Schreiber
infinity-Chern-Simons theory -- action functionals

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

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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 $U(1):=ℝ/\mathbb{Z}$.

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 $\exp (iL(-)):A_{\mathrm{conn}}\to B^{n}U(1{)}_{\mathrm{conn}}$ over an object $\Sigma$ is the composite of the induced morphism on cocycles with the 0-truncation morphism

$$\exp (i{S}_c(-){)}_{\mathrm{disc}}:H(\Sigma ,A_{\mathrm{conn}})\stackrel{\exp (i{L}_c)(-)}{\to }H(\Sigma ,B^{n}U(1{)}_{\mathrm{conn}})\to {\tau }_{\le n-\dim \Sigma }H(\Sigma ,B^{n}U(1{)}_{\mathrm{conn}}).$$\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

$$\exp (i{S}_c(-){)}_{\mathrm{smooth}}:[\Sigma ,A_{\mathrm{conn}}]\stackrel{\exp (i{L}_c)(-)}{\to }[\Sigma ,B^{n}U(1{)}_{\mathrm{conn}}]\to {\tau }_{\le n-\dim \Sigma }[\Sigma ,B^{n}U(1{)}_{\mathrm{conn}}]\to \mathrm{Conc}{\tau }_{\le n-\dim \Sigma }[\Sigma ,B^{n}U(1{)}_{\mathrm{conn}}],$$\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 $\hookrightarrow$ Smooth∞Grpd $=:H$ is a closed smooth manifold of dimension $\dim \Sigma \le n$ then

$${\tau }_{\le n-\dim \Sigma }H(\Sigma ,B^{n}U(1{)}_{\mathrm{conn}})\simeq B^{n-\dim \Sigma }U(1{)}_{\mathrm{disc}}\simeq K(U(1),n-\dim \Sigma )\in \mathrm{\infty }\mathrm{Grpd}$$\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

$${\tau }_{\le n-\dim \Sigma }[\Sigma ,B^{n}U(1{)}_{\mathrm{conn}}]\simeq B^{n-\dim \Sigma }U(1{)}_{\mathrm{disc}}\in \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd},$$\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 $B^{n-\dim \Sigma }U(1{)}_{\mathrm{disc}}$ is the discrete ∞-groupoid corresponding to the Eilenberg-MacLane space $K(U(1),n-\dim \Sigma )$, whereas in the second – smooth – version $B^{n-\dim \Sigma }U(1)$ 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 $\dim \Sigma \le n$ we have by this proposition that $H(\Sigma ,{♭}_{\mathrm{dR}}{B}^{n+1}ℝ)\simeq H_{\mathrm{dR}}^{n+1}(\Sigma )\simeq *$. It then follows by this proposition that we have an equivalence

$$H_{\mathrm{diff}}(\Sigma ,B^{n}U(1))\simeq H_{\mathrm{flat}}(\Sigma ,B^{n}U(1))=:H(\Pi (\Sigma ),B^{n}U(1))$$\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 $(\Pi ⊣\mathrm{Disc}⊣\Gamma )$-adjunction it follows that this is equivalently

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

$$\cdots \simeq C^n(\Sigma ,U(1)).$$\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

$${\pi }_i{H}_{\mathrm{diff}}(\Sigma ,B^{n}U(1))\stackrel{\simeq }{\to }{H}^{n-i}(\Sigma ,U(1)).$$\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

$${\int }_{(-)}(-):H^{n-i}(\Sigma ,U(1))\stackrel{}{\to }{\mathrm{Hom}}_{\mathrm{Ab}}(H_{n-i}(\Sigma ,\mathbb{Z}),U(1))$$\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 $U(1)$ to the pairing map

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

sits inside an exact sequence

(1)$$0\to {\mathrm{Ext}}^1(H_{n-i-1}(\Sigma ,\mathbb{Z}),U(1))\to H^{n-i}(\Sigma ,U(1))\stackrel{}{\to }{\mathrm{Hom}}_{\mathrm{Ab}}(H_{n-i}(\Sigma ,\mathbb{Z}),U(1))\to 0,$$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 $U(1)$ is an injective $\mathbb{Z}$-module we have

$${\mathrm{Ext}}^1(-,U(1))=0.$$Ext^1(-,U(1))=0 \,.

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

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

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

$${\pi }_i{H}_{\mathrm{diff}}(\Sigma ,B^{n}U(1))=0\mathrm{for}i<(n-\dim \Sigma ).$$\pi_i \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n U(1)) =0 \;\;\;\; for i\lt (n-dim \Sigma) \,.

For $i=(n-\dim \Sigma )$, instead, $H_{n-i}(\Sigma ,\mathbb{Z})\simeq \mathbb{Z}$, since $\Sigma$ is a closed $\dim \Sigma$-manifold and so

$${\pi }_{(n-\dim \Sigma )}{H}_{\mathrm{diff}}(\Sigma ,B^{n}U(1))\simeq U(1).$$\pi_{(n-dim\Sigma)} \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n U(1)) \simeq U(1) \,.

Remark

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

$$\exp (iS):H(\Sigma ,A_{\mathrm{conn}})\stackrel{\exp (iL)}{\to }H(\Sigma ,B^{n}U(1{)}_{\mathrm{conn}})\stackrel{{\int }_{[\Sigma ]}(-)}{\to }U(1)$$\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

$$S(-)={\int }_{\Sigma }L(-).$$S(-) = \int_\Sigma L(-) \,.

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

Remark

The objects of $[\Sigma ,A_{\mathrm{conn}}]$ constitute the configuration space of the $\mathrm{\infty }$-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 $U(1)$-valued function on configuration space .

Corollary

In codimension 0 the smooth action functional is a morphism

$$\exp (i{S}_c(-){)}_{\mathrm{smooth}}:[\Sigma ,A_{\mathrm{conn}}]\to U(1)\in \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$$\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 $\mathrm{\infty }$-Chern-Simons action functional by ordinary integration over ordinary differential form data.

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

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

for its Lie integration.

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

$$c:=\exp (\mu ):A\to B^{n}U(1).$$\mathbf{c} := \exp(\mu) : A \to \mathbf{B}^n U(1) \,.

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

$$\exp (\mathrm{cs}):A_{\mathrm{conn}}\to B^{n}U(1{)}_{\mathrm{conn}},$$\exp(cs) : A_{conn} \to \mathbf{B}^n U(1)_{conn} \,,

in $H=$ Smooth∞Grpd, where

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

is the coefficient object for ∞-connections with values in $𝔞$ and $B^{n}U(1{)}_{\mathrm{conn}}$ 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 $\hookrightarrow$ Smooth∞Grpd a smooth manifold of dimension $\dim \Sigma \le n$, the induced morphism

$$\exp (i{L}_c(-)):H(\Sigma ,A_{\mathrm{conn}})\to H(\Sigma ,B^{n}U(1{)}_{\mathrm{conn}})$$\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 $\mathrm{\infty }$-Chern-Simons theory defined by $\mathrm{cs}\in W(𝔞)$ 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 $A\in {\Omega }^{•}(U,𝔞)$, to the Chern-Simons form $L(\varphi {\mid }_U)=L(A)=\mathrm{cs}(A)\in {\Omega }^n(\Sigma )$.

Proposition

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

$$\exp (iS(-)):A\mapsto \exp (i{\int }_{\Sigma }\mathrm{CS}(A)).$$\exp(i S(-)) : A \mapsto \exp(i \int_\Sigma CS(A)) \,.

References

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_{\mathrm{\infty }}$-connections (web)

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

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes

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

• Urs Schreiber, differential cohomology in a cohesive topos