This is a sub-entry of infinity-Chern-Simons theory. See there for context.
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
and that of the Weil algebra
and
Let $P_{a b} r^a \wedge r^b \in W(\mathfrak{g})$ be the Killing form invariant polynomial. This being invariant
is equivalent to the fact that the coefficients
are skew-symmetric in $a$ and $b$, and therefore skew in all three indices.
A Chern-Simons element for the Killing form invariant polynomial $\langle -, - \rangle = P(-,-)$ is
In particular the Killing form $\langle -,-\rangle$ is in transgression with the degree 3-cocycle
We compute
Under a Lie algebra-valued form
this Chern-Simons element is sent to
If $\mathfrak{g}$ is a matrix Lie algebra then the Killing form is the trace and this is equivalently
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 $S_{CS} : \Omega^1(\Sigma, \mathfrak{g}) \to \mathbb{R}$
is the standard action functional of Chern-Simons theory.
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
The presymplectic structure on this space is
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.
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.
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.
We have
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;
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
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):
where $F_A$ is the ordinary curvature 2-form of $A$.
This is from (SSSI).
For $\{t_a\}$ a basis of $\mathfrak{g}_1$ and $\{b_i\}$ a basis of $\mathfrak{g}_2$ we have
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
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
Therefore
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.
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 $\mathfrak{string} := \mathfrak{so}_\mu$.
(…)
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.
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).
We consider the case where the target space object $\mathbf{B}G$ is a discrete ∞-groupoid.
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.
The background field for $\infty$-Dijkgraaf-Witten theory is necessarily flat.
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
and therefore, using the $(\Pi \dashv Disc)$-zig-zag identity, the constant path inclusion
is an equivalence. Therefore the intrinsic de Rham cohomology of $\mathbf{B}G$ is trivial
and so the intrinsic universal curvature class
is trivial.
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.
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.
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
By the $(Disc \dashv \Gamma)$-adjunction we have
For $G$ discrete
the intrinsic de Rham cohomology of $\mathbf{B}G$ is trivial
all $G$-principal bundles have a unique flat connection
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
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 $\Sigma$
is the action functional of Dijkgraaf-Witten theory.
By proposition \ref{IntrinsicIntegrationTheorem} 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).
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.
(…)
See
For the moment see closed string field 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.
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
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
This $\infty$-Lie theoretic structure is essentially what in the literature is mostly considered in terms of symplectic dg-geometry :
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
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:
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 $X$
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
and hence the Lie derivative
So in the above notation we have in particular
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
where $\{x^a\}$ is a basis of generators and $\mathrm{deg}(x^a)$ the degree of a generator.
We write
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
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.
On a graded manifold every closed differential form $\omega$ of positive grade $n$ is exact: the form
satisfies
Using this differential geometric language we can now capture something very close to def. 1 in more traditional symplectic geometry terms.
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
as usual for symplectic forms, and in addition
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.
The function algebra of a symplectic dg-manifold $(X,\omega)$ of grade $n$ is naturally equipped with a Poisson bracket
which decreases grade by $n$. On a local coordinate patch this is given by
where $\{\omega^{a b}\}$ is the inverse matrix to $\{\omega_{a b}\}$.
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
There is a full and faithful embedding of symplectic dg-manifolds of grade $n$ into symplectic Lie $n$-algebroids.
The dg-manifold itself is identified with an $L_\infty$-algebroid as in observation 6. 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.
Let $(\mathfrak{P},\omega)$ be a symplectic Lie $n$-algebroid for positive $n$ in the image of the embedding of prop. 10. Then it carries the canonical $L_\infty$-algebroid cocycle
which moreover is the Hamiltonian, according to def. 9, of $d_{\mathrm{CE}(\mathfrak{P})}$.
Our central observation now is the following.
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
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
by Cartan calculus. Using this we compute the first summand in $d_{\mathrm{W}(\mathfrak{P})} ( \iota_{\epsilon} \omega + \pi )$:
The second summand is simply
since $\pi$ is a cocycle.
For $(\mathfrak{P}. \omega)$ 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:
We work in local coordinates $\{x^a\}$ where
and the Chern-Simons element is
We want to substitute here $\mathbf{d} = d_{\mathrm{W}}- d_{\mathrm{CE}}$. Notice that in coordinates the equation
becomes
Therefore
Hence
This means that for $\Sigma$ an $(n+1)$-dimensional manifold and
a $\mathfrak{P}$-valued differential form on $\Sigma$ we have
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. 13 gives rise to the following dictionary of concepts\
The covariant phase space of AKSZ theory with target $(\mathfrak{P}, \omega)$ is the space of those ∞-Lie algebroid-valued forms $A$ whose curvature $(n+1)$-form $F_A$ vanishes
The presymplectic structure on this space is
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.
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
is the familiar term for the action functional of the particle (missing the kinetic term, which makes it “topological”).
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
where $\pi^{i j} := \{x^i , x^j\}$ and
The Hamiltonian cocycle from prop. 11 is
and the Chern-Simons element from prop. 12 is
In terms of $d_{\mathrm{W}}$ instead of $\mathbf{d}$ this is
So for
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
This is the Lagrangean of the Poisson $\sigma$-model \cite{CattaneoFelder}.
We show how the ordinary Chern-Simons form arises from this perspective. So let $\mathfrak{a} = \mathfrak{g}$ be a semisimple Lie algebra 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
where $C^a{}_{b c} := t^a([t_b,t_c])$ and
where $P_{ab} := \langle t_a, t_b \rangle$. The Hamiltonian cocycle $\pi$ from prop. 11 is
Therefore in this case the Chern-Simons element from def. 12 becomes
This is indeed the familiar standard choice of Chern-Simons element on a Lie algebra. Notice that evaluated on a $\mathfrak{g}$-valued form
this is
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
and hence
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
Hence
(…)
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
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
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.
The notion of Chern-Simons elements for $L_\infty$-algebras and the associated $\imnfty$-Chern-Simons Lagrangians is due to
The induced construction of the ∞-Chern-Weil homomorphism with special attention to the Chern-Simons circle 3-bundle and the Chern-Simons circle 7-bundle is in
In the general context of cohesive (∞,1)-toposes $\infty$-Chern-Simons theory is discussed in section 4.3 of
The case of the AKSZ sigma-model is discussed in
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)