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

Notice that in terms of this the differential of the Chevalley-Eilenberg 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 traditional incarnation 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.

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

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

The presymplectic structure on this space is

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

Proof

This is a special case of prop. , prop. in view of corollary , 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 $\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g})_1$ 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 $\langle -\rangle_{\mathfrak{g}_1} \in inv(\mathfrak{g}_1)$ on $\mathfrak{g}_1$ is a Chern-Simons element on $\mathfrak{g}$, exhibiting a transgression to a trivial ∞-Lie algebra cocycle;

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

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

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

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

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

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

Therefore

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

7d Chern-Simons functionals

7d $String$-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 $\langle -,-,-,-\rangle$ be the canonical quaternaty invariant polynomial on the special orthogonal Lie algebra $\mathfrak{so}$. This lifts directly also to an invariant polynomial on the string Lie 2-algebra $\mathfrak{string} := \mathfrak{so}_\mu$.

(…)

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 $\infty$-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).

$\infty$-Dijkgraaf-Witten theory

We consider the case where the target space object $\mathbf{B}G$ is a discrete ∞-groupoid.

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

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

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

Remark

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

Proof

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

$$\begin{aligned} \mathbf{\Pi} \mathbf{B}G & \simeq Disc \Pi Disc B G \\ & \simeq Disc B G \\ & \simeq \mathbf{B}G \end{aligned}$$

and therefore, using the $(\Pi \dashv Disc)$-zig-zag identity, the constant path inclusion

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

is an equivalence. Therefore the intrinsic de Rham cohomology of $\mathbf{B}G$ is trivial

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

and so the intrinsic universal curvature class

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

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 $Smooth \infty Grpd$: the Dijkgraaf-Witten action functional is that induced from applying the $\infty$-Chern-Simons homomorphism to a characteristic class of the form $Disc B G \to \mathbf{B}^3 U(1)$, for $Disc : \infty Grpd \to Smooth \infty Grpd$ the canonical embedding of discrete ∞-groupoids into all smooth ∞-groupoids.

Lagrangian

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

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

Proposition

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

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

Proof

By the $(Disc \dashv \Gamma)$-adjunction we have

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

Proposition

For $G$ discrete

• the intrinsic de Rham cohomology of $\mathbf{B}G$ is trivial

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

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

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

Proof

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

It follows that for $G$ discrete

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

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

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

Proposition

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

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

is the action functional of Dijkgraaf-Witten theory.

Proof

By proposition the morphism is given by evaluation of the pullback of the cocycle $\alpha : B G \to B^3 U(1)$ along a given $\nabla : \Pi(\Sigma) \to B G$, 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 $\mathfrak{P}$ 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 $\mathfrak{P}$.

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

Lagrangian

Definition

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

This means that

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

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

  with $\{\omega_{a b} \in \mathbb{R} \hookrightarrow C^\infty(X)\}$ and $deg(x^a) + deg(x^b) = n$;

• the coefficient matrix $\{\omega_{a b}\}$ has an inverse;

• we have

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

This $\infty$-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 $\mathfrak{a}$ as a graded manifold $X$ whose global function ring is the graded algebra underlying the Chevalley-Eilenberg algebra

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

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:

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

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

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

but the de Rham differential is just $\mathbf{d}$, not the full differential $d_{\mathrm{W}(\mathfrak{a})} = \mathbf{d} + d_{\mathrm{CE}(\mathfrak{a})}$ 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 $L_\infty$-algebroids. Notably for $v$ any vector field on $X$ there is the contraction derivation

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

and hence the Lie derivative

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

So in the above notation we have in particular

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

Definition

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

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

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

We write

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

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

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

Remark.

• This implies that for $x^i$ an element of grade $n$ on $U$, the 1-form $\mathbf{d}x^i$ 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 $\epsilon$ 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

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

satisfies

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

Using this differential geometric language we can now capture something very close to def. 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: $\mathcal{L}_v \omega = 0$.

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

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

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

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

where $\{\omega^{a b}\}$ is the inverse matrix to $\{\omega_{a b}\}$.

Observation

For $f \in C^\infty(X)$ and $v \in \Gamma(T X)$ we say that f$is a Hamiltonian for$v_ or equivalently that _v$is the [[nLab:Hamiltonian vector field]] of$f$ if

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

Proposition

There is a of symplectic dg-manifolds of grade $n$ into symplectic Lie $n$-algebroids.

Proof

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

It remains to observe that the $L_\infty$-algebroid $\mathfrak{a}$ 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 $gt n$ and the latter then implies that all such elements vanish.

Proposition

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

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

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

Proof

The required condition $\mathbf{d}\pi = \iota_v \omega$ from def. holds by observation .

Our central observation now is the following.

Proposition

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

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

Proof

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

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

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

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

The second summand is simply

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

since $\pi$ is a cocycle.

Proposition

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

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

Proof

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

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

and the Chern-Simons element is

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

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

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

becomes

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

Therefore

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

Hence

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

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

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

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

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

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

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

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

Covariant phase space

Proposition

The of with target $(\mathfrak{P}, \omega)$ is the space of those $A$ whose $(n+1)$-form $F_A$ vanishes

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

The on this space is

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

Proof

This is a special case of prop. , prop. in view of corollary , using that, by definition of , $\omega$ is a binary and non-degenerate .

$n=0$ – The topological particle

For $X$ a 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”).

$n=1$ – The Poisson $\sigma$-model

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

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

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

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

The Hamiltonian cocycle from prop. is

$$\begin{aligned} \pi &= \iota_v \iota_\epsilon \omega \\ &= \iota_v \partial_i \wedge \mathbf{d}x^i \\ & = \partial_i \wedge [\iota_v,\mathbf{d}]x^i \\ &= -\partial_i \wedge [\mathbf{d},\iota_v]x^i \\ &= + \partial_i \pi^{ij}\partial_j \end{aligned}$$

and the Chern-Simons element from prop. is

$$\begin{aligned} \mathrm{cs} &= \iota_\epsilon \omega + \pi \\ &= \partial_i \wedge \mathbf{d}x^i + \pi^{ij}\partial_i \partial_j \end{aligned} \,.$$

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

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

So for

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

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

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

This is the Lagrangean of the Poisson $\sigma$-model [CattaneoFelder].

$n=2$ – Ordinary Chern-Simons theory

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

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

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

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

where $P_{ab} := \langle t_a, t_b \rangle$. The Hamiltonian cocycle $\pi$ from prop. is

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

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

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

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

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

this is

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

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

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

and hence

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

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 $d_{\mathrm{W}(\mathfrak{g})}$ we can alternatively write

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

Hence

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

$n=2$ – The Courant $\sigma$-model

(…)

Supergravity

Lagrangians

Remark

By this discussion, what is called the cosmo-cocycle condition in the is the condition that while the L_ is not quite a , we have nevertheless that its

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

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

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

$$F_A = 0$$

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 with all fields vanishing is a (“cosmological”) solution.

References

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

• , , , $L_\infty$-connections (web)

The induced construction of the with special attention to the and the is in

• , , ,

In the general context of $\infty$-Chern-Simons theory is discussed in section 4.3 of

• ,

The case of the is discussed in

• , , ,

Discussion of s is in

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

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