# Schreiber infinity-Chern-Simons theory -- covariant phase space

This is a sub-entry of ∞-Chern-Simons theory. See there for background and context.

# Contents

## The covariant phase space

We describe the covariant phase spaces of ∞-Chern-Simons theory:

1. the space of solutions to its Euler-Lagrange equations of motion,

2. the canonical presymplectic structure on that space

3. 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.

### Variation

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:

###### Definition
• 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 $\theta : U(1) \to \mathbf{\flat}_{dR} \mathbf{B}U(1)$

$d \exp(i S_{\mathbf{c}}(-))_{smooth} : [\Sigma, A_{conn}] \stackrel{\exp(i S_{\mathbf{c}})(-)}{\to} U(1) \stackrel{\theta}{\to} \mathbf{\flat}_{dR} \mathbf{B}\mathbb{R} \simeq \Omega^1_{cl}(-)$

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

$\array{ P &\to& [\Sigma, A_{conn}] \\ \downarrow && \downarrow^{\mathrlap{0}} \\ [\Sigma, A_{conn}] &\stackrel{d \exp(i S_{\mathbf{c}}(-))}{\to}& T^* [\Sigma, A_{conn}] }$

we call the covariant phase space of $\mathbf{c}$-Chern-Simons theory over $\Sigma$.

### Equations of motion and presymplectic covariant phase space

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.

###### Proposition

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

$\langle -, F_A \wedge F_A \wedge \cdots \wedge F_A \rangle = 0 \,\,\, \in \mathfrak{g}^* \otimes \Omega^\bullet(\Sigma) \,,$

where $F_A$ denotes the (in general inhomogeneous) curvature form of $A$.

###### Proof

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

$F_{\hat A}(t) = F_{\hat A(t)} + (\frac{d}{d t}\hat A)(t) \wedge d t$

so that

$\iota_{\partial_t} \hat A = 0$

and

$\iota_{\partial_t} F_{\hat A} = \frac{d}{d t}\hat A .$

For the given path $\hat A$ of fields we may write

$\delta A := \left(\frac{d}{d t}A\right)(0) \,.$

By definition $A$ is critical if

$\left(\frac{d}{d t} \int_{\Sigma} cs(A)\right)_{t = 0} = 0$

for all extensions $\hat A$ of $A$. Using Cartan's magic formula and the Stokes theorem the left hand expression is

\begin{aligned} \left(\frac{d}{d t}\int_{\Sigma} cs(\hat A)\right)_{t = 0} & = \left(\int_{\Sigma} \frac{d}{d t} cs(\hat A)\right)_{t = 0} \\ & = \left( \int_{\Sigma} d \iota_{\partial t} cs(\hat A) + \int_{\Sigma} \iota_{\partial_t} d cs(\hat A) \right)_{t = 0} \\ & = \left( \int_{\Sigma} d_\Sigma ( \iota_{\partial t} cs(\hat A)) + \int_{\Sigma} \iota_{\partial_t} \langle F_{\hat A} \wedge \cdots \wedge F_{\hat A} \rangle \right)_{t = 0} \\ & = \left( \int_{\partial \Sigma} \iota_{\partial t} cs(\hat A) + n \int_{\Sigma} \langle (\frac{d}{d t} \hat A) \wedge \cdots \wedge F_A \rangle \right)_{t = 0} \\ & = \left( n \int_{\Sigma} \langle (\frac{d}{d t}\hat A) \wedge \cdots F_{\hat A} \rangle \right)_{t = 0} \end{aligned} \,,

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.

###### Proposition

The canonical presymplectic potential on the space of solutions is

$\theta_{A}(\delta A) \propto \int_{\partial \Sigma} (cs - \mu) (A, F_A|\delta A) \,,$

where $\delta A$ is inserted, termwise, for a curvature form.

###### Proof

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

\begin{aligned} \theta : \delta A & \mapsto \left( \int_{\partial \Sigma} \iota_{\partial t} cs(\hat A) \right)_{t = 0} \\ & = \int_{\partial \Sigma} (cs-\mu)(A, F_A | \delta A) \end{aligned} \,.

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.

###### Corollary/Example

For $\langle -,-\rangle$ a binary and non-degenerated invariant polynomial (as for ordinary Chern-Simons theory) the equations of motion are

$F_A = 0$

and the presymplectic structure on the space of solutions is

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

### Symmetries

We discuss the gauge symmetry of the $\infty$-Chern-Simons action functionals.

#### Gauge transformations of $L_\infty$-algebroid valued forms

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.

(…)

#### Global gauge transformations

Suppose there exists $v \in \Gamma(T \mathfrak{a})$ such that

$\iota_v \langle - \rangle = 0 \,.$

Then by the above the “constant” transformation

$\delta A = v$

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.

#### Diffeomorphism invariance

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 $[\Sigma, A_{conn}]$:

###### Proposition

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:

$\array{ && \stackrel{[\phi,A_{conn}]}{\to} \\ & \nearrow &\Downarrow^{\mathrlap{\simeq}}&\searrow & \\ [\Sigma,A_{conn}]_{crit} &&\underset{id}{\to}&& [\Sigma,A_{conn}]_{crit} } \,.$
###### Proof

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

\begin{aligned} \mathcal{L}_v A & = d_{dR} \iota_v A + \iota_v d_{dR} A \\ & = \nabla_A \lambda \end{aligned} \,,

where

$\lambda := \iota_v A$

is the gauge parameter and $\nabla_A$ is the covariant derivative

$\nabla_A \lambda = d_{dR} \lambda + [A \wedge \lambda] + [A \wedge A \wedge \lambda] + \cdots$

derived at connection on a principal infinity-bundle in the subsection infinitesimal gauge transformations. As discussed there, this are the infinitesimal gauge transformations in $[\Sigma, A_{conn}]$.