infinity-Chern-Simons theory -- action functionals

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


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):=/U(1) := \mathbb{R}/\mathbb{Z}.

Integration of the Lagrangian

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


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

exp(iS c()) disc:H(Σ,A conn)exp(iL c)()H(Σ,B nU(1) conn)τ ndimΣH(Σ,B nU(1) 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(iS c()) smooth:[Σ,A conn]exp(iL c)()[Σ,B nU(1) conn]τ ndimΣ[Σ,B nU(1) conn]Concτ ndimΣ[Σ,B nU(1) 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.


Let n1n \geq 1. If Σ\Sigma \in SmoothMfd \hookrightarrow Smooth∞Grpd =:H=: \mathbf{H} is a closed smooth manifold of dimension dimΣn dim \Sigma \leq n then

τ ndimΣH(Σ,B nU(1) conn)B ndimΣU(1) discK(U(1),ndimΣ)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


τ ndimΣ[Σ,B nU(1) conn]B ndimΣU(1) discSmoothGrpd, \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 ndimΣU(1) discB^{n-dim \Sigma} U(1)_{disc} is the discrete ∞-groupoid corresponding to the Eilenberg-MacLane space K(U(1),ndimΣ)K(U(1), n-dim \Sigma), whereas in the second – smooth – version B ndimΣU(1)\mathbf{B}^{n- dim \Sigma} U(1) is the smooth circle n-group.


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 dimΣndim \Sigma \leq n we have by this proposition that H(Σ, dRB n+1)H dR n+1(Σ)*H(\Sigma, \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}) \simeq H^{n+1}_{dR}(\Sigma) \simeq *. It then follows by this proposition that we have an equivalence

H diff(Σ,B nU(1))H flat(Σ,B nU(1))=:H(Π(Σ),B nU(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 Σ\Sigma, and using the definition Π=DiscΠ\mathbf{\Pi} = Disc \Pi and by the (ΠDiscΓ)(\Pi \dashv Disc \dashv \Gamma)-adjunction it follows that this is equivalently

Grpd(Π(Σ),ΓB nU(1)) Grpd(Π(Σ),B nU(1) disc), \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 nU(1) discB^n U(1)_{disc} is an Eilenberg-MacLane space K(U(1),n)\cdots \simeq K(U(1), n). By this proposition we have under :GrpdTop|-| : \infty Grpd \simeq Top a weak homotopy equivalence Π(Σ)Σ|\Pi(\Sigma)| \simeq \Sigma. Therefore the cocycle \infty-groupoid is that of ordinary cohomology

C n(Σ,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

π iH diff(Σ,B nU(1))H ni(Σ,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

()():H ni(Σ,U(1))Hom Ab(H ni(Σ,),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 ω\omega in singular cohomology with coefficients in U(1)U(1) to the pairing map

[c] [c]ω [c] \mapsto \int_{[c]} \omega

sits inside an exact sequence

(1)0Ext 1(H ni1(Σ,),U(1))H ni(Σ,U(1))Hom Ab(H ni(Σ,),U(1))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)U(1) is an injective \mathbb{Z}-module we have

Ext 1(,U(1))=0. Ext^1(-,U(1))=0 \,.

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

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

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

π iH diff(Σ,B nU(1))=0fori<(ndimΣ). \pi_i \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n U(1)) =0 \;\;\;\; for i\lt (n-dim \Sigma) \,.

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

π (ndimΣ)H diff(Σ,B nU(1))U(1). \pi_{(n-dim\Sigma)} \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n U(1)) \simeq U(1) \,.

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

exp(iS):H(Σ,A conn)exp(iL)H(Σ,B nU(1) conn) [Σ]()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 Σ\Sigma in order to obtain the action

S()= ΣL(). S(-) = \int_\Sigma L(-) \,.

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


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


In codimension 0 the smooth action functional is a morphism

exp(iS c()) smooth:[Σ,A conn]U(1)SmoothGrpd \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\mathfrak{a} be an ∞-Lie algebroid. We shall write

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

for its Lie integration.

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

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

If μ\mu is in transgression with an invariant polynomial W(𝔞)\langle -\rangle \in W(\mathfrak{a}) and csW(𝔞)cs \in W(\mathfrak{a}) is a Chern-Simons element exhibiting that transgression, then the above discussion constructs from this an ∞-Chern-Weil homomorphism

exp(cs):A connB nU(1) conn, \exp(cs) : A_{conn} \to \mathbf{B}^n U(1)_{conn} \,,

in H=\mathbf{H} = Smooth∞Grpd, where

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

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

For Σ\Sigma \in SmoothMfd \hookrightarrow Smooth∞Grpd a smooth manifold of dimension dimΣndim \Sigma \leq n, the induced morphism

exp(iL c()):H(Σ,A conn)H(Σ,B nU(1) 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 AA-principal ∞-bundles to that of circle n-bundles with connection we may interpret as the Lagrangian of the \infty-Chern-Simons theory defined by csW(𝔞)cs \in W(\mathfrak{a}) over Σ\Sigma: it sends a field configuration ϕ:ΣA conn\phi : \Sigma \to A_{conn}, which is locally on UΣU \subset \Sigma given by a ∞-Lie algebroid valued differential form AΩ (U,𝔞)A \in \Omega^\bullet(U, \mathfrak{a}), to the Chern-Simons form L(ϕ U)=L(A)=cs(A)Ω n(Σ)L(\phi|_U) = L(A) = cs(A) \in \Omega^n(\Sigma).


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

exp(iS()):Aexp(i ΣCS(A)). \exp(i S(-)) : A \mapsto \exp(i \int_\Sigma CS(A)) \,.


The notion of Chern-Simons elements for L L_\infty-algebras and the associated imnfty\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 (