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

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

$$\exp(i S_{\mathbf{c}}(-))_{disc} : \mathbf{H}(\Sigma, A_{conn}) \stackrel{\exp(i L_{\mathbf{c}})(-)}{\to} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \to \tau_{\leq n-dim \Sigma} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \,.$$

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

$$\exp(i S_{\mathbf{c}}(-))_{smooth} : [\Sigma, A_{conn}] \stackrel{\exp(i L_{\mathbf{c}})(-)}{\to} [\Sigma, \mathbf{B}^n U(1)_{conn}] \to \tau_{\leq n-dim \Sigma} [\Sigma, \mathbf{B}^n U(1)_{conn}] \to Conc\tau_{\leq n-dim \Sigma} [\Sigma, \mathbf{B}^n U(1)_{conn}] \,,$$

where the last morphism is concretification.

Theorem

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

$$\tau_{\leq n-dim \Sigma} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \simeq B^{n - dim \Sigma} U(1)_{disc} \simeq K(U(1), n-dim \Sigma) \;\;\;\;\; \in \infty Grpd$$

and

$$\tau_{\leq n-dim \Sigma} [\Sigma, \mathbf{B}^n U(1)_{conn}] \simeq \mathbf{B}^{n - dim \Sigma} U(1)_{disc} \;\;\;\;\; \in Smooth \infty Grpd \,,$$

where in the first – discrete – version $B^{n-dim \Sigma} U(1)_{disc}$ is the discrete ∞-groupoid corresponding to the Eilenberg-MacLane space $K(U(1), n-dim \Sigma)$, whereas in the second – smooth – version $\mathbf{B}^{n- dim \Sigma} U(1)$ 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 $dim \Sigma \leq n$ we have by this proposition that $H(\Sigma, \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}) \simeq H^{n+1}_{dR}(\Sigma) \simeq *$. It then follows by this proposition that we have an equivalence

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

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

$$\begin{aligned} \cdots & \simeq \infty Grpd(\Pi(\Sigma), \Gamma \mathbf{B}^n U(1)) \\ & \simeq \infty Grpd(\Pi(\Sigma), B^n U(1)_{disc}) \end{aligned} \,,$$

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

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

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

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

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

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

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

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

sits inside an exact sequence

$$0 \to Ext^1(H_{n-i-1}(\Sigma,\mathbb{Z}),U(1)) \to H^{n-i}(\Sigma,U(1)) \stackrel{}{\to} Hom_{Ab}(H_{n-i}(\Sigma,\mathbb{Z}),U(1)) \to 0 \,,$$

But since $U(1)$ is an injective $\mathbb{Z}$-module we have

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

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

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

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

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

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

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

Remark

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

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

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

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

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

Remark

The objects of $[\Sigma, A_{conn}]$ constitute the configuration space of the $\infty$-Chern-Simons theory, the morphisms are the gauge transformations, the n-morphisms are the order-$n$ 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

$$\exp(i S_{\mathbf{c}}(-))_{smooth} : [\Sigma, A_{conn}] \to U(1) \;\;\; \in Smooth \infty Grpd$$

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 $\mathfrak{a}$ be an ∞-Lie algebroid. We shall write

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

for its Lie integration.

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

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

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

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

in $\mathbf{H} =$ Smooth∞Grpd, where

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

is the coefficient object for ∞-connections with values in $\mathfrak{a}$ and $\mathbf{B}^n U(1)_{conn}$ 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 $dim \Sigma \leq n$, the induced morphism

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

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 $cs \in W(\mathfrak{a})$ over $\Sigma$: it sends a field configuration $\phi : \Sigma \to A_{conn}$, which is locally on $U \subset \Sigma$ given by a ∞-Lie algebroid valued differential form $A \in \Omega^\bullet(U, \mathfrak{a})$, to the Chern-Simons form $L(\phi|_U) = L(A) = cs(A) \in \Omega^n(\Sigma)$.

Proposition

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

$$\exp(i S(-)) : A \mapsto \exp(i \int_\Sigma CS(A)) \,.$$

References

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_\infty$-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 $\infty$-Chern-Simons theory is discussed in section 4.3 of

• Urs Schreiber, differential cohomology in a cohesive topos