This is a sub-entry of ∞-Chern-Simons theory. See there for background and context.
We describe the covariant phase spaces of ∞-Chern-Simons theory:
the space of solutions to its Euler-Lagrange equations of motion,
the canonical presymplectic structure on that space
and – eventually – its reduction to a genuine symplectic structure after homotopically dividing out gauge transformations: after passage to a Lagrangian submanifold of the derived critical locus.
By the discussion at ∞-Chern-Simons theory -- action functionals we have that the action functional is itself a representative of a characteristic class, in degree 0, in the cohesive (∞,1)-topos Smooth∞Grpd. Accordingly, it has itself a differential refinement:
We call $C := [\Sigma, A_{conn}]$ the configuration space of the $\infty$-Chern-Simons theory over $\Sigma$.
The postcomposition of the smooth action functional with the universal curvature characteristic form $\theta : U(1) \to \mathbf{\flat}_{dR} \mathbf{B}U(1)$
we call the Euler-Lagrange equations of $\mathbf{c}$-Chern-Simons theory over $\Sigma$.
The critical locus the action functional, hence the homotopy fiber $P \to C$ of $d \exp(i S_{\mathbf{c}}(-))$ regarded as a section of the cotangent bundle, hence the (∞,1)-pullback
we call the covariant phase space of $\mathbf{c}$-Chern-Simons theory over $\Sigma$.
Given any local action functional, its Euler-Lagrange equations determine the corresponding covariant phase space canonically equipped with a presymplectic structure.
We determine the presymplectic covaraint phase space of the explicit action functional presentation discussed here.
Let $\mathfrak{g}$ be an L-∞ algebra with $n$-ary invariant polynomial $\langle -, -, \cdots, -\rangle$. Then the ∞-connections $A$ with values in $\mathfrak{g}$ that satisfy the equations of motion of the corresponding $\infty$-Chern-Simons theory are precisely those for which
where $F_A$ denotes the (in general inhomogeneous) curvature form of $A$.
Let $\hat A \in \Omega(\Sigma \times I, \mathfrak{g})$ be a 1-parameter variation of $\hat A(t = 0) := A$, that vanishes on the boundary $\partial \Sigma$. Here we write $t : [0,1] \to \mathbb{R}$ for the canonical coordinate on the interval.
Notice that the curvature is
so that
and
For the given path $\hat A$ of fields we may write
By definition $A$ is critical if
for all extensions $\hat A$ of $A$. Using Cartan's magic formula and the Stokes theorem the left hand expression is
where we used that by assumption $\iota_{\partial t} F_{\hat A}$ and hence $\iota_{\partial_t} cs(\hat A)$ vanishes on $\partial \Sigma$. This yields the equations of motion as claimed.
The canonical presymplectic potential on the space of solutions is
where $\delta A$ is inserted, termwise, for a curvature form.
Comparing in the proof of prop. 1 the structure of the boundary term with the formula for the presymplectic structure discussed at covariant phase space we see that the presymplectic potential is
Here we are using that $\iota_{\partial t} \hat A = 0$ and $\iota_{\partial_t} F{\hat A}(t = 0) = \delta A$. Therefore the cocycle summand $\mu$ in the Chern-Simons element drops out.
For $\langle -,-\rangle$ a binary and non-degenerated invariant polynomial (as for ordinary Chern-Simons theory) the equations of motion are
and the presymplectic structure on the space of solutions is
We discuss the gauge symmetry of the $\infty$-Chern-Simons action functionals.
The genuine gauge transformations of L-∞-algebroid valued differential forms? are symmetries of the $\infty$-Chern-Simons action functional, by the invariance property of invariant polynomials.
(…)
Suppose there exists $v \in \Gamma(T \mathfrak{a})$ such that
Then by the above the “constant” transformation
is a symmetry of the the $\infty$-Chern-Simons action.
These spurious global symmetries are absent precisely if $\langle - \rangle$ is n-plectic, hence if $(\mathfrak{a}, \langle - \rangle)$ constitutes a higher symplectic geometry.
The ∞-Chern-Simons theory action functional $\exp(i S(-)) : [\Sigma, A_{conn}] \to U(1)$ is manifestly invariant under diffeomorphisms $\phi : \Sigma \to \Sigma$. But only in special cases does this invariance not add to the ghost-structure of the BV-BRST complex on top of the gauge ghosts contained already in $[\Sigma, A_{conn}]$:
If the invariant polynomial $\langle -\rangle$ that defines the $\infty$-Chern-Simons theory is binary and non-degenerate, then on covariant phase space $[\Sigma, A_{conn}]_{crit}$ every diffeomorphism $\phi : \Sigma \stackrel{\simeq}{\to} \Sigma$ connected to the identity is related by a gauge transformation to the identity:
The assumption that the invariant polynomial $\langle-,-\rangle$ is binary and invariant implies with corollary 1 that the equations of motion are $F_A = 0$.
Let $v$ be the vector field generating the diffeomorphism. Then for $A : * \to [\Sigma,A_{conn}]_{crit}$ a field configuration its iamge under the gauge transformation $[\phi,A_{conn}]$ is $\exp(\mathcal{L}_v) A$, where $\mathcal{L}_v$ is the Lie derivative along $v$. By Cartan's magic formula and the equations of motion we have
where
is the gauge parameter and $\nabla_A$ is the covariant derivative
derived at connection on a principal infinity-bundle in the subsection infinitesimal gauge transformations. As discussed there, this are the infinitesimal gauge transformations in $[\Sigma, A_{conn}]$.
See also the references at ∞-Chern-Simons theory.
A discussion of Chern-Simons theory for higher degree invariant polynomials (but on ordinary Lie algebras) is for instance in
Last revised on September 2, 2011 at 13:22:04. See the history of this page for a list of all contributions to it.