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 [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)
Last revised on January 22, 2013 at 13:16:35. See the history of this page for a list of all contributions to it.