Schreiber
infinity-Chern-Simons theory -- examples

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

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Examples

1d Chern-Simons functionals

• 1-dimensional Chern-Simons theory

2d Chern-Simons functionals

Poisson $\sigma$-model

• Poisson sigma-model

3d Chern-Simons functionals

Ordinary Chern-Simons theory

Lagrangian

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

Notice that in terms of this the differential of the Chevalley-Eilenberg algebra is

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

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

and

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

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

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

$$C_{abc}:=P_{aa\prime }{C}^{a\prime }{}_{bc}$$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 $⟨-,-⟩=P(-,-)$ is

$$\begin{array}{rl}\mathrm{cs}& =P_{ab}{t}^a\wedge (d_{W(𝔤)}{t}^b)+\frac{1}{3}{P}_{aa\prime }{C}^{a\prime }{}_{bc}{t}^a\wedge t^b\wedge t^c\\ & =P_{ab}{t}^a\wedge r^b-\frac{1}{6}{P}_{aa\prime }{C}^{a\prime }{}_{bc}{t}^a\wedge t^b\wedge t^c\end{array}.$$\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 $⟨-,-⟩$ is in transgression with the degree 3-cocycle

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

Proof

We compute

$$\begin{array}{rl}{d}_{W(𝔤)}(P_{ab}{t}^a\wedge r^b+\frac{1}{2}{P}_{aa\prime }{C}^{a\prime }{}_{bc}{t}^a\wedge t^b\wedge t^c)& =P_{ab}{r}^a\wedge r^b\\ & -\frac{1}{2}{P}_{ab}{C}^a{}_{de}{t}^d\wedge t^e\wedge r^b\\ & +P_{ab}{C}^b{}_{de}{t}^a\wedge t^d\wedge r^e\\ & -\frac{3}{6}{P}_{aa\prime }{C}^{a\prime }{}_{bc}{t}^a\wedge t^b\wedge r^c\\ & =P_{ab}{r}^a\wedge r^b\\ & +\frac{1}{2}{C}_{abc}{t}^a\wedge t^b\wedge r^c\\ & -\frac{1}{2}{C}_{abc}{t}^a\wedge t^b\wedge r^c\\ & =P_{ab}{r}^a\wedge r^b\end{array}.$$\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

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

this Chern-Simons element is sent to

$$\mathrm{cs}(A)=P_{ab}{A}^a\wedge d{A}^b+\frac{1}{3}{C}_{abc}{A}^a\wedge A^b\wedge A^c.$$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 $𝔤$ is a matrix Lie algebra then the Killing form is the trace and this is equivalently

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

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

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

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

is the standard action functional of Chern-Simons theory.

Covariant phase space

Proposition

The covariant phase space of ordinary Chern-Simons theory is the space of those Lie algebra valued form $A$ whose curvature 2-form $F_A$ vanishes

$$P=\{A\in {\Omega }^1(\Sigma ,𝔤)\mid F_A=0\}.$$P = \{A \in \Omega^1(\Sigma, \mathfrak{g}) | F_A = 0\} \,.

The presymplectic structure on this space is

$$\omega :(\delta A_1,\delta A_2)\mapsto {\int }_{\partial \Sigma }⟨\delta A_1,\delta A_2⟩.$$\omega : (\delta A_1, \delta A_2) \mapsto \int_{\partial \Sigma} \langle \delta A_1, \delta A_2 \rangle \,.

Proof

This is a special case of prop. \ref{TheEquationsOfMotion}, prop. \ref{PresymplecticStructure} in view of corollary \ref{CovariantPhaseSpaceForBinaryNonDegenerateInvariantPolynomial}, using that the Killing form is a binary and non-degenerate invariant polynomial.

Obstruction theory

The circle n-bundle with connection given by ordinary Chern-Simons theory is known as the Chern-Simons circle 3-bundle . The non-triviality of its underlying class is the obstruction to lifting the $G$-principal bundle to a string structure. The non-triviality of its connection is the obstruction to having a differential string structure. In general it defines a twisted differential string structure.

Courant $\sigma$-model

• Courant sigma-model

4d Chern-Simons functionals

BF-theory and topological Yang-Mills theory

Let $𝔤=({𝔤}_2\stackrel{\partial }{\to }𝔤{)}_1$ be a strict Lie 2-algebra, an L-∞ algebra concentrated in the lowesr two degrees and with the only nontrivial bracket being the binary one.

Proposition

We have

1. every invariant polynomial $⟨-{⟩}_{{𝔤}_1}\in \mathrm{inv}({𝔤}_1)$ on ${𝔤}_1$ is a Chern-Simons element on $𝔤$, exhibiting a transgression to a trivial ∞-Lie algebra cocycle;

2. for ${𝔤}_1$ a semisimple Lie algebra and $⟨-{⟩}_{{𝔤}_1}$ the Killing form, the corresponding Chern-Simons action functional on L-∞ algebra valued forms

   $${\Omega }^{•}(X)\stackrel{(A,B)}{\leftarrow }W({𝔤}_2\to {𝔤}_1)\stackrel{(⟨-{⟩}_{{𝔤}_1},d_W⟨-{⟩}_{{𝔤}_1})}{\leftarrow }W(b^{n-1}ℝ)$$\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):

   $${\mathrm{CS}}_{⟨-{⟩}_{{𝔤}_1}}(A,B)=⟨F_A\wedge F_A{⟩}_{{𝔤}_1}-2⟨F_A\wedge \partial B{⟩}_{{𝔤}_1}+2⟨\partial B\wedge \partial B{⟩}_{{𝔤}_1},$$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 ${𝔤}_1$ and $\{b_i\}$ a basis of ${𝔤}_2$ we have

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

Therefore with $⟨-{⟩}_{{𝔤}_1}=P_{a_1\cdots a_n}d{r}^{a_1}\wedge \cdots d{t}^{a_n}$ we have

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

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

Now for $(A,B)\in {\Omega }^1(U\times {\Delta }^k,𝔤)$ an ∞-Lie algebra-valued form, we have that the 2-form curvature is

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

Therefore

$$\begin{array}{rl}{\mathrm{CS}}_{⟨-{⟩}_{{𝔤}_1}}(A,B)& =⟨F_{(A,B)}^1{⟩}_{{𝔤}_1}\\ & =⟨F_A\wedge F_A{⟩}_{{𝔤}_1}-2⟨F_A\wedge \partial B{⟩}_{{𝔤}_1}+2⟨\partial B\wedge \partial B{⟩}_{{𝔤}_1}\end{array}.$$\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} \,.

7d Chern-Simons functionals

7d $\mathrm{String}$-Chern-Simons theory

In heterotic string theory Chern-Simons circle 3-bundles appear as the ordinary differential cohomology-incarnation of magnetic charge of NS-fivebranes that twists the Kalb-Ramond field as described by the Green-Schwarz mechanism.

At least when this twist vanishes there is expected to be an electric-magnetic dual description with a Chern-Simons circle 7-bundle.

Lagrangian

Let $⟨-,-,-,-⟩$ be the canonical quaternaty invariant polynomial on the special orthogonal Lie algebra $\mathrm{𝔰𝔬}$. This lifts directly also to an invariant polynomial on the string Lie 2-algebra $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}:={\mathrm{𝔰𝔬}}_{\mu }$.

(…)

Obstruction theory

The circle n-bundle with connection given by 7-dimensional Chern-Simons theory is the Chern-Simons circle 7-bundle . The non-triviality of its underlying class is the obstruction to lifting the string 2-group-principal 2-bundle to a fivebrane structure. The non-triviality of its 7-connection is the obstruction to having a differential fivebrane structure. In general it defines a twisted differential fivebrane structure.

Higher dimensional abelian Chern-Simons theory

The line Lie n-algebra carries a canonical invariant polynomial. The $\mathrm{\infty }$-Chern-Simons theory associated with this data is in the literature known as abelian higher dimensional Chern-Simons theory.

Also every higher-degree invariant polynomial on an ordinary Lie algebra gives rise to a higher dimensional Chern-Simons action functional. An concrete example of this is considered (below).

$\mathrm{\infty }$-Dijkgraaf-Witten theory

We consider the case where the target space object $BG$ is a discrete ∞-groupoid.

$$BG:=\mathrm{Disc}BG$$\mathbf{B}G := Disc B G

with $BG$ the delooping of an ∞-group $G\in$ Grpd $\simeq$ Top.

As we discuss below, $\mathrm{\infty }$-Chern-Simons theory for this setup subsumes and generalizes Dijkgraaf-Witten theory (and the Yetter model in next higher dimension). Therefore we speak of $\mathrm{\infty }$-Dijkgraaf-Witten theory.

Remark

The background field for $\mathrm{\infty }$-Dijkgraaf-Witten theory is necessarily flat.

Proof

By the $(\Pi ⊣\mathrm{Disc}⊣\Gamma )$-adjoint triple of the ambient cohesive (∞,1)-topos and usinf that $\mathrm{Disc}$ is a full and faithful (∞,1)-functor we have

$$\begin{array}{rl}\Pi BG& \simeq \mathrm{Disc}\Pi \mathrm{Disc}BG\\ & \simeq \mathrm{Disc}BG\\ & \simeq BG\end{array}$$\begin{aligned} \mathbf{\Pi} \mathbf{B}G & \simeq Disc \Pi Disc B G \\ & \simeq Disc B G \\ & \simeq \mathbf{B}G \end{aligned}

and therefore, using the $(\Pi ⊣\mathrm{Disc})$-zig-zag identity, the constant path inclusion

$$BG\to \Pi BG$$\mathbf{B}G \to \mathbf{\Pi} \mathbf{B}G

is an equivalence. Therefore the intrinsic de Rham cohomology of $BG$ is trivial

$$\begin{array}{rl}{H}_{\mathrm{dR}}(BG,B^{n}U(1))& \simeq H(\Pi (BG),B^{n}U(1))\prod _{H(BG,B^{n}U(1))}*\\ & \simeq *\end{array}$$\begin{aligned} \mathbf{H}_{dR}(\mathbf{B}G, \mathbf{B}^n U(1)) & \simeq \mathbf{H}(\mathbf{\Pi}(\mathbf{B}G), \mathbf{B}^n U(1)) \prod_{\mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1))} * \\ & \simeq * \end{aligned}

and so the intrinsic universal curvature class

$$\mathrm{curv}:H(BG,B^{n}U(1))\to H_{\mathrm{dR}}(BG,B^{n}U(1))$$curv : \mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1)) \to \mathbf{H}_{dR}(\mathbf{B}G, \mathbf{B}^n U(1))

is trivial.

3d Dijkgraaf-Witten theory

Dijkgraaf-Witten theory is the analog of Chern-Simons theory for discrete structure groups. We show that this becomes a precise statement in $\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$: the Dijkgraaf-Witten action functional is that induced from applying the $\mathrm{\infty }$-Chern-Simons homomorphism to a characteristic class of the form $\mathrm{Disc}BG\to B^{3}U(1)$, for $\mathrm{Disc}:\mathrm{\infty }\mathrm{Grpd}\to \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ the canonical embedding of discrete ∞-groupoids into all smooth ∞-groupoids.

Lagrangian

Let $G\in \mathrm{Grp}\to \mathrm{\infty }\mathrm{Grpd}\stackrel{\mathrm{Disc}}{\to }\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ be a discrete group regarded as an ∞-group object in discrete ∞-groupoids and hence as a smooth ∞-groupoid with discrete smooth cohesion. Write $BG=K(G,1)\in \mathrm{\infty }\mathrm{Grpd}$ for its delooping in ∞Grpd and $BG=\mathrm{Disc}BG$ for its delooping in Smooth∞Grpd.

We also write $\Gamma B^{n}U(1)\simeq K(U(1),n)$. Notice that this is different from $B^{n}U(1)\simeq \Pi BU(1)$, reflecting the fact that $U(1)$ has non-discrete smooth structure.

Proposition

For $G$ a discrete group, morphisms $BG\to B^{n}U(1)$ correspond precisely to cocycles in the ordinary group cohomology of $G$ with coefficients in the discrete group underlying the circle group

$${\pi }_0\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(BG,B^{n}U(1))\simeq H_{\mathrm{Grp}}^n(G,U(1)).$$\pi_0 Smooth\infty Grpd(\mathbf{B}G, \mathbf{B}^n U(1)) \simeq H^n_{Grp}(G,U(1)) \,.

Proof

By the $(\mathrm{Disc}⊣\Gamma )$-adjunction we have

$$\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(BG,B^{n}U(1))\simeq \mathrm{\infty }\mathrm{Grpd}(BG,K(U(1),n)).$$Smooth\infty Grpd(\mathbf{B}G, \mathbf{B}^n U(1)) \simeq \infty Grpd(B G, K(U(1),n)) \,.

Proposition

For $G$ discrete

• the intrinsic de Rham cohomology of $BG$ is trivial

  $$\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(BG,{♭}_{\mathrm{dR}}{B}^{n}U((1))\simeq *;$$Smooth \infty Grpd(\mathbf{B}G, \mathbf{\flat}_{dR}\mathbf{B}^n U((1)) \simeq * ;

• all $G$-principal bundles have a unique flat connection

  $$\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(X,BG)\simeq \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(\Pi (X),BG).$$Smooth\infty Grpd(X, \mathbf{B}G) \simeq Smooth\infty Grpd(\Pi(X), \mathbf{B}G) \,.

Proof

By the $(\mathrm{Disc}⊣\Gamma )$-adjunction and using that $\Gamma \circ {♭}_{\mathrm{dR}}K\simeq *$ for all $K$.

It follows that for $G$ discrete

• any characteristic class $c:BG\to B^{n}U(1)$ is a group cocycle;

• the $\mathrm{\infty }$-Chern-Weil homomorphism coincides with postcomposition with this class

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

Proposition

For $G$ discrete and $c:BG\to B^{3}U(1)$ any group 3-cocycle, the $\mathrm{\infty }$-Chern-Simons theory action functional on a 3-dimensional manifold $\Sigma$

$$\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(\Pi (\Sigma ),BG)\to U(1)$$Smooth\infty Grpd(\mathbf{\Pi}(\Sigma), \mathbf{B}G) \to U(1)

is the action functional of Dijkgraaf-Witten theory.

Proof

By proposition \ref{IntrinsicIntegrationTheorem} the morphism is given by evaluation of the pullback of the cocycle $\alpha :BG\to B^{3}U(1)$ along a given $\nabla :\Pi (\Sigma )\to BG$, on the fundamental homology class of $\Sigma$. This is the definition of the Dijkgraaf-Witten action (for instance equation (1.2) in FreedQuinn).

Obstruction theory

The flat Dijkgraaf-Witten circle 3-bundle on $\Sigma$ is the obstruction to lifting the $G$-principal bundle to a $\hat{G}$-principal 2-bundle, where $\hat{G}$ is the discrete 2-group classified by the group 3-cocycle.

(…)

4d Yetter model

See

• Yetter model

Closed string field theory

For the moment see closed string field theory .

AKSZ theory

We consider symplectic Lie n-algebroids $𝔓$ equipped with a canonical invariant polynomial and show that the ∞-Chern-Simons action functional associated to this data is the action functionalof the AKSZ theory sigma model with target space $𝔓$.

This is taken from (FRS11). See there for more details.

Lagrangian

Definition

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

This means that

• on each coordinate chart $U\to X$ of the base manifold $X$ of $𝔓$, there is a basis $\{x^a\}$ for $\mathrm{CE}(𝔞{\mid }_U)$ such that

  $$\omega ={\omega }_{ab}d{x}^a\wedge d{x}^b$$\omega = \omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b

  with $\{{\omega }_{ab}\in ℝ\hookrightarrow C^{\mathrm{\infty }}(X)\}$ and $\mathrm{deg}(x^a)+\mathrm{deg}(x^b)=n$;

• the coefficient matrix $\{{\omega }_{ab}\}$ has an inverse;

• we have

  $$d_{\mathrm{W}(𝔓)}\omega =d_{\mathrm{CE}(𝔓)}\omega +d\omega =0.$$d_{\mathrm{W}(\mathfrak{P})} \omega = d_{\mathrm{CE}(\mathfrak{P})} \omega + \mathbf{d} \omega = 0 \,.

This $\mathrm{\infty }$-Lie theoretic structure is essentially what in the literature is mostly considered in terms of symplectic dg-geometry :

Observation

We may think of an L-infinity-algebroid $𝔞$ as a graded manifold $X$ whose global function ring is the graded algebra underlying the Chevalley-Eilenberg algebra

$$C^{\mathrm{\infty }}(X):=\mathrm{CE}(𝔞)$$C^\infty(X) := \mathrm{CE}(\mathfrak{a})

and which is equipped with a vector field $v_X$ of grade 1 whose graded Lie bracket with itself vanishes $[v_X,v_X]=0$, given, as a derivation, by the
differential on the Chevalley-Eilenberg algebra:

$$v_X:=d_{\mathrm{CE}(𝔞)}:\mathrm{CE}(𝔞)\to \mathrm{CE}(𝔞).$$v_X := d_{\mathrm{CE}(\mathfrak{a})} : \mathrm{CE}(\mathfrak{a}) \to \mathrm{CE}(\mathfrak{a}) \,.

The pair $(X,v)$ is a differential graded manifold . In this perspective the graded algebra underlying the Weil algebra of $𝔞$ is the de Rham complex of $X$

$${\Omega }^{•}(X):=\mathrm{W}(𝔞),$$\Omega^\bullet(X) := \mathrm{W}(\mathfrak{a}) \,,

but the de Rham differential is just $d$, not the full differential $d_{\mathrm{W}(𝔞)}=d+d_{\mathrm{CE}(𝔞)}$ on the Weil algebra. The latter is thus a twisted de Rham differential on $X$.

From this perspective all standard constructions of Cartan calculus usefully apply to $L_{\mathrm{\infty }}$-algebroids. Notably for $v$ any vector field on $X$ there is the contraction derivation

$${\iota }_v:{\Omega }^{•}(X)\to {\Omega }^{•-1}(X)$$\iota_v : \Omega^\bullet(X) \to \Omega^{\bullet -1}(X)

and hence the Lie derivative

$${ℒ}_v:=[d,{\iota }_v]:{\Omega }^{•}(X)\to {\Omega }^{•}(X).$$\mathcal{L}_v := [\mathbf{d}, \iota_{v}] : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.

So in the above notation we have in particular

$$d_{\mathrm{W}(𝔞)}=d+{ℒ}_{v_X}:\mathrm{W}(𝔞)\to \mathrm{W}(𝔞).$$d_{\mathrm{W}(\mathfrak{a})} = \mathbf{d} + \mathcal{L}_{v_X} : \mathrm{W}(\mathfrak{a}) \to \mathrm{W}(\mathfrak{a}) \,.

Definition

For $X$ a dg-manifold, let $ϵ\in \Gamma (TX)$ be the vector field which over any coordinate patch $U\to X$ is given by the formula

$$ϵ{\mid }_U=\sum _a\mathrm{deg}(x^a)x^a\frac{\partial }{\partial x^a},$$\epsilon|_U = \sum_a \mathrm{deg}(x^a) x^a \frac{\partial}{\partial x^a} \,,

where $\{x^a\}$ is a basis of generators and $\mathrm{deg}(x^a)$ the degree of a generator.

We write

$$N:=[d,{\iota }_{ϵ}]$$N := [\mathbf{d}, \iota_\epsilon]

for the Lie derivative of this vector field. The grade of a homogeneous element $\alpha$ in ${\Omega }^{•}(X)$ is the unique natural number $n\in \mathbb{N}$ with

$${ℒ}_{ϵ}\alpha =N\alpha =n\alpha .$$\mathcal{L}_\epsilon \alpha = N \alpha = n \alpha \,.

Remark.

• This implies that for $x^i$ an element of grade $n$ on $U$, the 1-form $d{x}^i$ is also of grade $n$. This is why we speak of grade (as in “graded manifold”) instead of degree here.

• The above is indeed well-defined: on overlaps of patches the $\{x^a\}$ of positive degree/grade transform by a degreewise linear transformation, which manifestly preserves ${\sum }_a\mathrm{deg}(x^a)x^a\frac{\partial }{\partial x^a}$. Notice that the ordinary (degree-0 coordinates) do not appear in this formula. And indeed the vector field locally defined by ${\sum }_a{x}^a\frac{\partial }{\partial x^a}$ (thus including the coordinates of grade 0) does not in general exist globally.

The existence of $ϵ$ implies the following useful statement, which is a trivial variant of what in grade 0 would be the standard Poincare lemma.

Observation

On a graded manifold every closed differential form $\omega$ of positive grade $n$ is exact: the form

$$\lambda :=\frac{1}{n}{\iota }_{ϵ}\omega$$\lambda := \frac{1}{n} \iota_\epsilon \omega

satisfies

$$d\lambda =\omega .$$\mathbf{d}\lambda = \omega \,.

Using this differential geometric language we can now capture something very close to def. 1 in more traditional symplectic geometry terms.

Definition

A symplectic dg-manifold of grade $n\in \mathbb{N}$ is a dg-manifold $(X,v)$ equipped with 2-form $\omega \in {\Omega }^2(X)$ which is

• \item non-degenerate;
• closed;

as usual for symplectic forms, and in addition

• of grade $n$;
• $v$-invariant: ${ℒ}_v\omega =0$.

Example. It follows that a symplectic dg-manifold of grade 0 is the same as an ordinary symplectic manifold. In the following we are mostly interested in the case of positive grade.

Observation

The function algebra of a symplectic dg-manifold $(X,\omega )$ of grade $n$ is naturally equipped with a Poisson bracket

$$\{-,-\}:C^{\mathrm{\infty }}(X)\otimes C^{\mathrm{\infty }}(X)\to C^{\mathrm{\infty }}(X)$$\{-,-\} : C^\infty(X)\otimes C^\infty(X) \to C^\infty(X)

which decreases grade by $n$. On a local coordinate patch this is given by

$$\{f,g\}=\frac{\partial f}{\partial x^a}{\omega }^{ab}\frac{\partial g}{\partial x^b},$$\{f,g\} = \frac{\partial f}{\partial x^a} \omega^{a b} \frac{\partial g}{\partial x^b} \,,

where $\{{\omega }^{ab}\}$ is the inverse matrix to $\{{\omega }_{ab}\}$.

Observation

For $f\in C^{\mathrm{\infty }}(X)$ and $v\in \Gamma (TX)$ we say that f$\mathrm{is}a\mathrm{Hamiltonian}\mathrm{for}$v_ or equivalently that _v$\mathrm{is}\mathrm{the}\mathrm{chunk}86766920\mathrm{wikichunklinkchunk}\mathrm{of}$f$ if

$$df={\iota }_v\omega .$$\mathbf{d}f = \iota_v \omega \,.

Proposition

There is a full and faithful embedding of symplectic dg-manifolds of grade $n$ into symplectic Lie $n$-algebroids.

Proof

The dg-manifold itself is identified with an $L_{\mathrm{\infty }}$-algebroid as in observation 6. For $\omega \in {\Omega }^2(X)$ a symplectic form, the conditions $d\omega =0$ and ${ℒ}_v\omega =0$ imply $(d+ℒv)\omega =0$ and hence that under the identification ${\Omega }^{•}(X)\simeq \mathrm{W}(𝔞)$ this is an invariant polynomial on $𝔞$.

It remains to observe that the $L_{\mathrm{\infty }}$-algebroid $𝔞$ is in fact a Lie $n$-algebroid. This is implied by the fact that $\omega$ is of grade $n$ and non-degenerate: the former condition implies that it has no components in elements of grade $\mathrm{gt}n$ and the latter then implies that all such elements vanish.

Proposition

Let $(𝔓,\omega )$ be a symplectic Lie $n$-algebroid for positive $n$ in the image of the embedding of prop. 10. Then it carries the canonical $L_{\mathrm{\infty }}$-algebroid cocycle

$$\pi :=\frac{1}{n+1}{\iota }_{ϵ}{\iota }_v\omega \in \mathrm{CE}(𝔓)$$\pi := \frac{1}{n+1} \iota_\epsilon \iota_v \omega \in \mathrm{CE}(\mathfrak{P})

which moreover is the Hamiltonian, according to def. 9, of $d_{\mathrm{CE}(𝔓)}$.

Proof

The required condition $d\pi ={\iota }_v\omega$ from def. 9 holds by observation 7.

Our central observation now is the following.

Proposition

The cocycle $\pi$ from prop. 11 is in transgression with the invariant polynomial $n\omega$. A Chern-Simons element witnessing the transgression according to def. \ref{TransgressionAndCSElements} is

$$\mathrm{cs}={\iota }_{ϵ}\omega +\pi .$$\mathrm{cs} = \iota_\epsilon \omega + \pi \,.

Proof

It is clear that $i^{*}\mathrm{cs}=\pi$. So it remains to check that $d_{\mathrm{W}(𝔓)}\mathrm{cs}=n\omega$. Notice that

$$[d_{\mathrm{CE}(𝔓)},{\iota }_{ϵ}]=[{ℒ}_v,{\iota }_{ϵ}]={\iota }_{[v,ϵ]}=-{\iota }_v$$[d_{\mathrm{CE}(\mathfrak{P})}, \iota_\epsilon] = [\mathcal{L}_v, \iota_\epsilon] = \iota_{[v,\epsilon]} = - \iota_{v}

by Cartan calculus. Using this we compute the first summand in $d_{\mathrm{W}(𝔓)}({\iota }_{ϵ}\omega +\pi )$:

$$\begin{array}{rl}{d}_{\mathrm{W}(𝔓)}{\iota }_{ϵ}\omega & =(d+d_{\mathrm{CE}(𝔓)}){\iota }_{ϵ}\omega \\ & =n\omega +[d_{\mathrm{CE}(𝔓)},{\iota }_{ϵ}]\omega \\ & =n\omega -{\iota }_v\omega \\ & =n\omega -d\pi \end{array}.$$\begin{aligned} d_{\mathrm{W}(\mathfrak{P})} \iota_{\epsilon} \omega & = ( \mathbf{d} + d_{\mathrm{CE}(\mathfrak{P})} ) \iota_\epsilon \omega \\ &= n \omega + [d_{\mathrm{CE}(\mathfrak{P})}, \iota_\epsilon] \omega \\ &= n\omega - \iota_v \omega \\ & = n \omega - \mathbf{d}\pi \end{aligned} \,.

The second summand is simply

$$d_{\mathrm{W}(𝔓)}\pi =d\pi$$d_{\mathrm{W}(\mathfrak{P})} \pi = \mathbf{d}\pi

since $\pi$ is a cocycle.

Proposition

For $(𝔓.\omega )$ a symplectic Lie $n$-algebroid coming from a symplectic dg-manifold by prop. 10, the higher Chern-Simons action functional associated with its canonical Chern-Simons element $\mathrm{cs}$ from prop. 12 is the AKSZ Lagrangean:

$$L_{\mathrm{AKSZ}}=\mathrm{cs}.$$L_{\mathrm{AKSZ}} = \mathrm{cs} \,.

Proof

We work in local coordinates $\{x^a\}$ where

$$\omega =\frac{1}{2}{\omega }_{ab}d{x}^a\wedge d{x}^b$$\omega = \frac{1}{2}\omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b

and the Chern-Simons element is

$$\mathrm{cs}=\sum _a{\omega }_{ab}\mathrm{deg}(x^a)x^a\wedge d{x}^b+\pi .$$\mathrm{cs} = \sum_a \omega_{a b} \mathrm{deg}(x^a) x^a \wedge \mathbf{d}x^b + \pi \,.

We want to substitute here $d=d_{\mathrm{W}}-d_{\mathrm{CE}}$. Notice that in coordinates the equation

$$d\pi ={\iota }_v\omega$$\mathbf{d}\pi = \iota_v \omega

becomes

$$\begin{array}{rl}d{x}^a\frac{\partial \pi }{\partial x^a}& ={\omega }_{ab}{v}^a\wedge d{x}^b\\ & ={\omega }_{ab}d{x}^a\wedge v^b\end{array}.$$\begin{aligned} \mathbf{d}x^a \frac{\partial \pi}{\partial x^a} & = \omega_{a b} v^a \wedge \mathbf{d} x^b \\ & = \omega_{a b} \mathbf{d}x^a \wedge v^b \end{aligned} \,.

Therefore

$$\begin{array}{rl}\sum _a{\omega }_{ab}\mathrm{deg}(x^a)x^a\wedge d_{\mathrm{CE}}{x}^b& =\sum _a{\omega }_{ab}\mathrm{deg}(x^a)x^a\wedge v^b\\ & =\sum _a\mathrm{deg}(x^a)x^a\frac{\partial \pi }{\partial x^a}\\ & =(n+1)\pi \end{array}.$$\begin{aligned} \sum_a \omega_{a b} \mathrm{deg}(x^a) x^a \wedge d_{\mathrm{CE}} x^b & = \sum_a \omega_{a b} \mathrm{deg}(x^a) x^a \wedge v^b \\ & = \sum_a \mathrm{deg}(x^a)x^a \frac{\partial \pi}{\partial x^a} \\ &= (n+1) \pi \end{aligned} \,.

Hence

$$\mathrm{cs}=\sum _{ab}\mathrm{deg}(x^a){\omega }_{ab}{x}^a\wedge d{x}^b-n\pi .$$\mathrm{cs} = \sum_{a b} \mathrm{deg}(x^a) \,\omega_{a b} x^a \wedge \mathbf{d}x^b - n \pi \,.

This means that for $\Sigma$ an $(n+1)$-dimensional manifold and

$${\Omega }^{•}(\Sigma )\leftarrow \mathrm{W}(𝔓):X$$\Omega^\bullet(\Sigma) \leftarrow \mathrm{W}(\mathfrak{P}) : X

a $𝔓$-valued differential form on $\Sigma$ we have

$$\begin{array}{rl}\mathrm{cs}(X)& =\sum _{a,b}\mathrm{deg}(x^a){\omega }_{ab}{X}^a\wedge d_{\mathrm{dR}}{X}^b-n\Pi (X)\end{array}.$$\begin{aligned} \mathrm{cs}(X) &= \sum_{a,b} \mathrm{deg}(x^a)\,\omega_{a b} X^a \wedge d_{\mathrm{dR}} X^b - n \Pi(X) \end{aligned} \,.

This is indeed $L_{\mathrm{AKSZ}}(X)$.

Remark The AKSZ $\sigma$-model action functional interpretation of $\mathrm{\infty }$-Chern-Weil functionals for binary invariant polynomials on $L_{\mathrm{\infty }}$-algebroids from prop. 13 gives rise to the following dictionary of concepts\

$$\begin{array}{ccc}\mathrm{Chern}-\mathrm{Weil}\mathrm{theory}& & \mathrm{quantum}\mathrm{field}\mathrm{theory}\\ \\ \mathrm{cocycle}& \pi & \mathrm{Hamiltonian}\\ \\ \mathrm{transgression}\mathrm{element}& \mathrm{cs}& \mathrm{Lagrangean}\\ \\ \mathrm{curvature}\mathrm{characteristic}& \omega & \mathrm{symplectic}\mathrm{structure}\end{array}.$$\array{ Chern-Weil theory && quantum field theory \\ \\ cocycle & \pi & Hamiltonian \\ \\ transgression element & cs & Lagrangean \\ \\ curvature characteristic & \omega & symplectic structure } \,.

Covariant phase space

Proposition

The covariant phase space of AKSZ theory with target $(𝔓,\omega )$ is the space of those ∞-Lie algebroid-valued forms $A$ whose curvature $(n+1)$-form $F_A$ vanishes

$$P=\{A\in {\Omega }^1(\Sigma ,𝔤)\mid F_A=0\}.$$P = \{A \in \Omega^1(\Sigma, \mathfrak{g}) | F_A = 0\} \,.

The presymplectic structure on this space is

$$\omega :(\delta A_1,\delta A_2)\mapsto {\int }_{\partial \Sigma }\omega (\delta A_1,\delta A_2).$$\omega : (\delta A_1, \delta A_2) \mapsto \int_{\partial \Sigma} \omega(\delta A_1, \delta A_2 ) \,.

Proof

This is a special case of prop. \ref{TheEquationsOfMotion}, prop. \ref{PresymplecticStructure} in view of corollary \ref{CovariantPhaseSpaceForBinaryNonDegenerateInvariantPolynomial}, using that, by definition of symplectic Lie n-algebroid, $\omega$ is a binary and non-degenerate invariant polynomial.

$n=0$ – The topological particle

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

$${\int }_{ℝ}{\gamma }^{*}(p_{i}d{q}^i)$$\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”).

$n=1$ – The Poisson $\sigma$-model

Let $(X,\{-,-\})$ be a Poisson manifold. Over a Darboux chart the corresponding Poisson Lie algebroid has coordinates $\{x^i\}$ of degree 0 and ${\partial }_i$ of degree 1. We have

$$d_{\mathrm{W}}{x}^i=-{\pi }^{ij}{\partial }_j+d{x}^i$$d_{\mathrm{W}} x^i = -\pi^{i j}\mathbf{\partial}_j + \mathbf{d}x^i

where ${\pi }^{ij}:=\{x^i,x^j\}$ and

$$\omega =d{x}^i\wedge d{\partial }_i.$$\omega = \mathbf{d}x^i \wedge \mathbf{d}\partial_i \,.

The Hamiltonian cocycle from prop. 11 is

$$\begin{array}{rl}\pi & ={\iota }_v{\iota }_{ϵ}\omega \\ & ={\iota }_v{\partial }_i\wedge d{x}^i\\ & ={\partial }_i\wedge [{\iota }_v,d]x^i\\ & =-{\partial }_i\wedge [d,{\iota }_v]x^i\\ & =+{\partial }_i{\pi }^{\mathrm{ij}}{\partial }_j\end{array}$$\begin{aligned} \pi &= \iota_v \iota_\epsilon \omega \\ &= \iota_v \partial_i \wedge \mathbf{d}x^i \\ & = \partial_i \wedge [\iota_v,\mathbf{d}]x^i \\ &= -\partial_i \wedge [\mathbf{d},\iota_v]x^i \\ &= + \partial_i \pi^{ij}\partial_j \end{aligned}

and the Chern-Simons element from prop. 12 is

$$\begin{array}{rl}\mathrm{cs}& ={\iota }_{ϵ}\omega +\pi \\ & ={\partial }_i\wedge d{x}^i+{\pi }^{\mathrm{ij}}{\partial }_i{\partial }_j\end{array}.$$\begin{aligned} \mathrm{cs} &= \iota_\epsilon \omega + \pi \\ &= \partial_i \wedge \mathbf{d}x^i + \pi^{ij}\partial_i \partial_j \end{aligned} \,.

In terms of $d_{\mathrm{W}}$ instead of $d$ this is

$$\begin{array}{rl}\cdots & ={\partial }_i\wedge (d_{\mathrm{W}}-d_{\mathrm{CE}})x^i+{\pi }^{\mathrm{ij}}{\partial }_i{\partial }_j\\ & ={\partial }_i\wedge d{x}^i+2{\pi }^{\mathrm{ij}}{\partial }_i{\partial }_j\end{array}$$\begin{aligned} \cdots & = \partial_i \wedge (d_{\mathrm{W}} - d_{\mathrm{CE}}) x^i + \pi^{ij}\partial_i \partial_j \\ &= \partial_i \wedge \mathbf{d}x^i + 2 \pi^{ij}\partial_i \partial_j \end{aligned}

So for

$${\Omega }^{•}(\Sigma )\leftarrow \mathrm{W}(𝔓):(X,\eta )$$\Omega^\bullet(\Sigma) \leftarrow \mathrm{W}(\mathfrak{P}) : (X,\eta)

a Poisson-Lie algebroid valued differential form – which in components is a function $\varphi :\Sigma \to X$ and a 1-form $\eta \in {\Omega }^1(\Sigma ,{\varphi }^{*}{T}^{*}X)$ – the corresponding Chern-Simons form is

$$\mathrm{cs}(X,\eta )=⟨d_{\mathrm{dR}}X\wedge \eta ⟩+2\pi (\eta \wedge \eta ).$$\mathrm{cs}(X,\eta) = \langle d_{\mathrm{dR}}X \wedge \eta \rangle + 2 \pi(\eta \wedge \eta) \,.

This is the Lagrangean of the Poisson $\sigma$-model \cite{CattaneoFelder}.

$n=2$ – Ordinary Chern-Simons theory

We show how the ordinary Chern-Simons form arises from this perspective. So let $𝔞=𝔤$ be a semisimple Lie algebra and $\omega :=⟨-,-⟩\in \mathrm{W}(𝔤)$ its Killing form invariant polynomial. For $\{t^a\}$ a dual basis for $𝔤$ we have

$$d_{\mathrm{W}}{t}^a=-\frac{1}{2}{C}^a{}_{bc}{t}^a\wedge t^b+d{t}^a$$d_{\mathrm{W}} t^a = - \frac{1}{2}C^a{}_{b c} t^a \wedge t^b + \mathbf{d}t^a

where $C^a{}_{bc}:=t^a([t_b,t_c])$ and

$$\omega =\frac{1}{2}{P}_{ab}d{t}^a\wedge d{t}^b,$$\omega = \frac{1}{2} P_{a b} \mathbf{d}t^a \wedge \mathbf{d}t^b \,,

where $P_{\mathrm{ab}}:=⟨t_a,t_b⟩$. The Hamiltonian cocycle $\pi$ from prop. 11 is

$$\begin{array}{rl}\pi & =\frac{1}{2+1}{\iota }_{ϵ}{\iota }_v\omega \\ & =\frac{1}{3}{\iota }_v{\iota }_{ϵ}\omega \\ & =\frac{1}{3}{\iota }_v{P}_{ab}{t}^a\wedge d{t}^b\\ & =\frac{1}{3}{P}_{ab}{t}^a\wedge [{\iota }_v,d]t^b\\ & =-\frac{1}{3}{P}_{ab}{t}^a\wedge [d,{\iota }_v]t^b\\ & =-\frac{1}{3}{P}_{ab}{t}^a\wedge (-\frac{1}{2})C^b{}_{de}{t}^d\wedge t^e\\ & =+\frac{1}{6}{C}_{\mathrm{abc}}{t}^a\wedge t^b\wedge t^c\end{array}.$$\begin{aligned} \pi & = \frac{1}{2+1}\iota_\epsilon \iota_v \omega \\ & = \frac{1}{3} \iota_v \iota_\epsilon \omega \\ & = \frac{1}{3}\iota_v P_{a b} t^a \wedge \mathbf{d}t^b \\ & = \frac{1}{3} P_{a b} t^a \wedge [\iota_v,\mathbf{d}]t^b \\ & = -\frac{1}{3} P_{a b} t^a \wedge [\mathbf{d}, \iota_v]t^b \\ &= -\frac{1}{3} P_{a b} t^a \wedge (-\frac{1}{2})C^b{}_{d e} t^d \wedge t^e \\ & = +\frac{1}{6} C_{abc}t^a \wedge t^b \wedge t^c \end{aligned} \,.

Therefore in this case the Chern-Simons element from def. 12 becomes

$$\begin{array}{rl}\mathrm{cs}& ={\iota }_{ϵ}\omega +\pi \\ & =P_{ab}{t}^a\wedge d{t}^b+\frac{1}{6}{C}_{\mathrm{abc}}{t}^a\wedge t^b\wedge t^c\end{array}.$$\begin{aligned} \mathrm{cs} & = \iota_\epsilon \omega + \pi \\ & = P_{a b} t^a \wedge \mathbf{d}t^b + \frac{1}{6} C_{abc} t^a \wedge t^b \wedge t^c \end{aligned} \,.

This is indeed the familiar standard choice of Chern-Simons element on a Lie algebra. Notice that evaluated on a $𝔤$-valued form

$${\Omega }^{•}(\Sigma )\leftarrow \mathrm{W}(𝔤):A$$\Omega^\bullet(\Sigma) \leftarrow \mathrm{W}(\mathfrak{g}) : A

this is

$$\mathrm{cs}(A)=⟨A\wedge F_A⟩+\frac{1}{6}⟨A\wedge [A\wedge A]⟩.$$\mathrm{cs}(A) = \langle A \wedge F_A\rangle + \frac{1}{6}\langle A \wedge [A \wedge A]\rangle \,.

If $𝔤$ is a matrix Lie algebra then the Killing form is proportional to the trace of the matrix product: $⟨t_a,t_b⟩=\mathrm{tr}(t_a{t}_b)$. In this case we have

$$\begin{array}{rl}⟨A\wedge [A\wedge A]⟩& =A^a\wedge A^b\wedge A^c\mathrm{tr}(t_a(t_b{t}_c-t_c{t}_b))\\ & =2A^a\wedge A^b\wedge A^c\mathrm{tr}(t_a{t}_b{t}_c)\\ & =2\mathrm{tr}(A\wedge A\wedge A)\end{array}$$\begin{aligned} \langle A \wedge [A \wedge A]\rangle &= A^a \wedge A^b \wedge A^c \,\mathrm{tr}(t_a (t_b t_c - t_c t_b)) \\ &= 2 A^a \wedge A^b \wedge A^c \,\mathrm{tr}(t_a t_b t_c ) \\ &= 2 \,\mathrm{tr}(A \wedge A \wedge A) \end{aligned}

and hence

$$\mathrm{cs}(A)=\mathrm{tr}(A\wedge F_A)+\frac{1}{3}\mathrm{tr}(A\wedge A\wedge A).$$\mathrm{cs}(A) = \mathrm{tr}(A \wedge F_A) + \frac{1}{3}\,\mathrm{tr}(A \wedge A \wedge A) \,.

Often this is written in terms of the de Rham differential 2-form $d_{\mathrm{dR}}A$ instead of the curvature 2-form $F_A:=d_{\mathrm{dR}}A+\frac{1}{2}[A\wedge A]$. Since the former is the image under $A$ of $d_{\mathrm{W}(𝔤)}$ we can alternatively write

$$\begin{array}{rl}\mathrm{cs}& =P_{ab}{t}^a\wedge (d_{\mathrm{W}}-d_{\mathrm{CE}})t^b+\frac{1}{6}{C}_{\mathrm{abc}}{t}^a\wedge t^b\wedge t^c\\ & =P_{ab}{t}^a\wedge d_{\mathrm{W}}{t}^b-P_{\mathrm{ab}}{t}^a\wedge (-\frac{1}{2})C^b{}_{\mathrm{cd}}{t}^c\wedge t^d+\frac{1}{6}{C}_{\mathrm{abc}}{t}^a\wedge t^b\wedge t^c\\ & =P_{\mathrm{ab}}{t}^a\wedge d_{\mathrm{W}}{t}^b+(\frac{1}{2}+\frac{1}{6})C_{\mathrm{abc}}{t}^a\wedge t^b\wedge t^c\\ & =P_{\mathrm{ab}}{t}^a\wedge d_{\mathrm{W}}{t}^b+\frac{2}{3}{C}_{\mathrm{abc}}{t}^a\wedge t^b\wedge t^c\end{array}.$$\begin{aligned} \mathrm{cs} &= P_{a b} t^a \wedge (d_{\mathrm{W}} - d_{\mathrm{CE}})t^b + \frac{1}{6} C_{abc} t^a \wedge t^b \wedge t^c \\ &= P_{a b} t^a \wedge d_{\mathrm{W}} t^b - P_{ab} t^a \wedge (-\frac{1}{2})C^b{}_{cd}t^c \wedge t^d + \frac{1}{6} C_{abc} t^a \wedge t^b \wedge t^c \\ & = P_{ab} t^a \wedge d_{\mathrm{W}}t^b + (\frac{1}{2}+\frac{1}{6}) C_{abc} t^a \wedge t^b \wedge t^c \\ & = P_{ab} t^a \wedge d_{\mathrm{W}}t^b + \frac{2}{3}C_{abc} t^a \wedge t^b \wedge t^c \end{aligned} \,.

Hence

$$\mathrm{cs}(A)=⟨A\wedge d_{\mathrm{dR}}A⟩+\frac{2}{3}⟨A\wedge [A\wedge A]⟩.$$\mathrm{cs}(A) = \langle A \wedge d_{\mathrm{dR}} A\rangle + \frac{2}{3} \langle A \wedge [A \wedge A]\rangle \,.

$n=2$ – The Courant $\sigma$-model

(…)

Supergravity

• D'Auria-Fre formulation of supergravity

• Chern-Simons gravity

Lagrangians

Remark

By this discussion, what is called the cosmo-cocycle condition in the D'Auria-Fre formulation of supergravity is the condition that while the Lagrangian L_ is not quite a Chern-Simons form, we have nevertheless that its Weil algebra differential

$$d_{W(𝔤)}{L}_{\mathrm{sugra}}\in W(𝔤)\otimes {\lambda }^2({𝔤}^{*}[1])$$d_{W(\mathfrak{g})} L_{sugra} \in W(\mathfrak{g})\otimes \lambda^2 (\mathfrak{g}^*[1])

at least biliear in the curvatures (the shifted components).

While for this case the argument of prop. \ref{TheEquationsOfMotion} does not give a closed formula for the full equations of motions, but it still implies that field configurations $F$ with vanishing curvature do solve the equations of motion. Hence that

$$F_A=0$$F_A = 0

is a sufficient condition for $A$ to be a point in the covariant phase space.

This is supposedly the reason for the terms “cosmo cocycle” for this condiiton: it ensures in supergravity that that flat Minkowski space with all fields vanishing is a (“cosmological”) solution.

References

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

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

• Urs Schreiber, differential cohomology in a cohesive topos

The case of the AKSZ sigma-model is discussed in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, AKSZ Sigma-Models in Higher Chern-Weil Theory

Discussion of symplectic Lie n-algebroids is 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)