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

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

---

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

References

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

• Maximo Banados, Luis Garay, Marc Henneaux, Existence of local degrees of freedom for higher dimensional pure Chern-Simons theories Phys. Rev. D. vol 53, nr 2 (1996)