# 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 $C:=\left[\Sigma ,{A}_{\mathrm{conn}}\right]$ the configuration space of the $\infty$-Chern-Simons theory over $\Sigma$.

• The postcomposition of the smooth action functional with the universal curvature characteristic form $\theta :U\left(1\right)\to {♭}_{\mathrm{dR}}BU\left(1\right)$

$d\mathrm{exp}\left(i{S}_{c}\left(-\right){\right)}_{\mathrm{smooth}}:\left[\Sigma ,{A}_{\mathrm{conn}}\right]\stackrel{\mathrm{exp}\left(i{S}_{c}\right)\left(-\right)}{\to }U\left(1\right)\stackrel{\theta }{\to }{♭}_{\mathrm{dR}}Bℝ\simeq {\Omega }_{\mathrm{cl}}^{1}\left(-\right)$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 $c$-Chern-Simons theory over $\Sigma$.

• The critical locus the action functional, hence the homotopy fiber $P\to C$ of $d\mathrm{exp}\left(i{S}_{c}\left(-\right)\right)$ regarded as a section of the cotangent bundle, hence the (∞,1)-pullback

$\begin{array}{ccc}P& \to & \left[\Sigma ,{A}_{\mathrm{conn}}\right]\\ ↓& & {↓}^{0}\\ \left[\Sigma ,{A}_{\mathrm{conn}}\right]& \stackrel{d\mathrm{exp}\left(i{S}_{c}\left(-\right)\right)}{\to }& {T}^{*}\left[\Sigma ,{A}_{\mathrm{conn}}\right]\end{array}$\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 $c$-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 $𝔤$ be an L-∞ algebra with $n$-ary invariant polynomial $⟨-,-,\cdots ,-⟩$. Then the ∞-connections $A$ with values in $𝔤$ that satisfy the equations of motion of the corresponding $\infty$-Chern-Simons theory are precisely those for which

$⟨-,{F}_{A}\wedge {F}_{A}\wedge \cdots \wedge {F}_{A}⟩=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\in {𝔤}^{*}\otimes {\Omega }^{•}\left(\Sigma \right)\phantom{\rule{thinmathspace}{0ex}},$\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 $\stackrel{^}{A}\in \Omega \left(\Sigma ×I,𝔤\right)$ be a 1-parameter variation of $\stackrel{^}{A}\left(t=0\right):=A$, that vanishes on the boundary $\partial \Sigma$. Here we write $t:\left[0,1\right]\to ℝ$ for the canonical coordinate on the interval.

Notice that the curvature is

${F}_{\stackrel{^}{A}}\left(t\right)={F}_{\stackrel{^}{A}\left(t\right)}+\left(\frac{d}{dt}\stackrel{^}{A}\right)\left(t\right)\wedge dt$F_{\hat A}(t) = F_{\hat A(t)} + (\frac{d}{d t}\hat A)(t) \wedge d t

so that

${\iota }_{{\partial }_{t}}\stackrel{^}{A}=0$\iota_{\partial_t} \hat A = 0

and

${\iota }_{{\partial }_{t}}{F}_{\stackrel{^}{A}}=\frac{d}{dt}\stackrel{^}{A}.$\iota_{\partial_t} F_{\hat A} = \frac{d}{d t}\hat A .

For the given path $\stackrel{^}{A}$ of fields we may write

$\delta A:=\left(\frac{d}{dt}A\right)\left(0\right)\phantom{\rule{thinmathspace}{0ex}}.$\delta A := \left(\frac{d}{d t}A\right)(0) \,.

By definition $A$ is critical if

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

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

$\begin{array}{rl}{\left(\frac{d}{dt}{\int }_{\Sigma }\mathrm{cs}\left(\stackrel{^}{A}\right)\right)}_{t=0}& ={\left({\int }_{\Sigma }\frac{d}{dt}\mathrm{cs}\left(\stackrel{^}{A}\right)\right)}_{t=0}\\ & ={\left({\int }_{\Sigma }d{\iota }_{\partial t}\mathrm{cs}\left(\stackrel{^}{A}\right)+{\int }_{\Sigma }{\iota }_{{\partial }_{t}}d\mathrm{cs}\left(\stackrel{^}{A}\right)\right)}_{t=0}\\ & ={\left({\int }_{\Sigma }{d}_{\Sigma }\left({\iota }_{\partial t}\mathrm{cs}\left(\stackrel{^}{A}\right)\right)+{\int }_{\Sigma }{\iota }_{{\partial }_{t}}⟨{F}_{\stackrel{^}{A}}\wedge \cdots \wedge {F}_{\stackrel{^}{A}}⟩\right)}_{t=0}\\ & ={\left({\int }_{\partial \Sigma }{\iota }_{\partial t}\mathrm{cs}\left(\stackrel{^}{A}\right)+n{\int }_{\Sigma }⟨\left(\frac{d}{dt}\stackrel{^}{A}\right)\wedge \cdots \wedge {F}_{A}⟩\right)}_{t=0}\\ & ={\left(n{\int }_{\Sigma }⟨\left(\frac{d}{dt}\stackrel{^}{A}\right)\wedge \cdots {F}_{\stackrel{^}{A}}⟩\right)}_{t=0}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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 ${\iota }_{\partial t}{F}_{\stackrel{^}{A}}$ and hence ${\iota }_{{\partial }_{t}}\mathrm{cs}\left(\stackrel{^}{A}\right)$ vanishes on $\partial \Sigma$. This yields the equations of motion as claimed.

###### Proposition

The canonical presymplectic potential on the space of solutions is

${\theta }_{A}\left(\delta A\right)\propto {\int }_{\partial \Sigma }\left(\mathrm{cs}-\mu \right)\left(A,{F}_{A}\mid \delta A\right)\phantom{\rule{thinmathspace}{0ex}},$\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

$\begin{array}{rl}\theta :\delta A& ↦{\left({\int }_{\partial \Sigma }{\iota }_{\partial t}\mathrm{cs}\left(\stackrel{^}{A}\right)\right)}_{t=0}\\ & ={\int }_{\partial \Sigma }\left(\mathrm{cs}-\mu \right)\left(A,{F}_{A}\mid \delta A\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\iota }_{\partial t}\stackrel{^}{A}=0$ and ${\iota }_{{\partial }_{t}}F\stackrel{^}{A}\left(t=0\right)=\delta A$. Therefore the cocycle summand $\mu$ in the Chern-Simons element drops out.

###### Corollary/Example

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

${F}_{A}=0$F_A = 0

and the presymplectic structure on the space of solutions is

$\omega \left(\delta {A}_{1},\delta {A}_{2}\right)\propto {\int }_{\partial \Sigma }⟨\delta {A}_{1},\delta {A}_{2}⟩\phantom{\rule{thinmathspace}{0ex}}.$\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 $v\in \Gamma \left(T𝔞\right)$ such that

${\iota }_{v}⟨-⟩=0\phantom{\rule{thinmathspace}{0ex}}.$\iota_v \langle - \rangle = 0 \,.

Then by the above the “constant” transformation

$\delta A=v$\delta A = v

is a symmetry of the the $\infty$-Chern-Simons action.

These spurious global symmetries are absent precisely if $⟨-⟩$ is n-plectic, hence if $\left(𝔞,⟨-⟩\right)$ constitutes a higher symplectic geometry.

#### Diffeomorphism invariance

The ∞-Chern-Simons theory action functional $\mathrm{exp}\left(iS\left(-\right)\right):\left[\Sigma ,{A}_{\mathrm{conn}}\right]\to U\left(1\right)$ is manifestly invariant under diffeomorphisms $\varphi :\Sigma \to \Sigma$. 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 $\left[\Sigma ,{A}_{\mathrm{conn}}\right]$:

###### Proposition

If the invariant polynomial $⟨-⟩$ that defines the $\infty$-Chern-Simons theory is binary and non-degenerate, then on covariant phase space $\left[\Sigma ,{A}_{\mathrm{conn}}{\right]}_{\mathrm{crit}}$ every diffeomorphism $\varphi :\Sigma \stackrel{\simeq }{\to }\Sigma$ connected to the identity is related by a gauge transformation to the identity:

$\begin{array}{ccc}& & \stackrel{\left[\varphi ,{A}_{\mathrm{conn}}\right]}{\to }\\ & ↗& {⇓}^{\simeq }& ↘& \\ \left[\Sigma ,{A}_{\mathrm{conn}}{\right]}_{\mathrm{crit}}& & \underset{\mathrm{id}}{\to }& & \left[\Sigma ,{A}_{\mathrm{conn}}{\right]}_{\mathrm{crit}}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $⟨-,-⟩$ 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 $A:*\to \left[\Sigma ,{A}_{\mathrm{conn}}{\right]}_{\mathrm{crit}}$ a field configuration its iamge under the gauge transformation $\left[\varphi ,{A}_{\mathrm{conn}}\right]$ is $\mathrm{exp}\left({ℒ}_{v}\right)A$, where ${ℒ}_{v}$ is the Lie derivative along $v$. By Cartan's magic formula and the equations of motion we have

$\begin{array}{rl}{ℒ}_{v}A& ={d}_{\mathrm{dR}}{\iota }_{v}A+{\iota }_{v}{d}_{\mathrm{dR}}A\\ & ={\nabla }_{A}\lambda \end{array}\phantom{\rule{thinmathspace}{0ex}},$\begin{aligned} \mathcal{L}_v A & = d_{dR} \iota_v A + \iota_v d_{dR} A \\ & = \nabla_A \lambda \end{aligned} \,,

where

$\lambda :={\iota }_{v}A$\lambda := \iota_v A

is the gauge parameter and ${\nabla }_{A}$ is the covariant derivative

${\nabla }_{A}\lambda ={d}_{\mathrm{dR}}\lambda +\left[A\wedge \lambda \right]+\left[A\wedge A\wedge \lambda \right]+\cdots$\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 $\left[\Sigma ,{A}_{\mathrm{conn}}\right]$.