sigma \right)\simeq *$. It then follows by this proposition that we have an equivalence

${H}_{\mathrm{diff}}\left(\Sigma ,{B}^{n}U\left(1\right)\right)\simeq {H}_{\mathrm{flat}}\left(\Sigma ,{B}^{n}U\left(1\right)\right)=:H\left(\Pi \left(\Sigma \right),{B}^{n}U\left(1\right)\right)$\mathbf{H}_{diff}(\Sigma, \mathbf{B}^n U(1)) \simeq \mathbf{H}_{flat}(\Sigma, \mathbf{B}^n U(1)) =: \mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1))

with the flat differential cohomology on $\Sigma$, and using the definition $\Pi =\mathrm{Disc}\Pi$ and by the $\left(\Pi ⊣\mathrm{Disc}⊣\Gamma \right)$-adjunction it follows that this is equivalently

$\begin{array}{rl}\cdots & \simeq \infty \mathrm{Grpd}\left(\Pi \left(\Sigma \right),\Gamma {B}^{n}U\left(1\right)\right)\\ & \simeq \infty \mathrm{Grpd}\left(\Pi \left(\Sigma \right),{B}^{n}U\left(1{\right)}_{\mathrm{disc}}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\begin{aligned} \cdots & \simeq \infty Grpd(\Pi(\Sigma), \Gamma \mathbf{B}^n U(1)) \\ & \simeq \infty Grpd(\Pi(\Sigma), B^n U(1)_{disc}) \end{aligned} \,,

where ${B}^{n}U\left(1{\right)}_{\mathrm{disc}}$ is an Eilenberg-MacLane space $\cdots \simeq K\left(U\left(1\right),n\right)$. By this proposition we have under $\mid -\mid :\infty \mathrm{Grpd}\simeq \mathrm{Top}$ a weak homotopy equivalence $\mid \Pi \left(\Sigma \right)\mid \simeq \Sigma$. Therefore the cocycle $\infty$-groupoid is that of ordinary cohomology

$\cdots \simeq {C}^{n}\left(\Sigma ,U\left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq C^n(\Sigma, U(1)) \,.

By general abstract reasoning (recalled at cohomology and fiber sequence) it follows that we have for the homotopy groups an isomorphism

${\pi }_{i}{H}_{\mathrm{diff}}\left(\Sigma ,{B}^{n}U\left(1\right)\right)\stackrel{\simeq }{\to }{H}^{n-i}\left(\Sigma ,U\left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_i \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n U(1)) \stackrel{\simeq}{\to} H^{n-i}(\Sigma, U(1)) \,.

Now we invoke the universal coefficient theorem. This asserts that the morphism

${\int }_{\left(-\right)}\left(-\right):{H}^{n-i}\left(\Sigma ,U\left(1\right)\right)\stackrel{}{\to }{\mathrm{Hom}}_{\mathrm{Ab}}\left({H}_{n-i}\left(\Sigma ,ℤ\right),U\left(1\right)\right)$\int_{(-)}(-) : H^{n-i}(\Sigma,U(1)) \stackrel{}{\to} Hom_{Ab}(H_{n-i}(\Sigma,\mathbb{Z}),U(1))

which sends a cocycle $\omega$ in singular cohomology with coefficients in $U\left(1\right)$ to the pairing map

$\left[c\right]↦{\int }_{\left[c\right]}\omega$[c] \mapsto \int_{[c]} \omega

sits inside an exact sequence

(1)$0\to {\mathrm{Ext}}^{1}\left({H}_{n-i-1}\left(\Sigma ,ℤ\right),U\left(1\right)\right)\to {H}^{n-i}\left(\Sigma ,U\left(1\right)\right)\stackrel{}{\to }{\mathrm{Hom}}_{\mathrm{Ab}}\left({H}_{n-i}\left(\Sigma ,ℤ\right),U\left(1\right)\right)\to 0\phantom{\rule{thinmathspace}{0ex}},$0 \to Ext^1(H_{n-i-1}(\Sigma,\mathbb{Z}),U(1)) \to H^{n-i}(\Sigma,U(1)) \stackrel{}{\to} Hom_{Ab}(H_{n-i}(\Sigma,\mathbb{Z}),U(1)) \to 0 \,,

But since $U\left(1\right)$ is an injective $ℤ$-module we have

${\mathrm{Ext}}^{1}\left(-,U\left(1\right)\right)=0\phantom{\rule{thinmathspace}{0ex}}.$Ext^1(-,U(1))=0 \,.

This means that the integration/pairing map ${\int }_{\left(-\right)}\left(-\right)$ is an isomorphism

(2)${\int }_{\left(-\right)}\left(-\right):{H}^{n-i}\left(\Sigma ,U\left(1\right)\right)\simeq {\mathrm{Hom}}_{\mathrm{Ab}}\left({H}_{n-i}\left(\Sigma ,ℤ\right),U\left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\int_{(-)}(-) : H^{n-i}(\Sigma,U(1)) \simeq Hom_{Ab}(H_{n-i}(\Sigma,\mathbb{Z}),U(1)) \,.

For $i<\left(n-\mathrm{dim}\Sigma \right)$, the right hand is zero, so that

${\pi }_{i}{H}_{\mathrm{diff}}\left(\Sigma ,{B}^{n}U\left(1\right)\right)=0\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{for}i<\left(n-\mathrm{dim}\Sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_i \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n U(1)) =0 \;\;\;\; for i\lt (n-dim \Sigma) \,.

For $i=\left(n-\mathrm{dim}\Sigma \right)$, instead, ${H}_{n-i}\left(\Sigma ,ℤ\right)\simeq ℤ$, since $\Sigma$ is a closed $\mathrm{dim}\Sigma$-manifold and so

${\pi }_{\left(n-\mathrm{dim}\Sigma \right)}{H}_{\mathrm{diff}}\left(\Sigma ,{B}^{n}U\left(1\right)\right)\simeq U\left(1\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_{(n-dim\Sigma)} \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n U(1)) \simeq U(1) \,.
###### Remark

This proof also shows that for $\mathrm{dim}\Sigma =n$ we may think of the composite

$\mathrm{exp}\left(iS\right):H\left(\Sigma ,{A}_{\mathrm{conn}}\right)\stackrel{\mathrm{exp}\left(iL\right)}{\to }H\left(\Sigma ,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right)\stackrel{{\int }_{\left[\Sigma \right]}\left(-\right)}{\to }U\left(1\right)$\exp(i S) : \mathbf{H}(\Sigma, A_{conn}) \stackrel{\exp(i L)}{\to} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \stackrel{\int_{[\Sigma]}(-)}{\to} U(1)

as being indeed given by integrating the Lagrangian over $\Sigma$ in order to obtain the action

$S\left(-\right)={\int }_{\Sigma }L\left(-\right)\phantom{\rule{thinmathspace}{0ex}}.$S(-) = \int_\Sigma L(-) \,.

We will see precise versions of this statement in the following examples.

###### Remark

The objects of $\left[\Sigma ,{A}_{\mathrm{conn}}\right]$ constitute the configuration space of the $\infty$-Chern-Simons theory, the morphisms are the gauge transformations, the n-morphisms are the order-$n$ gauge transformation. The above smooth action functional is therefore a gauge invariant smooth $U\left(1\right)$-valued function on configuration space .

###### Corollary

In codimension 0 the smooth action functional is a morphism

$\mathrm{exp}\left(i{S}_{c}\left(-\right){\right)}_{\mathrm{smooth}}:\left[\Sigma ,{A}_{\mathrm{conn}}\right]\to U\left(1\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in \mathrm{Smooth}\infty \mathrm{Grpd}$\exp(i S_{\mathbf{c}}(-))_{smooth} : [\Sigma, A_{conn}] \to U(1) \;\;\; \in Smooth \infty Grpd

with values in the object underlying the smooth circle group.

### Explicit presentation by Lie integration

We use the presentation of the ∞-Chern-Weil homomorphism (see there) by Lie integration of L-∞ algebra cocycles to give an explicit presentation of of the $\infty$-Chern-Simons action functional by ordinary integration over ordinary differential form data.

Let $𝔞$ be an ∞-Lie algebroid. We shall write

$A:={\mathrm{cosk}}_{n}\mathrm{exp}\left(𝔞\right)\in \mathrm{Smooth}\infty \mathrm{Grpd}$A := \mathbf{cosk}_n \exp(\mathfrak{a}) \in Smooth\infty Grpd

for its Lie integration.

Similarly, for $\mu \in \mathrm{CE}\left(𝔞\right)$ an n-cocycle on $𝔞$, its Lie integration is a representative of a characteristic class

$c:=\mathrm{exp}\left(\mu \right):A\to {B}^{n}U\left(1\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{c} := \exp(\mu) : A \to \mathbf{B}^n U(1) \,.

If $\mu$ is in transgression with an invariant polynomial $⟨-⟩\in W\left(𝔞\right)$ and $\mathrm{cs}\in W\left(𝔞\right)$ is a Chern-Simons element exhibiting that transgression, then the above discussion constructs from this an ∞-Chern-Weil homomorphism

$\mathrm{exp}\left(\mathrm{cs}\right):{A}_{\mathrm{conn}}\to {B}^{n}U\left(1{\right)}_{\mathrm{conn}}\phantom{\rule{thinmathspace}{0ex}},$\exp(cs) : A_{conn} \to \mathbf{B}^n U(1)_{conn} \,,

in $H=$ Smooth∞Grpd, where

${A}_{\mathrm{conn}}:={\mathrm{cosk}}_{n}\mathrm{exp}\left(𝔞{\right)}_{\mathrm{conn}}$A_{conn} := \mathbf{cosk}_n \exp(\mathfrak{a})_{conn}

is the coefficient object for ∞-connections with values in $𝔞$ and ${B}^{n}U\left(1{\right)}_{\mathrm{conn}}$ is the objected presented under the Dold-Kan correspondence by the Deligne complex in degree $n$: the coefficient object for circle n-bundles with connection.

For $\Sigma \in$ SmoothMfd $↪$ Smooth∞Grpd a smooth manifold of dimension $\mathrm{dim}\Sigma \le n$, the induced morphism

$\mathrm{exp}\left(i{L}_{c}\left(-\right)\right):H\left(\Sigma ,{A}_{\mathrm{conn}}\right)\to H\left(\Sigma ,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right)$\exp(i L_{\mathbf{c}}(-)) : \mathbf{H}(\Sigma, A_{conn}) \to \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})

from the discrete ∞-groupoid of ∞-connections on $A$-principal ∞-bundles to that of circle n-bundles with connection we may interpret as the Lagrangian of the $\infty$-Chern-Simons theory defined by $\mathrm{cs}\in W\left(𝔞\right)$ over $\Sigma$: it sends a field configuration $\varphi :\Sigma \to {A}_{\mathrm{conn}}$, which is locally on $U\subset \Sigma$ given by a ∞-Lie algebroid valued differential form $A\in {\Omega }^{•}\left(U,𝔞\right)$, to the Chern-Simons form $L\left(\varphi {\mid }_{U}\right)=L\left(A\right)=\mathrm{cs}\left(A\right)\in {\Omega }^{n}\left(\Sigma \right)$.

###### Proposition

In codimension 0 the corresponding $\infty$-Chern-Simons functional sends $𝔤$-calued differential forms $A$ to the ordinary integral

$\mathrm{exp}\left(iS\left(-\right)\right):A↦\mathrm{exp}\left(i{\int }_{\Sigma }\mathrm{CS}\left(A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\exp(i S(-)) : A \mapsto \exp(i \int_\Sigma CS(A)) \,.

## References

The notion of Chern-Simons elements for ${L}_{\infty }$-algebras and the associated $imnfty$-Chern-Simons Lagrangians is due to

The induced construction of the ∞-Chern-Weil homomorphism with special attention to the Chern-Simons circle 3-bundle and the Chern-Simons circle 7-bundle is in

In the general context of cohesive (∞,1)-toposes $\infty$-Chern-Simons theory is discussed in section 4.3 of

Revised on October 27, 2011 02:59:58 by Urs Schreiber (77.61.2.209)