infinity-Chern-Simons theory -- action functionals

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

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

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 $\exp(i L(-)) : A_{conn} {\to} \mathbf{B}^n U(1)_{conn}$ 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.

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.

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

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

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.

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* .

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.

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 $L(\phi|_U) = L(A) = cs(A) \in \Omega^n(\Sigma)$.

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

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

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

Last revised on October 27, 2011 at 02:59:58. See the history of this page for a list of all contributions to it.