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 .
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 over an object is the composite of the induced morphism on cocycles with the 0-truncation morphism
Refined to a morphism of smooth ∞-groupoids the action functional is the corresponding composite on internal hom-objects
where the last morphism is concretification.
Let . If SmoothMfd Smooth∞Grpd is a closed smooth manifold of dimension then
where in the first – discrete – version is the discrete ∞-groupoid corresponding to the Eilenberg-MacLane space , whereas in the second – smooth – version 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 we have by this proposition that . It then follows by this proposition that we have an equivalence
with the flat differential cohomology on , and using the definition and by the -adjunction it follows that this is equivalently
where is an Eilenberg-MacLane space . By this proposition we have under a weak homotopy equivalence . Therefore the cocycle -groupoid is that of ordinary cohomology
By general abstract reasoning (recalled at cohomology and fiber sequence) it follows that we have for the homotopy groups an isomorphism
Now we invoke the universal coefficient theorem. This asserts that the morphism
which sends a cocycle in singular cohomology with coefficients in to the pairing map
sits inside an exact sequence
But since is an injective -module we have
This means that the integration/pairing map is an isomorphism
For , the right hand is zero, so that
For , instead, , since is a closed -manifold and so
In codimension 0 the smooth action functional is a morphism
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 -Chern-Simons action functional by ordinary integration over ordinary differential form data.
Let be an ∞-Lie algebroid. We shall write
for its Lie integration.
Similarly, for an n-cocycle on , its Lie integration is a representative of a characteristic class
If is in transgression with an invariant polynomial and is a Chern-Simons element exhibiting that transgression, then the above discussion constructs from this an ∞-Chern-Weil homomorphism
in Smooth∞Grpd, where
is the coefficient object for ∞-connections with values in and is the objected presented under the Dold-Kan correspondence by the Deligne complex in degree : the coefficient object for circle n-bundles with connection.
For SmoothMfd Smooth∞Grpd a smooth manifold of dimension , the induced morphism
from the discrete ∞-groupoid of ∞-connections on -principal ∞-bundles to that of circle n-bundles with connection we may interpret as the Lagrangian of the -Chern-Simons theory defined by over : it sends a field configuration , which is locally on given by a ∞-Lie algebroid valued differential form , to the Chern-Simons form .
In codimension 0 the corresponding -Chern-Simons functional sends -calued differential forms to the ordinary integral
The notion of Chern-Simons elements for -algebras and the associated -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 -Chern-Simons theory is discussed in section 4.3 of