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:=[Σ,A conn]C := [\Sigma, A_{conn}] the configuration space of the \infty-Chern-Simons theory over Σ\Sigma.

  • The postcomposition of the smooth action functional with the universal curvature characteristic form θ:U(1) dRBU(1)\theta : U(1) \to \mathbf{\flat}_{dR} \mathbf{B}U(1)

    dexp(iS c()) smooth:[Σ,A conn]exp(iS c)()U(1)θ dRBΩ cl 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 c\mathbf{c}-Chern-Simons theory over Σ\Sigma.

  • The critical locus the action functional, hence the homotopy fiber PCP \to C of dexp(iS c())d \exp(i S_{\mathbf{c}}(-)) regarded as a section of the cotangent bundle, hence the (∞,1)-pullback

    P [Σ,A conn] 0 [Σ,A conn] dexp(iS c()) T *[Σ,A conn] \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\mathbf{c}-Chern-Simons theory over Σ\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\mathfrak{g} be an L-∞ algebra with nn-ary invariant polynomial ,,,\langle -, -, \cdots, -\rangle. Then the ∞-connections AA with values in 𝔤\mathfrak{g} that satisfy the equations of motion of the corresponding \infty-Chern-Simons theory are precisely those for which

,F AF AF A=0𝔤 *Ω (Σ), \langle -, F_A \wedge F_A \wedge \cdots \wedge F_A \rangle = 0 \,\,\, \in \mathfrak{g}^* \otimes \Omega^\bullet(\Sigma) \,,

where F AF_A denotes the (in general inhomogeneous) curvature form of AA.

Proof

Let A^Ω(Σ×I,𝔤)\hat A \in \Omega(\Sigma \times I, \mathfrak{g}) be a 1-parameter variation of A^(t=0):=A\hat A(t = 0) := A, that vanishes on the boundary Σ\partial \Sigma. Here we write t:[0,1]t : [0,1] \to \mathbb{R} for the canonical coordinate on the interval.

Notice that the curvature is

F A^(t)=F A^(t)+(ddtA^)(t)dt F_{\hat A}(t) = F_{\hat A(t)} + (\frac{d}{d t}\hat A)(t) \wedge d t

so that

ι tA^=0 \iota_{\partial_t} \hat A = 0

and

ι tF A^=ddtA^. \iota_{\partial_t} F_{\hat A} = \frac{d}{d t}\hat A .

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

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

By definition AA is critical if

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

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

(ddt Σcs(A^)) t=0 =( Σddtcs(A^)) t=0 =( Σdι tcs(A^)+ Σι tdcs(A^)) t=0 =( Σd Σ(ι tcs(A^))+ Σι tF A^F A^) t=0 =( Σι tcs(A^)+n Σ(ddtA^)F A) t=0 =(n Σ(ddtA^)F A^) t=0, \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 ι tF A^\iota_{\partial t} F_{\hat A} and hence ι tcs(A^)\iota_{\partial_t} cs(\hat A) vanishes on Σ\partial \Sigma. This yields the equations of motion as claimed.

Proposition

The canonical presymplectic potential on the space of solutions is

θ A(δA) Σ(csμ)(A,F A|δA), \theta_{A}(\delta A) \propto \int_{\partial \Sigma} (cs - \mu) (A, F_A|\delta A) \,,

where δA\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

θ:δA ( Σι tcs(A^)) t=0 = Σ(csμ)(A,F A|δA). \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 ι tA^=0\iota_{\partial t} \hat A = 0 and ι tFA^(t=0)=δA\iota_{\partial_t} F{\hat A}(t = 0) = \delta A. Therefore the cocycle summand μ\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 F_A = 0

and the presymplectic structure on the space of solutions is

ω(δA 1,δA 2) ΣδA 1,δA 2. \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 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 vΓ(T𝔞)v \in \Gamma(T \mathfrak{a}) such that

ι v=0. \iota_v \langle - \rangle = 0 \,.

Then by the above the “constant” transformation

δA=v \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 (𝔞,)(\mathfrak{a}, \langle - \rangle) constitutes a higher symplectic geometry.

Diffeomorphism invariance

The ∞-Chern-Simons theory action functional exp(iS()):[Σ,A conn]U(1)\exp(i S(-)) : [\Sigma, A_{conn}] \to U(1) is manifestly invariant under diffeomorphisms ϕ:ΣΣ\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 [Σ,A conn][\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 [Σ,A conn] crit[\Sigma, A_{conn}]_{crit} every diffeomorphism ϕ:ΣΣ\phi : \Sigma \stackrel{\simeq}{\to} \Sigma connected to the identity is related by a gauge transformation to the identity:

[ϕ,A conn] [Σ,A conn] crit id [Σ,A conn] crit. \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=0F_A = 0.

Let vv be the vector field generating the diffeomorphism. Then for A:*[Σ,A conn] critA : * \to [\Sigma,A_{conn}]_{crit} a field configuration its iamge under the gauge transformation [ϕ,A conn][\phi,A_{conn}] is exp( v)A\exp(\mathcal{L}_v) A, where v\mathcal{L}_v is the Lie derivative along vv. By Cartan's magic formula and the equations of motion we have

vA =d dRι vA+ι vd dRA = Aλ, \begin{aligned} \mathcal{L}_v A & = d_{dR} \iota_v A + \iota_v d_{dR} A \\ & = \nabla_A \lambda \end{aligned} \,,

where

λ:=ι vA \lambda := \iota_v A

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

Aλ=d dRλ+[Aλ]+[AAλ]+ \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 [Σ,A conn][\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)
Revised on September 2, 2011 13:22:04 by Urs Schreiber (131.211.239.252)