nLab Chern-Simons element

Idea

A Chern-Simons element on an L-∞ algebroid (named after Shiing-shen Chern and James Simons who considered this for semisimple Lie algebras) is an element of its Weil algebra that exhibits a transgression between an ∞-Lie algebroid cocycle and an invariant polynomial.

It is construct that arises in the presentation of the ∞-Chern-Weil homomorphism by an ∞-anafunctor of simplicial presheaves.

Definition

We discuss ∞-Lie algebras and ∞-Lie algebroids $\mathfrak{a}$ of finite type in terms of their Chevalley-Eilenberg algebras $CE(\mathfrak{a})$. For $\infty$-Lie algebras these are objects in the category dgAlg of dg-algebras (over a given ground field). For $\infty$-Lie algebroids these are dg-algebras equipped with a lift of the degree-0 algebra to an algebra over a given Fermat theory $T$ and such that the differential is a $T$-derivation in this degree. (See ∞-Lie algebroid for details). We shall write in the following $dgAlg$ also for the category of dg-algebras with this extra structure and leave the Fermat theory $T$ implicit.

Definition

For $\mathfrak{g}$ an ∞-Lie algebra or more generally ∞-Lie algebroid, $\mu \in CE(\mathfrak{g})$ a ∞-Lie algebra cocycle (a closed element of the Chevalley-Eilenberg algebra) and $\langle - \rangle \in W(\mathfrak{g})$ an invariant polynomial, a Chern-Simons element exhibiting the transgression between the two is an element

$$cs \in W(\mathfrak{g})$$

such that

1. we have $d_{W(\mathfrak{g})} cs = \langle -\rangle$

2. and $cs|_{CE(\mathfrak{g})} = \mu$

where the restriction is along the canonical morphism $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

Remark

Notice that a degree-$n$ ∞-Lie algebroid cocycle $\mu$ is equivalently a morphism

$$CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathbb{R}) : \mu$$

and an invariant polynomial of degree $n+1$ is equivalently a morphism

$$inv(\mathfrak{g}) \leftarrow inv(b^{n-1}\mathbb{R}) = CE(b^n \mathbb{R}) : \langle - \rangle$$

in dgAlg.

Observation

A Chern-Simons element $cs$ an $\mathfrak{a}$ witnessing the transgression of $\langle - \rangle$ to $\mu$ is equivalently a morphism

$$W(\mathfrak{g}) \leftarrow W(b^{n-1} \mathbb{R}) : cs$$

such that we have a commuting diagram in dgAlg

(1)$$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1} \mathbb{R}) &&& cocycle \\ \uparrow && \uparrow^{\mathrlap{i^*}} \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{n-1} \mathbb{R}) &&& Chern-Simons\;element \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& inv(b^{n-1} \mathbb{R}) &&& invariant\;polynomial } \,,$$

where the vertical morphisms are the canonical ones.

Remark

If we think of

• $W(\mathfrak{g})$ as differential forms on the total space of the universal $G$-bundles;

• $CE(\mathfrak{g})$ as differential forms on the fiber

• $inv(\mathfrak{g})$ as differential forms on the base space

then the abov expresses the classical notion of transgression of forms from the fiber to the base of a fibe bundle (for instance Borel, section 9).

Properties

Uniqueness

Observation

For a given transgressive cocycle $\mu$ and transgressing invariant polynomial $\langle - \rangle$ the set of Chern-Simons elements witnessing the transgression is a torsor (based over the point and) over the additive group

$$\{\omega \in W(\mathfrak{g}) | d_{W(\mathfrak{g})} \omega = 0, i^* \omega = 0\}$$

of Chern-Simons elements for vanishing cocycle and vanishing invariant polynomial.

Canonical $\infty$-Chern-Simons elements

Since the Weil algebra of an L-∞ algebra has trivial cohomolgy in positive degree, every invariant polynomial $\langle -,\cdots, -\rangle$ has a Chern-Simons element and there is a standard formula for it.

Proposition

Let $\mathfrak{g}$ be an L-∞ algebra with $k$-ary brackets $[-,\cdots, -]_k : \mathfrak{g}^{\otimes k} \to \mathfrak{g}$ and equipped with a quadratic invariant polynomial $\langle -,-\rangle$.

A Chern-Simons element for $\langle-,-\rangle$ is given by the formula

$$cs(A) = \langle A, d_{dR} A\rangle + \sum_{k = 1}^\infty \frac{2}{(k+1)!} \langle A, [A,\cdots,A]_k\rangle \,,$$

where $A : W(\mathfrak{g}) \to \Omega^\bullet(\Sigma)$ is any $\mathfrak{g}$-valued form

Proof

There is a canonical contracting homotopy

$$\tau : W(\mathfrak{g}) \to W(\mathfrak{g})$$

satisfying $[d_W, \tau] = Id$

and the above element is

$$cs = \tau \langle -,-\rangle \,.$$

To see this, let $\{t_a\}$ be a basis and $\{t^a\}$ the dual basis. Then the differential of the Chevalley-Eilenberg algebra can be written

$$d_{CE(\mathfrak{g})} t^a = - \sum_{k = 1}^\infty \frac{1}{k!} [t_{a_1}, t_{a_2}, \cdots, t_{a_k}]^a \,\, t^{a_1} \wedge t^{a_2}\wedge \cdots t^{a_k} \,,$$

where

$$[-,-, \cdots, -] : \mathfrak{g}^{\otimes_k} \to \mathfrak{g}$$

is the corresponding $k$-ary bracket.

Write

$$P_{a b} := \langle t_a , t_b\rangle \,,$$

for the components of the invariant polynomial in this basis.

Then the claim is that

$$cs = 2 P_{a b} t^a \wedge d_{W(\mathfrak{g})} t^b + \sum_{k = 1}^\infty C_{a b_1, \cdots, b_k} \,\, t^a \wedge t^{b_1} \wedge \cdots \wedge t^{b_k} \,,$$

where the coefficients are

$$C_{a b_1, \cdots, b_k} := \frac{1}{(k+1)!} (P_{a b} [t_{b_1}, \cdots, t_{b_k}]^b)$$

Write $F(\mathfrak{g})$ for the free dg-algebra on the graded vector space $\mathfrak{g}^*$. In terms of the above basis this is generated from $\{t^a, \mathbf{d}t^a\}$. As discussed at Weil algebra, there is a dg-algebra isomorphism

$$F(\mathfrak{g}) \stackrel{\simeq}{\to} W(\mathfrak{g})$$

given by sending $t^a \mapsto t^a$ and $\mathbf{d}t^a \mapsto d_{CE} t^a + r^a$.

Let $h : F(\mathfrak{g}) \to F(\mathfrak{g})$ be the derivation which on generators is defined by

$$h : t^a \mapsto 0$$

$$h : \mathbf{d}t^a \mapsto t^a \,.$$

Notice that this is not the homotopy that exhibits the triviality of $Id_{F(\mathfrak{g}^*)}$, rather that homotopy is $\frac{1}{L} h$, where $L$ is the word length operator for element in $F(\mathfrak{g}^*)$ in terms of the generators $\{t^a , \mathbf{d}t^a\}$.

Therefore the homotopy $\tau$ is the composite top morphism in the diagram

$$\array{ W(\mathfrak{g}) &\stackrel{\tau}{\to}& W(\mathfrak{g}) \\ \downarrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}} \\ F(\mathfrak{g}) &\stackrel{\frac{1}{L} h}{\to}& F(\mathfrak{g}) } \,.$$

Unwinding this, we find

$$\begin{aligned} cs & := \tau \left( P_{a b} r^{a} \wedge r^b \right) \\ & = P_{a b} \tau \left( d_{W(\mathfrak{g})} t^a + \sum_{k = 1}^\infty [t_{a_1}, \cdots, t_{a_k}]^a t^{a_1} \wedge \dots t^{a_k} \right) \wedge \left( d_{W(\mathfrak{g})} t^b + \sum_{k = 1}^\infty [t_{b_1}, \cdots, t_{b_k}]^b t^{b_1} \wedge \dots t^{b_k} \right) \\ & = P_{a b} t^a \wedge d_{W(\mathfrak{g})} t^b + \sum_{k = 1}^\infty \frac{2}{k! (k+1) } P_{a b} [t_{b_1}, \cdots t_{b_k}]^b t^{b_1} \wedge \cdots \wedge t^{b_k} \end{aligned} \,.$$

Example

We consider the ordinary Chern-Simons element as an example of this formula: let $\mathfrak{g}$ be a semisimple Lie algebra and $\langle -,-\rangle$ the Killing form invariant polynomial. Then the above computation gives

$$\begin{aligned} cs & = \tau \left( P_{a b} r^a \wedge r^b \right) \\ & = P_{a b}\tau \left(d_W t^a + \frac{1}{2}C^a{}_{a_1 a_2}t^{a_1} \wedge t^{a_2} \right) \wedge \left(d_W t^b + \frac{1}{2}C^b{}_{b_1 b_2}t^{b_1} \wedge t^{b_2} \right) \\ & = P_{a b} t^a \wedge d_W t^b + \frac{2}{2! 3} P_{a b} t^a \wedge C^b_{b_1 b_2} t^{b_1} \wedge t^{b_2} \\ & = P_{a b} t^a \wedge d_W t^b + \frac{1}{3} C_{a b c} t^a \wedge t^{b} \wedge t^{c} \end{aligned} \,.$$

Origin and relation to other concepts

We discuss the general abstract structures of which Chern-Simons elements are presentations and how they are related to other structures.

The term Chern-Simons element alludes to the term Chern-Simons form and Chern-Simons theory. In the following we explain the relation.

As presentations for the $\infty$-Chern-Weil homomorphism

We explain here briefly how Chern-Simons elements provide a presentation of a generalization of the Chern-Weil homomorphism – the ∞-Chern-Weil homomorphism in cohesive (∞,1)-topos theory – in the sense in which (∞,1)-toposes have presentations by a model structure on simplicial presheaves.

To warm up, we start with considering a traditional setup of Lie groupoid theory. Recall that for $G$ a Lie group, we may form its delooping Lie groupoid dnoted $*//G$ or $\mathbf{B}G$. Then with $X$ any smooth manifold, we have that the groupoid of morphisms of Lie groupoids $X \to \mathbf{B}G$ is equivalent to that of $G$-principal bundles on $X$:

$$SmoothGrpd(X, \mathbf{B}G) \simeq G Bund(X) \,.$$

Here we are thinking of Lie groupoids as differentiable stacks, hence as objects in the (2,1)-topos

$$SmoothGrpd := Sh_{(2,1)}(SmoothMfd)$$

of stacks/(2,1)-sheaves on the site SmoothMfd (equivalently on its small dense subsite CartSp of Cartesian spaces). (This is discussed in detail at principal bundle ).

There is a differential refinement of the Lie groupoid $\mathbf{B}G$, to the smooth groupoid

$$\mathbf{B}G_{conn} := SmoothGrpd(\mathbf{P}_1(-), \mathbf{B}G) \,,$$

where $\mathbf{P}_1(X)$ is the path groupoid of $X$. This is the (2,1)-sheaf given by the (2,1)-sheafification of the assignment that sends a smooth manifold $U$ to the groupoid of Lie algebra-valued 1-forms on $U$.

There is a corresponding natural equivalence

$$SmoothGrpd(X, \mathbf{B}G_{conn}) \simeq G Bund_{conn}(X)$$

of morphisms into $\mathbf{B}G_{conn}$ with the groupoid of $G$-principal bundles with with connection on $X$. (This is described in detail at connection on a bundle ).

In particular if $G = U(1)$ is the circle group, a morphism $X\to \mathbf{B}U(1)_{conn}$ is a circle bundle with connection. This

This allows already to consider a simple case of a characteristic class and its refinement to a differential characteristic class: Let $U$ be the unitary group. There is a canonical morphism of Lie groupoids $\mathbf{c}_1 : \mathbf{B}U \to \mathbf{B}U(1)$ given by the determinant. This – or rather its image in cohomology

$$\mathbf{c}_1 : SmoothGrpd(- ,\mathbf{B}U) \to SmoothGrpd(-, \mathbf{B} U(1))$$

is a smooth representative of the characteristic class called the first Chern class . Its differential refinement is the evident morphism

$$\hat \mathbf{c}_1 : \mathbf{B}U_{conn} \to \mathbf{B}U(1)_{conn}$$

that sends a $\mathfrak{u}$-valued differential form to the trace of its Lie algebra value. Postcomposition with this is the refined Chern-Weil homomorphism

$$\array{ SmoothGrpd(X, \mathbf{B}U)_{conn} &\stackrel{\hat \mathbf{c}_1}{\to}& SmoothGrpd(X, \mathbf{B}U(1)_{conn}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ U Bund_\nabla(X) &\to& U(1) Bund_\nabla(X) }$$

with values in circle bundles with connection, hence in degree-2 ordinary differential cohomology.

It is this kind of construction on Lie groupoids that we now want to generalize to a notion of smooth ∞-groupoids, to see that Chern-Simons elements are a means to constructi morphisms akind to the differential first Chern-class $\hat \mathbf{c}_1$.

A general abstract context for higher geometry equipped with differential cohomology is a cohesive (∞,1)-topos $\mathbf{H}$ of ∞-groupoids equipped with cohesive structure , such as smooth cohesive structure.

An example for such is the ∞-stack-analog of the stack-(2,1)-topos over SmoothMfd: the ∞-stack (∞,1)-topos Smooth∞Grpd $:= \hat Sh_{(\infty,1)}(SmoothMfd)$.

In that context we have for instance all the higher deloopings of $U(1)$: the circle Lie (n+1)-groups

$$\mathbf{B}^n U(1) \in Smooth\infty Grpd \,.$$

This is such that the evident generalizations of the above classification statements hold: we have that morphisms $X \to \mathbf{B}^n U(1)$ form an n-groupoid

$$Smooth\infty Grpd(X, \mathbf{B}^n U(1)) \simeq U(1) (n-1)Bund(X)$$

equivalent to that of circle n-bundles/$(n-1)$-bundle gerbes on $X$.

If here $X = \mathbf{B}G$ is again the delooping of a Lie group, this means that now also the higher characteristic classes are represented by morphisms

$$\mathbf{B}G \to \mathbf{B}^n U(1) \,.$$

For instance for $G = Spin$ the spin group, the first fractional Pontryagin class has a smooth incarnation given by a morphism of the form

$$\frac{1}{2}\mathbf{p}_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1)$$

corresponding under the above equivalence to the ordinary Chern-Simons circle 3-bundle on $\mathbf{B}G$.

Every cohesive (∞,1)-topos comes canonically and essentially uniquely equipped with all the intrinsic structure that we need for the discussion of a refinement of this to differential characteristic classes:

There is an endo-(∞,1)-adjunction

$$(\mathbf{\Pi} \dashv \mathbf{\flat}) : Smooth\infty Grpd \to Smooth \infty Grpd$$

where

• $\mathbf{\Pi}(X)$ is the path ∞-groupoid of a smooth ∞-groupoid $X$;

• $\mathbf{\flat}\mathbf{B}G$ is the coefficient object for flat ∞-connections on $G$-principal ∞-bundles.

A morphism $\mathbf{\Pi}(X) \to \mathbf{B}^n U(1)$ encodes the flat higher parallel transport of a flat circle n-bundle with connection, and we have that the n-groupoid of morphisms

$$Smooth \infty Grpd(\mathbf{\Pi}(X), \mathbf{B}^n U(1)) \simeq U(1) n Bund_{\nabla_{flat}}(X)$$

is that of flat circle n-bundles with connection/ (n-1)-bundle gerbes with connection.

We observe that a trivial circle $n$-bundle with connection is equivalently just a globally defined differential n-form. Therefore if we define the modified (∞,1)-adjunction

$$(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR}) : */Smooth\infty Grpd \stackrel{\leftarrow}{\to} Smooth\infty Grpd$$

by forming the (∞,1)-pullback

$$\mathbf{\flat}_{dR}\mathbf{B}^n U(1) := * \prod_{\mathbf{B}^n U(1)} \mathbf{\flat} \mathbf{B}^n U(1) \,,$$

which is the coefficient object for trivial principal $\infty$-bundles equipped with flat $\infty$-connection, one finds (discussed in detail here) that morphisms $X \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ correspond to trivial circle bundle with connection, hence to cocycles in de Rham cohomology of $X$;

$$\pi_0 Smooth \infty Grpd(X, \mathbf{\flat}_{dR} \mathbf{B}^n U(1)) = \left\{ \array{ H_{dR}^n(X) & n \geq 2 \\ \Omega^1_{cl}(X) & n = 1 } \right. \,.$$

This now allows us to construct differential refinements:

one can show (detailed discussion is here) that there are canonical cocycles

$$curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}$$

in the degree $(n+1)$-de Rham cohomology of $\mathbf{B}^n U(1)$: these are the universal curvature characteristic forms on $\mathbf{B}^n U(1)$.

Then for $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ any smooth characteristic class, the corresponding (unrefined) differential characteristic class is simply the composite

$$\mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1) \,.$$

The (unrefined) ∞-Chern-Weil homomorphism is postcomposition with this morphism:

$$(\mathbf{c}_{dR})_* : H^1(X,G) \to H_{dR}^{n+1}(X) \,.$$

This is finally where the Chern-Simons elements come in:

Chern-Simons elements are a means to present the composite morphism $\mathbf{c}_{dR}$ of smooth ∞-groupoids by an ∞-anafunctor between smooth Kan complexes.

This presentation we describe in the next section.

(In fact a bit more is true: the serve to present the refinement of $\mathbf{c}_{dR}$ to a morphism $\hat \mathbf{c}$ with values in ordinary differential cohomology. This we come to further below.)

Chern-Simons forms

We explain now how Chern-Simons elements arise as a presentation of a differential characteristic class $\mathbf{c}_{dR}$ by a span of simplicial presheaves.

At the heart of the presentation of differenial characteristic classes by morphisms of simplicial presheaves is a differential refinement of the Lie integration of L-∞ algebras and ∞-Lie algebroid: for $\mathfrak{g}$ an ordinary Lie algebra, one finds that the 3-coskeleton of the simplicial presheaf that assigns flat vertical $\mathfrak{g}$-Lie algebra valued 1-forms

$$\exp(\mathfrak{g}) : (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{si, vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) } \right\}$$

is the delooping of the simply connected Lie group $G$ integrating $\mathfrak{g}$

$$\mathbf{cosk}_3 \exp(\mathfrak{g}) \stackrel{\simeq}{\to} \mathbf{B}G \,.$$

Similarly the Lie integration of the line Lie n-algebra $b^{n-1}\mathbb{R}$

$$\exp(b^{n-1}\mathbb{R}) : (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n-1}\mathbb{R}) } \right\}$$

is the $n$-fold delooping of $\mathbb{R}$:

$$\exp(b^{n-1}\mathbb{R}) \stackrel{\simeq}{\to} \mathbf{B}^n \mathbb{R} \,.$$

Moreover, for $\mu : \mathfrak{g} \to b^{n-1}\mathbb{R}$ a degree-$n$ cocycle in Lie algebra cohomology, simple postcomoposition gives its image under Lie integration

$$\exp(\mu) : \exp(\mathfrak{g}) \to \exp(b^{n-1}\mathbb{R}) \,.$$

Under coskeletization on the left this carves out the periods of $\mu$ as a lattice in $\mathbb{R}$, which typically is the integers, so that this descends to degree $n$-cocycle in Lie group cohomology with coefficients in $U(1) \simeq \mathbb{R}/\mathbb{Z}$

$$\exp(\mu) : \mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\mathbb{Z} \,.$$

(See Lie group cohomology and smooth ∞-groupoid for discussion of the refined notion of Lie group cohomology arising here.)

The differential refinement of these construction is based on the following fact (discussed in detail here)

1. the object $\mathbf{B}^n U(1) \in$ Smooth∞Grpd is equivalently presented by a quotient of the presheaf of Kan complexes given by

   $$\mathbf{B}^n U(1)_{diff} : (U \in SmoothMfd, [k] \in \Delta) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{\omega}{\leftarrow}& W(b^{n-1} \mathbb{R}) } \right\} \,,$$

   where on the right we have the set of horizontal morphisms in dgAlg that make a commuting diagram with the canonical vertical morphisms as indicated.

   $$\array{ \mathbf{B}^n U(1)_{diff} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n U(1) } \,.$$

   (We may think of a morphism of simplicial presheaves $X \to \mathbf{B}^n U(1)_{diff}$ as a circle n-bundle/$(n-1)$-bundle gerbe equipped with a pseudo-connection . )

   Notice that the bottom morphism here encodes precisely a degree-$n$ differential form $\omega$ on $U \times \Delta^k$,

2. The morphism $curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$ is presented on this by the ∞-anafunctor

   $$\array{ \mathbf{B}^n U(1)_{diff} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{sim} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n U(1) }$$

   by the map that sends such a form $\omega$ to its curvature $d \omega$. If the pseudo-connections that we are dealing with are genuine connections the curvature is a basic form down on $U$ and this means diagrammatically that it forms the pasting composite

   $$\array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{\omega}{\leftarrow}& W(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{d \omega}{\leftarrow}& inv(b^{n-1} \mathbb{R}) }$$

   and then picks out the bottom horizontal morphism.

Therefore our task of presenting $\mathbf{c}_{dR}$ amounts to computing the composition of ∞-anafunctors

$$\array{ && \mathbf{B}^n U(1)_{diff} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{sim} \\ && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G &\stackrel{\exp(\mu)}{\to}& \mathbf{B}^n U(1) }$$

To do se we need to complete componentwise to commuting diagrams. To this end we first complete the assignment of $\exp(\mathfrak{g})$ to a diagram

(2)$$(U,[k]) \mapsto \left\{ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &&& transition\;function \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &&& connection \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A \rangle}{\leftarrow}& inv(\mathfrak{g}) &&& curvature\;characteristic\;form } \right\} \,.$$

Here in the middle row an unrestricted $\mathfrak{g}$-∞-Lie algebra valued differential form appears, which is the local $\mathfrak{g}$-∞-connection. And in the lower row all its curvature characteristic forms appear, obtained by evaluating the curvature $F_A$ in the invariant polynomials on $\mathfrak{g}$.

This is such that a choice of Chern-Simons element witnessing the transgression of an invariant polynomial to $\mu$ allows to refine $\exp(\mu)$ to

(3)$$\cdots \stackrel{\exp(\mu)_{conn}}{\mapsto} \left\{ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) &&& characteristic\;class \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{n-1}\mathbb{R}) &&& Chern-Simons\;form \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle -\rangle}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& inv(b^{n-1}\mathbb{R}) &&& curvature\;characteristic\;form } \right\} \,.$$

Here now the middle row is the evaluationn of the connection form inside the Chern-Simons element. This is the corresponding Chern-Simons form

$$\Omega^\bullet(U) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{cs}{\leftarrow} W(b^{n-1}\mathbb{R}) : CS(A)$$

of the $\mathfrak{g}$-connection evaluated in the given Chern-Simons element. Its curvature is the curvature characteristic form $\langle F_A \rangle$ appearing in the bottom line of the diagram, which is obtained by evaluating the $\mathfrak{g}$-valued curvature in the given invariant polynomial.

A more comprehensive account of this is at Chern-Weil homomorphism in Smooth∞Grpd.

Chern-Simons action functionals

By the above construction, every Chern-Simons element $cs \in W(\mathfrak{a})$ of degree $d$ on an ∞-Lie algebroid $\mathfrak{a}$ induces an action functional on the space of ∞-Lie algebroid valued forms on $\mathfrak{a}$ over a $d$-dimensional smooth manifold $\Sigma$

$$S_{cs} : \Omega(\Sigma, \mathfrak{a}) \to \mathbb{R}$$

given by

$$(A,B,C, \cdots) \mapsto \int_\Sigma CS(A,B,C, \cdots) \,.$$

This generalizes the action functional of ordinary Chern-Simons theory to general Chern-Simons elements. In the examples below is a list of various quantum field theories that arise as generalized Chern-Simons theories this way.

For more details see infinity-Chern-Simons theory.

Examples

On semisimple Lie algebra– Standard Chern-Simons action functional

Let $\mathfrak{g}$ be a semisimple Lie algebra. For the following computations, choose a basis $\{t^a\}$ of $\mathfrak{g}^*$ and let $\{r^a\}$ denotes the corresponding degree-shifted basis of $\mathfrak{g}^*[1]$.

Notice that in terms of this the differential of the CE-algebra is

$$d_{CE(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c$$

and that of the Weil algebra

$$d_{W(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c + r^a$$

and

$$d_{W(\mathfrak{g})} : r^a \mapsto -C^a{}_{b c} t^b \wedge r^c \,.$$

Let $P_{a b} r^a \wedge r^b \in W(\mathfrak{g})$ be the Killing form invariant polynomial. This being invariant

$$d_{W(\mathfrak{g})} P_{a b} r^a \wedge r^b = 2 P_{a b} C^{a}{}_{d e} t^d \wedge r^e \wedge r^b = 0$$

is equivalent to the fact that the coefficients

$$C_{a b c} := P_{a a'}C^{a'}{}_{b c}$$

are skew-symmetric in $a$ and $b$, and therefore skew in all three indices.

Proposition

A Chern-Simons element for the Killing form invariant polynomial $\langle -, - \rangle = P(-,-)$ is

$$\begin{aligned} cs &= P_{a b} t^a \wedge (d_{W(\mathfrak{g})} t^b) + \frac{1}{3} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c \\ & = P_{a b} t^a \wedge r^b - \frac{1}{6} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c \end{aligned} \,.$$

In particular the Killing form $\langle -,-\rangle$ is in transgression with the degree 3-cocycle

$$\mu = -\frac{1}{6}\langle -,[-,-]\rangle \,.$$

Proof

We compute

$$\begin{aligned} d_{W(\mathfrak{g})} \left( P_{a b} t^a \wedge r^b + \frac{1}{2}P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c \right) & = P_{a b} r^a \wedge r^b \\ & -\frac{1}{2} P_{a b} C^a{}_{d e} t^d \wedge t^e \wedge r^b \\ & + P_{a b} C^b{}_{d e} t^a \wedge t^d \wedge r^e \\ & - \frac{3}{6} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge r^c \\ & = P_{a b} r^a \wedge r^b \\ & + \frac{1}{2}C_{a b c} t^a \wedge t^b \wedge r^c \\ & - \frac{1}{2} C_{a b c} t^a \wedge t^b \wedge r^c \\ & = P_{a b } r^a \wedge r^b \end{aligned} \,.$$

Under a Lie algebra-valued form

$$\Omega^\bullet(X) \stackrel{}{\leftarrow} W(\mathfrak{g}) : A$$

this Chern-Simons element is sent to

$$cs(A) = P_{a b} A^a \wedge d A^b + \frac{1}{3} C_{a b c} A^a \wedge A^b \wedge A^c \,.$$

If $\mathfrak{g}$ is a matrix Lie algebra then the Killing form is the trace and this is equivalently

$$cs(A) = tr(A \wedge d A) + \frac{2}{3} tr(A \wedge A \wedge A) \,.$$

This is a familiar form of the standard Chern-Simons form in degree 3.

For $\Sigma$ a 3-dimensional smooth manifold the corresponding action functional $S_{CS} : \Omega^1(\Sigma, \mathfrak{g}) \to \mathbb{R}$

$$S_{CS} : A \mapsto \int_\Sigma cs(A)$$

is the standard action functional of Chern-Simons theory.

Higher CS-forms on semisimple Lie algebras

For $\mu \in CE(\mathfrak{g})$ any higher order cocycle, $CS_\mu(A)$ is the corresponding higher order Chern-Simons form.

Action functional for Chern-Simons (super-)gravity

Higher Chern-Simons elements on the Poincare Lie algebra $\mathfrak{g} = \mathfrak{iso}(d,1)$ or the super Poincare Lie algebra $\mathfrak{g} = \mathfrak{siso}(d,1)$ yield action functionals for gravity and supergravity.

(…)

See (Zanelli).

Fractional secondary Pontryagin classes

For instance for $\mu_7$ the 7-cocycle on a semisimple Lie algebra, $CS_{\mu_7}(A)$ is the corresponding Chern-Simons 7-form, corresponding to the second Pontryagin class.

Notice that this we may also think of as a 7-cocycle on the corresponding string Lie 2-algebra. As such it is the one that classifies the extension to the fivebrane Lie 6-algebra. The corresponding Chern-Simons 7-form appears as the local conneciton data in the Chern-Simons circle 7-bundle with connection that obstructions the lift from a differential string structure to a differential fivebrane structure.

On strict Lie 2-algebras – BF-theory action functional

Proposition

Let $\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g})_1$ be a strict Lie 2-algebra.

Then

• every invariant polynomial $\langle -\rangle_{\mathfrak{g}_1} \in inv(\mathfrak{g}_1)$ on $\mathfrak{g}_1$ is a Chern-Simons element on $\mathfrak{g}$, restricting to the trivial ∞-Lie algebra cocycle;

• for $\mathfrak{g}_1$ a semisimple Lie algebra and $\langle - \rangle_{\mathfrak{g}_1}$ the Killing form, the corresponding Chern-Simons action functional on ∞-Lie algebra valued forms

  $$\Omega^\bullet(X) \stackrel{(A,B)}{\leftarrow} W(\mathfrak{g}_2 \to \mathfrak{g}_1) \stackrel{(\langle - \rangle_{\mathfrak{g}_1}, d_W \langle - \rangle_{\mathfrak{g}_1} )}{\leftarrow} W(b^{n-1} \mathbb{R})$$

is the sum of the action functionals of topological Yang-Mills theory with BF-theory with cosmological constant (in the sense of gravity as a BF-theory):

$$CS_{\langle-\rangle_{\mathfrak{g}_1}}(A,B) = \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1} - 2\langle F_A \wedge \partial B\rangle_{\mathfrak{g}_1} + 2\langle \partial B \wedge \partial B\rangle_{\mathfrak{g}_1} \,,$$

where $F_A$ is the ordinary curvature 2-form of $A$.

This is from (SSSI).

Proof

For $\{t_a\}$ a basis of $\mathfrak{g}_1$ and $\{b_i\}$ a basis of $\mathfrak{g}_2$ we have

$$d_{W(\mathfrak{g})} : \sigma t^a \mapsto d_{W(\mathfrak{g}_1)} + \partial^a{}_i \sigma b^i \,.$$

Therefore with $\langle -\rangle_{\mathfrak{g}_1} = P_{a_1 \cdots a_n} \sigma r^{a_1} \wedge \cdots \sigma t^{a_n}$ we have

$$d_{W(\mathfrak{g})} \langle - \rangle_{\mathfrak{g}_1} = n P_{a_1 \cdots a_n}\partial^{a_1}{}_i \sigma b^{i} \wedge \cdots \sigma t^{a_n} \,.$$

The right hand is a polynomial in the shifted generators of $W(\mathfrak{g})$, and hence an invariant polynomial on $\mathfrak{g}$. Therefore $\langle - \rangle_{\mathfrak{g}_1}$ is a Chern-Simons element for it.

Now for $(A,B)$ an ∞-Lie algebra-valued form, we have that the 2-form curvature is

$$F_{(A,B)}^1 = F_A - \partial B \,.$$

Therefore

$$\begin{aligned} CS_{\langle -\rangle_{\mathfrak{g}_1}}(A,B) & = \langle F_{(A,B)}^1\rangle_{\mathfrak{g}_1} \\ & = \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1} - 2\langle F_A \wedge \partial B\rangle_{\mathfrak{g}_1} + 2\langle \partial B \wedge \partial B\rangle_{\mathfrak{g}_1} \end{aligned} \,.$$

On symplectic $\infty$-Lie algebroids – The AKSZ Lagrangian

A symplectic Lie n-algebroid $(\mathfrak{a}, \omega)$ is a Lie n-algebroid $\mathfrak{a}$ equipped with a binary non-degenerate invariant polynomial $\omega \in W(\mathfrak{a})$ of degree $n+2$.

The corresponding Chern-Simons elements of $\omega$ are the integrands for the action functionals of various TQFT sigma-models.

With $\{-,-\} : CE(\mathfrak{a}) \otimes CE(\mathfrak{a}) \to CE(\mathfrak{a})$ the graded Poisson bracket induced by $\omega$ we have (see Roytenberg) that there exists a ∞-Lie algebra cocycle $\mu \in CE(\mathfrak{a})$ such that

$$d_{CE(\mathfrak{a})} = \{\mu, -\} \,.$$

So in particular $\mu$ being a cocycle means that

$$d_{CE(\mathfrak{a})} \mu = \{\mu, \mu\} = 0 \,.$$

Proposition

The cocycle $\mu$ is in transgression with the invariant polynomial $\frac{n}{2}\omega$ via the Chern-Simons element

$$\begin{aligned} cs &= \frac{1}{2 }\iota_{\epsilon} \omega - \mu \end{aligned} \,,$$

where $\epsilon$ is the Euler vector field (Roytenberg).

Here $\mathbf{d}$ is the shift derivation in the Weil algebra, in that $d_{W(\mathfrak{a})} = d_{CE(\mathfrak{a})} + \mathbf{d}$.

Proof

To safe typing signs, we write as if all functions were even graded. By standard reasoning the computation holds true then also for arbitrary grading.

Observe that

1. on unshifted generators we have

   $$(\mathbf{d}x^b) \omega_{a b} \{x^a , -\} = \mathbf{d}$$

2. we have graded commutators

   • $[\mathbf{d}, \iota_v] = N$ (the degree operator)

   and

   • $[d_{CE(\mathfrak{a})}, \iota_v] = -d_{CE(\mathfrak{a})}\mathbf{d}^{-1}$.

   (as one checks on generators).

Therefore

$$\begin{aligned} d_{W(\mathfrak{a})} \frac{1}{2}\iota_{v} \omega &= [d_{W(\mathfrak{a})}, \iota_{v}] \frac{1}{2}\omega \\ & = (n - d_{CE(\mathfrak{a})} \mathbf{d}^{-1} ) \frac{1}{2}\omega \\ & = \frac{n}{2}\omega - \omega_{a b} \{\mu, x^a\} \mathbf{d}x^b \\ &= \frac{1}{2} \omega + \mathbf{d}\mu \end{aligned} \,,$$

where in the first line we used that by definition of invariant polynomial $d_{W(\mathfrak{a})} \omega = 0$. Similarly, using that by definition $d_{CE(\mathfrak{a})} \mu = 0$ we have

$$d_{W(\mathfrak{a})} \mu = \mathbf{d}\mu \,.$$

So in total we have

$$d_{W(\mathfrak{a})} (\frac{1}{2} \iota_\epsilon \omega - \mu) = \frac{1}{2}\omega \,.$$

Remark

In local coordinates where $\omega = \omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b$ we have

$$cs = n \omega_{a b} x^a \wedge \mathbf{d}x^b + \mu \,.$$

The Chern-Simons action functional corresponding to this Chern-Simons element on $\mathfrak{a}$ is that considered in AKSZ theory.

Below we spell out some low-dimensional cases explicitly.

Higher phase space —Hamiltonian and Lagrangian mechanics

The symplectic Lie $n$-algebroid $(\mathfrak{P}, \omega)$ may be thought of as an n-symplectic manifold that models the phase space of a physical system.

This means for $(\mathfrak{g},\langle-\rangle) = (\mathfrak{P}, \omega)$ a symplectic Lie $n$-algebroid, the general diagram (1) exhibiting the transgression between cocycles and invariant polynomials via Chern-Simons elements may be labeled in terms of Hamiltonian mechanics, Lagrangian mechanics and symplectic geometry as follows

(4)$$\array{ CE(\mathfrak{P}) &\stackrel{H}{\leftarrow}& CE(b^{n}\mathbb{R}) &&& Hamiltonian \\ \uparrow && \uparrow \\ W(\mathfrak{P}) &\stackrel{L}{\leftarrow}& W(b^n \mathbb{R}) &&& Lagrangian \\ \uparrow && \uparrow \\ inv(\mathfrak{P}) &\stackrel{\omega}{\leftarrow}& inv(b^n \mathbb{R}) &&& symplectic\;structure }$$

=–

See Hamiltonian, Lagrangian, symplectic structure.

On a symplectic manifold – The topological particle

For $X$ a smooth manifold we may regard its cotangent bundle $\mathfrak{a} = T^* X$ as a Lie 0-algebroid and the canonical 2-form $\omega \in W(\mathfrak{a}) = \Omega^\bullet(X)$ as a binary invariant polynomial in degree 2.

The Chern-Simons element is the canonical 1-form $\alpha$ which in local coordinates is $\alpha = p_i d q^i$.

The corresponding action functional on the line

$$\int_{\mathbb{R}} \gamma^* (p_i\, d q^i)$$

is the familiar term for the action functional of the particle (missing the kinetic term, which makes it “topological”).

On a Poisson Lie algebroid – The Poisson $\sigma$-model

Proposition

Let $\mathfrak{a} = \mathfrak{P}(X,\pi)$ by a Poisson Lie algebroid. This comes with the canonical invariant polynomial $\omega = \mathbf{d} \partial_i \wedge \mathbf{d} x^i$.

The corresponding ∞-Lie algebroid cocycle is

$$\mu_{\omega} = \pi$$

and a Chern-Simons element for this is

$$cs_\omega = \pi^{i j} \partial_i \wedge \partial_j + \partial_i \wedge \mathbf{d}x^i \,.$$

For $\Sigma$ a 2-dimensional smooth manifold the corresponding action functional on ∞-Lie algebroid-valued forms $S : \Omega^\bullet(X, \mathfrak{P}(X,\pi)) \to \mathbb{R}$ is the actional functional of the Poisson sigma-model

$$S : (X, \eta) \mapsto \int_\Sigma (\eta \wedge d X + \pi(\eta \wedge \eta) \,.$$

Proof

We compute in a local coordinte patch:

$$\begin{aligned} d_{W(\mathfrak{P}(X,\pi))} (\pi^{i j} \partial_i \wedge \partial_j + \partial_i \wedge \mathbf{d} x^i) = & \mathbf{d}x^k (\partial_k \pi^{i j}) \partial_i \wedge \partial_j \\ & + 2 \pi^{i j} (\mathbf{d}\partial_i) \wedge \partial_j \\ & - (\partial_i \pi^{j k}) \partial_j \wedge \partial_k \wedge \mathbf{d}x^i \\ & + (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i) \\ & + (-)(-) 2\pi^{i j} \partial_i \wedge \mathbf{d}\partial_j \\ = & (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i) \end{aligned} \,.$$

On higher extensions of the super Poincare Lie algebra – supergravity

See D'Auria-Fre formulation of supergravity for the moment.

Related concepts

• ∞-Lie algebroid cocycle

• Chern-Simons element

• invariant polynomial

References

A classical reference on transgression of differential forms from the fiber to the base of a fiber bundle is section 9 of.

• Armand Borel, Topology of Lie groups and characteristic classes Bull. Amer. Math. Soc. Volume 61, Number 5 (1955), 397-432. (EUCLID)

The general definition of Chern-Simons element on $\infty$-Lie algebras and $\infty$-Lie algebroids is definition 21 in

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_\infty$-connections (web)

The examples of the BF-theory invariant polynomials and Chern-Simons elements are in prop. 18 and def. 26 and the BF-action functional itself is extracted below proposition 28.

Dedicated discussion of $\infty$-Chern-Simons theory is at

• Domenico Fiorenza, Chris Rogers Urs Schreiber, ∞-Chern-Simons functionals

A comprehensive account is in

• Urs Schreiber, differential cohomology in a cohesive topos .

A survey of higher Chern-Simons elements and their action functionals as applied to gravity and supergravity is in

• Jorge Zanelli, Lecture notes on Chern-Simons (super-)gravities arXiv:0502193

Symplectic Lie $n$-algebroids are discussed in

• Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis (arXiv)

  On the structure of graded symplectic supermanifolds and Courant algebroids (arXiv)

A talk about the historical origins of the standard Chern-Simons forms see

• Jim Simons, Origin of Chern-Simons talk at Simons Center for Geometry and Physics (2011) (video)