This is a sub-entry of ∞-Chern-Simons theory . See there for background and details.
We describe some details of the construction of the ∞-Chern-Weil homomorphism induced by an invariant polynomial on an ∞-Lie algebroid . As discussed at Chern-Weil homomorphism and ∞-connections, this model for the -Chern-Weil homomorphism;
is controlled by a Chern-Simons element that exhibits the transgression of to an invariant polynomial ;
sends an -connection on an ∞-bundle given locally by an ∞-Lie algebroid valued differential form to the circle n-bundle with connection whose local connection -form is Chern-Simons form .
The Lagrangian of -Chern-Simons theory is then of the form . Therefore the identification of -Chern-Simons theories essentially reduces to the identificaton of higher Chern-Simons elements on ∞-Lie algebroids. In the following we survey the relevant theory for this construction.
Abstractly, an ∞-Lie algebroid is a synthetic-differential ∞-groupoid all whose k-morphisms are infinitesimal, for all . Concretely, these objects may be defined as formal duals of certain cosimplicial algebras and equivalrently but more conveniently, using the monoidal Dold-Kan correspondence, of graded-commutative semifree dg-algebras. This perspective we describe now.
We fix a commutative -algebra . If is graded -module, then the suspension of is denoted as where . We say has finite type if it is degree-wise finitely generated. Given a chain complex of finite type, its dual is the cochain complex . Hence if is concentrated in non-negative degrees, so is
Let be a chain complex of -modules of finite type concentrated in non-negative degrees. A Lie -algebroid structure on is a degree 1 -derivation on the graded commutative algebra such that: * is a differential: , * is compatible with : If is the canonical projection, then for all and .
Such a chain complex endowed with a Lie -algebroid structure is called Lie -algebroid. If is a Lie -algebroid, and the underlying complex is concentrated in degrees , then we say is a Lie -algebroid.
In analogy with the case in Lie theory, if is a Lie -algebroid, we call the DGCA the Chevalley-Eilenberg algebra of . Note that is in fact a quasi-free DGCA i.e. it is free as a graded commutative algebra. A morphism of Lie -algebroids is a dgca morphism .
(want to expand this example out explicitly later)
In differential geometry, a Lie algebroid is traditionally thought of as a vector bundle equipped with a bundle morphism to the tangent bundle and a Lie bracket on the space of sections satisfying some compatibility axioms. It is well-known that a Lie algebroid structure gives a degree 1 differential on the dgca , where and . Hence a Lie algebroid is a Lie -algebroid in the sense of def. 1. Similarly, Courant algebroids are also examples of Lie -algebroids (Roytenberg99).
The geometry of -algebroids
In general, there is a simple geometric interpretation of our Lie -algebroids. Given a Lie -algebroid , the degree 0 part of is a commutative -algebra, which we interpret as the formal dual to the space over which the algebroid is defined. If this commutative algebra is itself, then we have a Lie -algebroid over the point. It is easy to see this implies that is an -algebra.
The geometry behind our Lie -algebroids can be made more explicit by introducing the notion of a cosimplicial smooth algebra…..
In this section, we introduce the -analogues of the machinery found in classical Chern-Weil theory which we will then use to construct Chern-Simons elements for Lie -algebroids.
The Weil algebra of a Lie -algebroid is the dg-algebra
where the differential is defined to be
We may write this as
where is the shift operator and .
The Weil algebra is, in fact, the (quasi-free DGCA) mapping cone of the identity . Note that when is a Lie algebra, we recover the usual Weil algebra. It is straightforward to show that is isomorphic to a free differential algebra and therefore is contractible (c.f. Prop. 6 in SSS09). Hence we have:
If is a Lie -algebroid, then the cochain cohomology of is trivial.
We also adopt the familiar notion of a cocycle in the Chevalley-Eilenberg algebra:
An -cocycle on a Lie -algebroid is a degree element such that .
This next definition generalizes -invariant polynomials from Lie theory to ∞-Lie theory:
An invariant polynomial on a Lie -algebroid is an element
The space of invariant polynomials is denoted .
Clearly is a dg-sub algebra with the trivial differential. Note that if is a Lie algebra, then the -closed condition is equivalent to the usual -invariance.
Now we are ready to define -Chern-Simons elements. Recall that there is a canonical surjection which restricts to the identity on .
Let be a cocycle on a Lie -algebroid and an invariant polynomial. A Chern-Simons element for and is an element such that
If a Chern-Simons element exists for and , then, in analogy with the classical case for fiber bundles, we say is in transgression with . Note that we have a short exact sequence of complexes:
Since , the properties that a Chern-Simons element must satisfy can be understood in terms of the connecting homomorphism of the corresponding long exact sequence in cohomology. In particular, if and , then is a Chern-Simons element iff we have
A Chern-Simons element witnessing the transgression from to is equivalently a commuting diagram of the form
in dgAlg. On the other hand, an n-connection with values in a Lie -algebroid is a cocycle
with coefficients in the simplicial presheaf that sends CartSp to the -coskeleton of the simplicial set, which in degree is the set of commuting diagrams
such that the curvature forms of the ∞-Lie algebroid valued differential forms on with values in in the middle are horizontal.
If is an ∞-Lie algebroid cocycle of degree , then the ∞-Chern-Weil homomorphism operates by sending an -connection given by a Cech cocycle with values in simplicial sets of such commuting diagrams to the obvious pasting composite
Under the map to the coskeleton the group of such cocycles for line -bundle with connection is quotiented by the discrete group of periods of , such that the -Chern-Weil homomorphism is given by sending the -connections presented by (10) to
This is a cocycle for a circle n-bundle with connection, whose connection -form is locally given by the Chern-Simons form . This is the Lagrangian of the -Chern-Simons theory defined by and evaluated on the given -connection. If is a smooth manifold of dimension , then the higher holonomy of this circle -bundle over is the value of the Chern-Simons action. After a suitable gauge transformation this is given by the integral
This integration process we discuss in more detail in the following section.
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