Schreiber
infinity-Chern-Simons theory -- examples

This is a sub-entry of infinity-Chern-Simons theory. See there for context.


Contents

Examples

1d Chern-Simons functionals

2d Chern-Simons functionals

Poisson σ\sigma-model

3d Chern-Simons functionals

Ordinary Chern-Simons theory

Lagrangian

Let 𝔤\mathfrak{g} be a semisimple Lie algebra. For the following computations, choose a basis {t a}\{t^a\} of 𝔤 *\mathfrak{g}^* and let {r a}\{r^a\} denote the corresponding degree-shifted basis of 𝔤 *[1]\mathfrak{g}^*[1].

Notice that in terms of this the differential of the Chevalley-Eilenberg algebra is

d CE(𝔤):t a12C a bct bt c d_{CE(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c

and that of the Weil algebra

d W(𝔤):t a12C a bct bt c+r a d_{W(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c + r^a

and

d W(𝔤):r aC a bct br c. d_{W(\mathfrak{g})} : r^a \mapsto -C^a{}_{b c} t^b \wedge r^c \,.

Let P abr ar bW(𝔤)P_{a b} r^a \wedge r^b \in W(\mathfrak{g}) be the Killing form invariant polynomial. This being invariant

d W(𝔤)P abr ar b=2P abC a det dr er b=0 d_{W(\mathfrak{g})} P_{a b} r^a \wedge r^b = 2 P_{a b} C^{a}{}_{d e} t^d \wedge r^e \wedge r^b = 0

is equivalent to the fact that the coefficients

C abc:=P aaC a bc C_{a b c} := P_{a a'}C^{a'}{}_{b c}

are skew-symmetric in aa and bb, and therefore skew in all three indices.

Proposition

A Chern-Simons element for the Killing form invariant polynomial ,=P(,) \langle -, - \rangle = P(-,-) is

cs =P abt a(d W(𝔤)t b)+13P aaC a bct at bt c =P abt ar b16P aaC a bct at bt c. \begin{aligned} cs &= P_{a b} t^a \wedge (d_{W(\mathfrak{g})} t^b) + \frac{1}{3} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c \\ & = P_{a b} t^a \wedge r^b - \frac{1}{6} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c \end{aligned} \,.

In particular the Killing form ,\langle -,-\rangle is in transgression with the degree 3-cocycle

μ=16,[,]. \mu = -\frac{1}{6}\langle -,[-,-]\rangle \,.
Proof

We compute

d W(𝔤)(P abt ar b+12P aaC a bct at bt c) =P abr ar b 12P abC a det dt er b +P abC b det at dr e 36P aaC a bct at br c =P abr ar b +12C abct at br c 12C abct at br c =P abr ar b. \begin{aligned} d_{W(\mathfrak{g})} \left( P_{a b} t^a \wedge r^b + \frac{1}{2}P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c \right) & = P_{a b} r^a \wedge r^b \\ & -\frac{1}{2} P_{a b} C^a{}_{d e} t^d \wedge t^e \wedge r^b \\ & + P_{a b} C^b{}_{d e} t^a \wedge t^d \wedge r^e \\ & - \frac{3}{6} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge r^c \\ & = P_{a b} r^a \wedge r^b \\ & + \frac{1}{2}C_{a b c} t^a \wedge t^b \wedge r^c \\ & - \frac{1}{2} C_{a b c} t^a \wedge t^b \wedge r^c \\ & = P_{a b } r^a \wedge r^b \end{aligned} \,.

Under a Lie algebra-valued form

Ω (X)W(𝔤):A \Omega^\bullet(X) \stackrel{}{\leftarrow} W(\mathfrak{g}) : A

this Chern-Simons element is sent to

cs(A)=P abA adA b+13C abcA aA bA c. cs(A) = P_{a b} A^a \wedge d A^b + \frac{1}{3} C_{a b c} A^a \wedge A^b \wedge A^c \,.

If 𝔤\mathfrak{g} is a matrix Lie algebra then the Killing form is the trace and this is equivalently

cs(A)=tr(AdA)+23tr(AAA). cs(A) = tr(A \wedge d A) + \frac{2}{3} tr(A \wedge A \wedge A) \,.

This is a traditional incarnation of the standard Chern-Simons form in degree 3.

For Σ\Sigma a 3-dimensional smooth manifold the corresponding action functional S CS:Ω 1(Σ,𝔤)S_{CS} : \Omega^1(\Sigma, \mathfrak{g}) \to \mathbb{R}

S CS:A Σcs(A) S_{CS} : A \mapsto \int_\Sigma cs(A)

is the standard action functional of Chern-Simons theory.

Covariant phase space
Proposition

The covariant phase space of ordinary Chern-Simons theory is the space of those Lie algebra valued form AA whose curvature 2-form F AF_A vanishes

P={AΩ 1(Σ,𝔤)|F A=0}. P = \{A \in \Omega^1(\Sigma, \mathfrak{g}) | F_A = 0\} \,.

The presymplectic structure on this space is

ω:(δA 1,δA 2) ΣδA 1,δA 2. \omega : (\delta A_1, \delta A_2) \mapsto \int_{\partial \Sigma} \langle \delta A_1, \delta A_2 \rangle \,.
Proof

This is a special case of prop. \ref{TheEquationsOfMotion}, prop. \ref{PresymplecticStructure} in view of corollary \ref{CovariantPhaseSpaceForBinaryNonDegenerateInvariantPolynomial}, using that the Killing form is a binary and non-degenerate invariant polynomial.

Obstruction theory

The circle n-bundle with connection given by ordinary Chern-Simons theory is known as the Chern-Simons circle 3-bundle . The non-triviality of its underlying class is the obstruction to lifting the GG-principal bundle to a string structure. The non-triviality of its connection is the obstruction to having a differential string structure. In general it defines a twisted differential string structure.

Courant σ\sigma-model

4d Chern-Simons functionals

BF-theory and topological Yang-Mills theory

Let 𝔤=(𝔤 2𝔤) 1\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g})_1 be a strict Lie 2-algebra, an L-∞ algebra concentrated in the lowesr two degrees and with the only nontrivial bracket being the binary one.

Proposition

We have

  1. every invariant polynomial 𝔤 1inv(𝔤 1)\langle -\rangle_{\mathfrak{g}_1} \in inv(\mathfrak{g}_1) on 𝔤 1\mathfrak{g}_1 is a Chern-Simons element on 𝔤\mathfrak{g}, exhibiting a transgression to a trivial ∞-Lie algebra cocycle;

  2. for 𝔤 1\mathfrak{g}_1 a semisimple Lie algebra and 𝔤 1\langle - \rangle_{\mathfrak{g}_1} the Killing form, the corresponding Chern-Simons action functional on L-∞ algebra valued forms

    Ω (X)(A,B)W(𝔤 2𝔤 1)( 𝔤 1,d W 𝔤 1)W(b n1) \Omega^\bullet(X) \stackrel{(A,B)}{\leftarrow} W(\mathfrak{g}_2 \to \mathfrak{g}_1) \stackrel{(\langle - \rangle_{\mathfrak{g}_1}, d_W \langle - \rangle_{\mathfrak{g}_1} )}{\leftarrow} W(b^{n-1} \mathbb{R})

    is the sum of the action functionals of topological Yang-Mills theory with BF-theory with cosmological constant (in the sense of gravity as a BF-theory):

    CS 𝔤 1(A,B)=F AF A 𝔤 12F AB 𝔤 1+2BB 𝔤 1, CS_{\langle-\rangle_{\mathfrak{g}_1}}(A,B) = \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1} - 2\langle F_A \wedge \partial B\rangle_{\mathfrak{g}_1} + 2\langle \partial B \wedge \partial B\rangle_{\mathfrak{g}_1} \,,

    where F AF_A is the ordinary curvature 2-form of AA.

This is from (SSSI).

Proof

For {t a}\{t_a\} a basis of 𝔤 1\mathfrak{g}_1 and {b i}\{b_i\} a basis of 𝔤 2\mathfrak{g}_2 we have

d W(𝔤):dt ad W(𝔤 1)+ a idb i. d_{W(\mathfrak{g})} : \mathbf{d} t^a \mapsto d_{W(\mathfrak{g}_1)} + \partial^a{}_i \mathbf{d} b^i \,.

Therefore with 𝔤 1=P a 1a ndr a 1dt a n\langle -\rangle_{\mathfrak{g}_1} = P_{a_1 \cdots a_n} \mathbf{d} r^{a_1} \wedge \cdots \mathbf{d} t^{a_n} we have

d W(𝔤) 𝔤 1=nP a 1a n a 1 idb idt a n. d_{W(\mathfrak{g})} \langle - \rangle_{\mathfrak{g}_1} = n P_{a_1 \cdots a_n}\partial^{a_1}{}_i \mathbf{d} b^{i} \wedge \cdots \mathbf{d} t^{a_n} \,.

The right hand is a polynomial in the shifted generators of W(𝔤)W(\mathfrak{g}), and hence an invariant polynomial on 𝔤\mathfrak{g}. Therefore 𝔤 1\langle - \rangle_{\mathfrak{g}_1} is a Chern-Simons element for it.

Now for (A,B)Ω 1(U×Δ k,𝔤)(A,B) \in \Omega^1(U \times \Delta^k, \mathfrak{g}) an ∞-Lie algebra-valued form, we have that the 2-form curvature is

F (A,B) 1=F AB. F_{(A,B)}^1 = F_A - \partial B \,.

Therefore

CS 𝔤 1(A,B) =F (A,B) 1 𝔤 1 =F AF A 𝔤 12F AB 𝔤 1+2BB 𝔤 1. \begin{aligned} CS_{\langle -\rangle_{\mathfrak{g}_1}}(A,B) & = \langle F_{(A,B)}^1\rangle_{\mathfrak{g}_1} \\ & = \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1} - 2\langle F_A \wedge \partial B\rangle_{\mathfrak{g}_1} + 2\langle \partial B \wedge \partial B\rangle_{\mathfrak{g}_1} \end{aligned} \,.

7d Chern-Simons functionals

7d StringString-Chern-Simons theory

In heterotic string theory Chern-Simons circle 3-bundles appear as the ordinary differential cohomology-incarnation of magnetic charge of NS-fivebranes that twists the Kalb-Ramond field as described by the Green-Schwarz mechanism.

At least when this twist vanishes there is expected to be an electric-magnetic dual description with a Chern-Simons circle 7-bundle.

Lagrangian

Let ,,,\langle -,-,-,-\rangle be the canonical quaternaty invariant polynomial on the special orthogonal Lie algebra 𝔰𝔬\mathfrak{so}. This lifts directly also to an invariant polynomial on the string Lie 2-algebra 𝔰𝔱𝔯𝔦𝔫𝔤:=𝔰𝔬 μ\mathfrak{string} := \mathfrak{so}_\mu.

(…)

Obstruction theory

The circle n-bundle with connection given by 7-dimensional Chern-Simons theory is the Chern-Simons circle 7-bundle . The non-triviality of its underlying class is the obstruction to lifting the string 2-group-principal 2-bundle to a fivebrane structure. The non-triviality of its 7-connection is the obstruction to having a differential fivebrane structure. In general it defines a twisted differential fivebrane structure.

Higher dimensional abelian Chern-Simons theory

The line Lie n-algebra carries a canonical invariant polynomial. The \infty-Chern-Simons theory associated with this data is in the literature known as abelian higher dimensional Chern-Simons theory.

Also every higher-degree invariant polynomial on an ordinary Lie algebra gives rise to a higher dimensional Chern-Simons action functional. An concrete example of this is considered (below).

\infty-Dijkgraaf-Witten theory

We consider the case where the target space object BG\mathbf{B}G is a discrete ∞-groupoid.

BG:=DiscBG \mathbf{B}G := Disc B G

with BGB G the delooping of an ∞-group GG \in Grpd \simeq Top.

As we discuss below, \infty-Chern-Simons theory for this setup subsumes and generalizes Dijkgraaf-Witten theory (and the Yetter model in next higher dimension). Therefore we speak of \infty-Dijkgraaf-Witten theory.

Remark

The background field for \infty-Dijkgraaf-Witten theory is necessarily flat.

Proof

By the (ΠDiscΓ)(\Pi \dashv Disc \dashv \Gamma)-adjoint triple of the ambient cohesive (∞,1)-topos and usinf that DiscDisc is a full and faithful (∞,1)-functor we have

ΠBG DiscΠDiscBG DiscBG BG \begin{aligned} \mathbf{\Pi} \mathbf{B}G & \simeq Disc \Pi Disc B G \\ & \simeq Disc B G \\ & \simeq \mathbf{B}G \end{aligned}

and therefore, using the (ΠDisc)(\Pi \dashv Disc)-zig-zag identity, the constant path inclusion

BGΠBG \mathbf{B}G \to \mathbf{\Pi} \mathbf{B}G

is an equivalence. Therefore the intrinsic de Rham cohomology of BG\mathbf{B}G is trivial

H dR(BG,B nU(1)) H(Π(BG),B nU(1)) H(BG,B nU(1))* * \begin{aligned} \mathbf{H}_{dR}(\mathbf{B}G, \mathbf{B}^n U(1)) & \simeq \mathbf{H}(\mathbf{\Pi}(\mathbf{B}G), \mathbf{B}^n U(1)) \prod_{\mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1))} * \\ & \simeq * \end{aligned}

and so the intrinsic universal curvature class

curv:H(BG,B nU(1))H dR(BG,B nU(1)) curv : \mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1)) \to \mathbf{H}_{dR}(\mathbf{B}G, \mathbf{B}^n U(1))

is trivial.

3d Dijkgraaf-Witten theory

Dijkgraaf-Witten theory is the analog of Chern-Simons theory for discrete structure groups. We show that this becomes a precise statement in SmoothGrpdSmooth \infty Grpd: the Dijkgraaf-Witten action functional is that induced from applying the \infty-Chern-Simons homomorphism to a characteristic class of the form DiscBGB 3U(1)Disc B G \to \mathbf{B}^3 U(1), for Disc:GrpdSmoothGrpdDisc : \infty Grpd \to Smooth \infty Grpd the canonical embedding of discrete ∞-groupoids into all smooth ∞-groupoids.

Lagrangian

Let GGrpGrpdDiscSmoothGrpdG \in Grp \to \infty Grpd \stackrel{Disc}{\to} Smooth \infty Grpd be a discrete group regarded as an ∞-group object in discrete ∞-groupoids and hence as a smooth ∞-groupoid with discrete smooth cohesion. Write BG=K(G,1)GrpdB G = K(G,1) \in \infty Grpd for its delooping in ∞Grpd and BG=DiscBG\mathbf{B}G = Disc B G for its delooping in Smooth∞Grpd.

We also write ΓB nU(1)K(U(1),n)\Gamma \mathbf{B}^n U(1) \simeq K(U(1), n). Notice that this is different from B nU(1)ΠBU(1)B^n U(1) \simeq \Pi \mathbf{B}U(1), reflecting the fact that U(1)U(1) has non-discrete smooth structure.

Proposition

For GG a discrete group, morphisms BGB nU(1)\mathbf{B}G \to \mathbf{B}^n U(1) correspond precisely to cocycles in the ordinary group cohomology of GG with coefficients in the discrete group underlying the circle group

π 0SmoothGrpd(BG,B nU(1))H Grp n(G,U(1)). \pi_0 Smooth\infty Grpd(\mathbf{B}G, \mathbf{B}^n U(1)) \simeq H^n_{Grp}(G,U(1)) \,.
Proof

By the (DiscΓ)(Disc \dashv \Gamma)-adjunction we have

SmoothGrpd(BG,B nU(1))Grpd(BG,K(U(1),n)). Smooth\infty Grpd(\mathbf{B}G, \mathbf{B}^n U(1)) \simeq \infty Grpd(B G, K(U(1),n)) \,.
Proposition

For GG discrete

  • the intrinsic de Rham cohomology of BG\mathbf{B}G is trivial

    SmoothGrpd(BG, dRB nU((1))*; Smooth \infty Grpd(\mathbf{B}G, \mathbf{\flat}_{dR}\mathbf{B}^n U((1)) \simeq * ;
  • all GG-principal bundles have a unique flat connection

    SmoothGrpd(X,BG)SmoothGrpd(Π(X),BG). Smooth\infty Grpd(X, \mathbf{B}G) \simeq Smooth\infty Grpd(\Pi(X), \mathbf{B}G) \,.
Proof

By the (DiscΓ)(Disc \dashv \Gamma)-adjunction and using that Γ dRK*\Gamma \circ \mathbf{\flat}_{dR} K \simeq * for all KK.

It follows that for GG discrete

  • any characteristic class c:BGB nU(1)\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1) is a group cocycle;

  • the \infty-Chern-Weil homomorphism coincides with postcomposition with this class

    H(Σ,BG)H(Σ,B nU(1)). \mathbf{H}(\Sigma, \mathbf{B}G) \to \mathbf{H}(\Sigma, \mathbf{B}^n U(1)) \,.
Proposition

For GG discrete and c:BGB 3U(1)\mathbf{c} : \mathbf{B}G \to \mathbf{B}^3 U(1) any group 3-cocycle, the \infty-Chern-Simons theory action functional on a 3-dimensional manifold Σ\Sigma

SmoothGrpd(Π(Σ),BG)U(1) Smooth\infty Grpd(\mathbf{\Pi}(\Sigma), \mathbf{B}G) \to U(1)

is the action functional of Dijkgraaf-Witten theory.

Proof

By proposition \ref{IntrinsicIntegrationTheorem} the morphism is given by evaluation of the pullback of the cocycle α:BGB 3U(1)\alpha : B G \to B^3 U(1) along a given :Π(Σ)BG\nabla : \Pi(\Sigma) \to B G, on the fundamental homology class of Σ\Sigma. This is the definition of the Dijkgraaf-Witten action (for instance equation (1.2) in FreedQuinn).

Obstruction theory

The flat Dijkgraaf-Witten circle 3-bundle on Σ\Sigma is the obstruction to lifting the GG-principal bundle to a G^\hat G-principal 2-bundle, where G^\hat G is the discrete 2-group classified by the group 3-cocycle.

(…)

4d Yetter model

See

Closed string field theory

For the moment see closed string field theory .

AKSZ theory

We consider symplectic Lie n-algebroids 𝔓\mathfrak{P} equipped with a canonical invariant polynomial and show that the ∞-Chern-Simons action functional associated to this data is the action functionalof the AKSZ theory sigma model with target space 𝔓\mathfrak{P}.

This is taken from (FRS11). See there for more details.

Lagrangian

Definition

A symplectic Lie n-algebroid (𝔓,ω)(\mathfrak{P}, \omega) is a Lie n-algebroid 𝔓\mathfrak{P} equipped with a binary non-degenerate invariant polynomial ωW(𝔓)\omega \in W(\mathfrak{P}) of degree n+2n+2.

This means that

  • on each coordinate chart UXU \to X of the base manifold XX of 𝔓\mathfrak{P}, there is a basis {x a}\{x^a\} for CE(𝔞| U)CE(\mathfrak{a}|_U) such that

    ω=ω abdx adx b \omega = \omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b

    with {ω abC (X)}\{\omega_{a b} \in \mathbb{R} \hookrightarrow C^\infty(X)\} and deg(x a)+deg(x b)=ndeg(x^a) + deg(x^b) = n;

  • the coefficient matrix {ω ab}\{\omega_{a b}\} has an inverse;

  • we have

    d W(𝔓)ω=d CE(𝔓)ω+dω=0. d_{\mathrm{W}(\mathfrak{P})} \omega = d_{\mathrm{CE}(\mathfrak{P})} \omega + \mathbf{d} \omega = 0 \,.

This \infty-Lie theoretic structure is essentially what in the literature is mostly considered in terms of symplectic dg-geometry :

Observation

We may think of an L-infinity-algebroid 𝔞\mathfrak{a} as a graded manifold XX whose global function ring is the graded algebra underlying the Chevalley-Eilenberg algebra

C (X):=CE(𝔞) C^\infty(X) := \mathrm{CE}(\mathfrak{a})

and which is equipped with a vector field v Xv_X of grade 1 whose graded Lie bracket with itself vanishes [v X,v X]=0[v_X, v_X] = 0, given, as a derivation, by the
differential on the Chevalley-Eilenberg algebra:

v X:=d CE(𝔞):CE(𝔞)CE(𝔞). v_X := d_{\mathrm{CE}(\mathfrak{a})} : \mathrm{CE}(\mathfrak{a}) \to \mathrm{CE}(\mathfrak{a}) \,.

The pair (X,v)(X,v) is a differential graded manifold . In this perspective the graded algebra underlying the Weil algebra of 𝔞\mathfrak{a} is the de Rham complex of XX

Ω (X):=W(𝔞), \Omega^\bullet(X) := \mathrm{W}(\mathfrak{a}) \,,

but the de Rham differential is just d\mathbf{d}, not the full differential d W(𝔞)=d+d CE(𝔞)d_{\mathrm{W}(\mathfrak{a})} = \mathbf{d} + d_{\mathrm{CE}(\mathfrak{a})} on the Weil algebra. The latter is thus a twisted de Rham differential on XX.

From this perspective all standard constructions of Cartan calculus usefully apply to L L_\infty-algebroids. Notably for vv any vector field on XX there is the contraction derivation

ι v:Ω (X)Ω 1(X) \iota_v : \Omega^\bullet(X) \to \Omega^{\bullet -1}(X)

and hence the Lie derivative

v:=[d,ι v]:Ω (X)Ω (X). \mathcal{L}_v := [\mathbf{d}, \iota_{v}] : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.

So in the above notation we have in particular

d W(𝔞)=d+ v X:W(𝔞)W(𝔞). d_{\mathrm{W}(\mathfrak{a})} = \mathbf{d} + \mathcal{L}_{v_X} : \mathrm{W}(\mathfrak{a}) \to \mathrm{W}(\mathfrak{a}) \,.
Definition

For XX a dg-manifold, let ϵΓ(TX)\epsilon \in \Gamma(T X) be the vector field which over any coordinate patch UXU \to X is given by the formula

ϵ| U= adeg(x a)x ax a, \epsilon|_U = \sum_a \mathrm{deg}(x^a) x^a \frac{\partial}{\partial x^a} \,,

where {x a}\{x^a\} is a basis of generators and deg(x a)\mathrm{deg}(x^a) the degree of a generator.

We write

N:=[d,ι ϵ] N := [\mathbf{d}, \iota_\epsilon]

for the Lie derivative of this vector field. The grade of a homogeneous element α\alpha in Ω (X)\Omega^\bullet(X) is the unique natural number nn \in \mathbb{N} with

ϵα=Nα=nα. \mathcal{L}_\epsilon \alpha = N \alpha = n \alpha \,.

Remark.

  • This implies that for x ix^i an element of grade nn on UU, the 1-form dx i\mathbf{d}x^i is also of grade nn. This is why we speak of grade (as in “graded manifold”) instead of degree here.

  • The above is indeed well-defined: on overlaps of patches the {x a}\{x^a\} of positive degree/grade transform by a degreewise linear transformation, which manifestly preserves adeg(x a)x ax a\sum_a \mathrm{deg}(x^a) x^a \frac{\partial}{\partial x^a}. Notice that the ordinary (degree-0 coordinates) do not appear in this formula. And indeed the vector field locally defined by ax ax a\sum_a x^a \frac{\partial}{\partial x^a} (thus including the coordinates of grade 0) does not in general exist globally.

The existence of ϵ\epsilon implies the following useful statement, which is a trivial variant of what in grade 0 would be the standard Poincare lemma.

Observation

On a graded manifold every closed differential form ω\omega of positive grade nn is exact: the form

λ:=1nι ϵω \lambda := \frac{1}{n} \iota_\epsilon \omega

satisfies

dλ=ω. \mathbf{d}\lambda = \omega \,.

Using this differential geometric language we can now capture something very close to def. 1 in more traditional symplectic geometry terms.

Definition

A symplectic dg-manifold of grade nn \in \mathbb{N} is a dg-manifold (X,v)(X,v) equipped with 2-form ωΩ 2(X)\omega \in \Omega^2(X) which is

  • \item non-degenerate;
  • closed;

as usual for symplectic forms, and in addition

  • of grade nn;
  • vv-invariant: vω=0\mathcal{L}_v \omega = 0.

Example. It follows that a symplectic dg-manifold of grade 0 is the same as an ordinary symplectic manifold. In the following we are mostly interested in the case of positive grade.

Observation

The function algebra of a symplectic dg-manifold (X,ω)(X,\omega) of grade nn is naturally equipped with a Poisson bracket

{,}:C (X)C (X)C (X) \{-,-\} : C^\infty(X)\otimes C^\infty(X) \to C^\infty(X)

which decreases grade by nn. On a local coordinate patch this is given by

{f,g}=fx aω abgx b, \{f,g\} = \frac{\partial f}{\partial x^a} \omega^{a b} \frac{\partial g}{\partial x^b} \,,

where {ω ab}\{\omega^{a b}\} is the inverse matrix to {ω ab}\{\omega_{a b}\}.

Observation

For fC (X)f \in C^\infty(X) and vΓ(TX)v \in \Gamma(T X) we say that fisaHamiltonianfor is a Hamiltonian for v_ or equivalently that _visthe[[nLab:Hamiltonianvectorfield]]of is the [[nLab:Hamiltonian vector field]] of f$ if

df=ι vω. \mathbf{d}f = \iota_v \omega \,.
Proposition

There is a full and faithful embedding of symplectic dg-manifolds of grade nn into symplectic Lie nn-algebroids.

Proof

The dg-manifold itself is identified with an L L_\infty-algebroid as in observation 6. For ωΩ 2(X)\omega \in \Omega^2(X) a symplectic form, the conditions dω=0\mathbf{d} \omega = 0 and vω=0\mathcal{L}_v \omega = 0 imply (d+v)ω=0(\mathbf{d}+ \mathcal{L}v)\omega = 0 and hence that under the identification Ω (X)W(𝔞)\Omega^\bullet(X) \simeq \mathrm{W}(\mathfrak{a}) this is an invariant polynomial on 𝔞\mathfrak{a}.

It remains to observe that the L L_\infty-algebroid 𝔞\mathfrak{a} is in fact a Lie nn-algebroid. This is implied by the fact that ω\omega is of grade nn and non-degenerate: the former condition implies that it has no components in elements of grade gtngt n and the latter then implies that all such elements vanish.

Proposition

Let (𝔓,ω)(\mathfrak{P},\omega) be a symplectic Lie nn-algebroid for positive nn in the image of the embedding of prop. 10. Then it carries the canonical L L_\infty-algebroid cocycle

π:=1n+1ι ϵι vωCE(𝔓) \pi := \frac{1}{n+1} \iota_\epsilon \iota_v \omega \in \mathrm{CE}(\mathfrak{P})

which moreover is the Hamiltonian, according to def. 9, of d CE(𝔓)d_{\mathrm{CE}(\mathfrak{P})}.

Proof

The required condition dπ=ι vω\mathbf{d}\pi = \iota_v \omega from def. 9 holds by observation 7.

Our central observation now is the following.

Proposition

The cocycle π\pi from prop. 11 is in transgression with the invariant polynomial nωn \omega. A Chern-Simons element witnessing the transgression according to def. \ref{TransgressionAndCSElements} is

cs=ι ϵω+π. \mathrm{cs} = \iota_\epsilon \omega + \pi \,.
Proof

It is clear that i *cs=πi^* \mathrm{cs} = \pi. So it remains to check that d W(𝔓)cs=nωd_{\mathrm{W}(\mathfrak{P})} \mathrm{cs} = n\omega. Notice that

[d CE(𝔓),ι ϵ]=[ v,ι ϵ]=ι [v,ϵ]=ι v [d_{\mathrm{CE}(\mathfrak{P})}, \iota_\epsilon] = [\mathcal{L}_v, \iota_\epsilon] = \iota_{[v,\epsilon]} = - \iota_{v}

by Cartan calculus. Using this we compute the first summand in d W(𝔓)(ι ϵω+π)d_{\mathrm{W}(\mathfrak{P})} ( \iota_{\epsilon} \omega + \pi ):

d W(𝔓)ι ϵω =(d+d CE(𝔓))ι ϵω =nω+[d CE(𝔓),ι ϵ]ω =nωι vω =nωdπ. \begin{aligned} d_{\mathrm{W}(\mathfrak{P})} \iota_{\epsilon} \omega & = ( \mathbf{d} + d_{\mathrm{CE}(\mathfrak{P})} ) \iota_\epsilon \omega \\ &= n \omega + [d_{\mathrm{CE}(\mathfrak{P})}, \iota_\epsilon] \omega \\ &= n\omega - \iota_v \omega \\ & = n \omega - \mathbf{d}\pi \end{aligned} \,.

The second summand is simply

d W(𝔓)π=dπ d_{\mathrm{W}(\mathfrak{P})} \pi = \mathbf{d}\pi

since π\pi is a cocycle.

Proposition

For (𝔓.ω)(\mathfrak{P}. \omega) a symplectic Lie nn-algebroid coming from a symplectic dg-manifold by prop. 10, the higher Chern-Simons action functional associated with its canonical Chern-Simons element cs\mathrm{cs} from prop. 12 is the AKSZ Lagrangean:

L AKSZ=cs. L_{\mathrm{AKSZ}} = \mathrm{cs} \,.
Proof

We work in local coordinates {x a}\{x^a\} where

ω=12ω abdx adx b \omega = \frac{1}{2}\omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b

and the Chern-Simons element is

cs= aω abdeg(x a)x adx b+π. \mathrm{cs} = \sum_a \omega_{a b} \mathrm{deg}(x^a) x^a \wedge \mathbf{d}x^b + \pi \,.

We want to substitute here d=d Wd CE\mathbf{d} = d_{\mathrm{W}}- d_{\mathrm{CE}}. Notice that in coordinates the equation

dπ=ι vω \mathbf{d}\pi = \iota_v \omega

becomes

dx aπx a =ω abv adx b =ω abdx av b. \begin{aligned} \mathbf{d}x^a \frac{\partial \pi}{\partial x^a} & = \omega_{a b} v^a \wedge \mathbf{d} x^b \\ & = \omega_{a b} \mathbf{d}x^a \wedge v^b \end{aligned} \,.

Therefore

aω abdeg(x a)x ad CEx b = aω abdeg(x a)x av b = adeg(x a)x aπx a =(n+1)π. \begin{aligned} \sum_a \omega_{a b} \mathrm{deg}(x^a) x^a \wedge d_{\mathrm{CE}} x^b & = \sum_a \omega_{a b} \mathrm{deg}(x^a) x^a \wedge v^b \\ & = \sum_a \mathrm{deg}(x^a)x^a \frac{\partial \pi}{\partial x^a} \\ &= (n+1) \pi \end{aligned} \,.

Hence

cs= abdeg(x a)ω abx adx bnπ. \mathrm{cs} = \sum_{a b} \mathrm{deg}(x^a) \,\omega_{a b} x^a \wedge \mathbf{d}x^b - n \pi \,.

This means that for Σ\Sigma an (n+1)(n+1)-dimensional manifold and

Ω (Σ)W(𝔓):X \Omega^\bullet(\Sigma) \leftarrow \mathrm{W}(\mathfrak{P}) : X

a 𝔓\mathfrak{P}-valued differential form on Σ\Sigma we have

cs(X) = a,bdeg(x a)ω abX ad dRX bnΠ(X). \begin{aligned} \mathrm{cs}(X) &= \sum_{a,b} \mathrm{deg}(x^a)\,\omega_{a b} X^a \wedge d_{\mathrm{dR}} X^b - n \Pi(X) \end{aligned} \,.

This is indeed L AKSZ(X)L_{\mathrm{AKSZ}}(X).

Remark The AKSZ σ\sigma-model action functional interpretation of \infty-Chern-Weil functionals for binary invariant polynomials on L L_\infty-algebroids from prop. 13 gives rise to the following dictionary of concepts\

ChernWeiltheory quantumfieldtheory cocycle π Hamiltonian transgressionelement cs Lagrangean curvaturecharacteristic ω symplecticstructure. \array{ Chern-Weil theory && quantum field theory \\ \\ cocycle & \pi & Hamiltonian \\ \\ transgression element & cs & Lagrangean \\ \\ curvature characteristic & \omega & symplectic structure } \,.

Covariant phase space

Proposition

The covariant phase space of AKSZ theory with target (𝔓,ω)(\mathfrak{P}, \omega) is the space of those ∞-Lie algebroid-valued forms AA whose curvature (n+1)(n+1)-form F AF_A vanishes

P={AΩ 1(Σ,𝔤)|F A=0}. P = \{A \in \Omega^1(\Sigma, \mathfrak{g}) | F_A = 0\} \,.

The presymplectic structure on this space is

ω:(δA 1,δA 2) Σω(δA 1,δA 2). \omega : (\delta A_1, \delta A_2) \mapsto \int_{\partial \Sigma} \omega(\delta A_1, \delta A_2 ) \,.
Proof

This is a special case of prop. \ref{TheEquationsOfMotion}, prop. \ref{PresymplecticStructure} in view of corollary \ref{CovariantPhaseSpaceForBinaryNonDegenerateInvariantPolynomial}, using that, by definition of symplectic Lie n-algebroid, ω\omega is a binary and non-degenerate invariant polynomial.

n=0n=0 – The topological particle

For XX a smooth manifold we may regard its cotangent bundle 𝔞=T *X\mathfrak{a} = T^* X as a Lie 0-algebroid and the canonical 2-form ωW(𝔞)=Ω (X)\omega \in W(\mathfrak{a}) = \Omega^\bullet(X) as a binary invariant polynomial in degree 2.

The Chern-Simons element is the canonical 1-form α\alpha which in local coordinates is α=p idq i\alpha = p_i d q^i.

The corresponding action functional on the line

γ *(p idq i) \int_{\mathbb{R}} \gamma^* (p_i\, d q^i)

is the familiar term for the action functional of the particle (missing the kinetic term, which makes it “topological”).

n=1n=1 – The Poisson σ\sigma-model

Let (X,{,})(X, \{-,-\}) be a Poisson manifold. Over a Darboux chart the corresponding Poisson Lie algebroid has coordinates {x i}\{x^i\} of degree 0 and i\partial_i of degree 1. We have

d Wx i=π ij j+dx i d_{\mathrm{W}} x^i = -\pi^{i j}\mathbf{\partial}_j + \mathbf{d}x^i

where π ij:={x i,x j}\pi^{i j} := \{x^i , x^j\} and

ω=dx id i. \omega = \mathbf{d}x^i \wedge \mathbf{d}\partial_i \,.

The Hamiltonian cocycle from prop. 11 is

π =ι vι ϵω =ι v idx i = i[ι v,d]x i = i[d,ι v]x i =+ iπ ij j \begin{aligned} \pi &= \iota_v \iota_\epsilon \omega \\ &= \iota_v \partial_i \wedge \mathbf{d}x^i \\ & = \partial_i \wedge [\iota_v,\mathbf{d}]x^i \\ &= -\partial_i \wedge [\mathbf{d},\iota_v]x^i \\ &= + \partial_i \pi^{ij}\partial_j \end{aligned}

and the Chern-Simons element from prop. 12 is

cs =ι ϵω+π = idx i+π ij i j. \begin{aligned} \mathrm{cs} &= \iota_\epsilon \omega + \pi \\ &= \partial_i \wedge \mathbf{d}x^i + \pi^{ij}\partial_i \partial_j \end{aligned} \,.

In terms of d Wd_{\mathrm{W}} instead of d\mathbf{d} this is

= i(d Wd CE)x i+π ij i j = idx i+2π ij i j \begin{aligned} \cdots & = \partial_i \wedge (d_{\mathrm{W}} - d_{\mathrm{CE}}) x^i + \pi^{ij}\partial_i \partial_j \\ &= \partial_i \wedge \mathbf{d}x^i + 2 \pi^{ij}\partial_i \partial_j \end{aligned}

So for

Ω (Σ)W(𝔓):(X,η) \Omega^\bullet(\Sigma) \leftarrow \mathrm{W}(\mathfrak{P}) : (X,\eta)

a Poisson-Lie algebroid valued differential form – which in components is a function ϕ:ΣX\phi: \Sigma \to X and a 1-form ηΩ 1(Σ,ϕ *T *X)\eta \in \Omega^1(\Sigma, \phi^* T^* X) – the corresponding Chern-Simons form is

cs(X,η)=d dRXη+2π(ηη). \mathrm{cs}(X,\eta) = \langle d_{\mathrm{dR}}X \wedge \eta \rangle + 2 \pi(\eta \wedge \eta) \,.

This is the Lagrangean of the Poisson σ\sigma-model [CattaneoFelder].

n=2n=2 – Ordinary Chern-Simons theory

We show how the ordinary Chern-Simons form arises from this perspective. So let 𝔞=𝔤\mathfrak{a} = \mathfrak{g} be a semisimple Lie algebra and ω:=,W(𝔤)\omega := \langle -,-\rangle\in \mathrm{W}(\mathfrak{g}) its Killing form invariant polynomial. For {t a}\{t^a\} a dual basis for 𝔤\mathfrak{g} we have

d Wt a=12C a bct at b+dt a d_{\mathrm{W}} t^a = - \frac{1}{2}C^a{}_{b c} t^a \wedge t^b + \mathbf{d}t^a

where C a bc:=t a([t b,t c])C^a{}_{b c} := t^a([t_b,t_c]) and

ω=12P abdt adt b, \omega = \frac{1}{2} P_{a b} \mathbf{d}t^a \wedge \mathbf{d}t^b \,,

where P ab:=t a,t bP_{ab} := \langle t_a, t_b \rangle. The Hamiltonian cocycle π\pi from prop. 11 is

π =12+1ι ϵι vω =13ι vι ϵω =13ι vP abt adt b =13P abt a[ι v,d]t b =13P abt a[d,ι v]t b =13P abt a(12)C b det dt e =+16C abct at bt c. \begin{aligned} \pi & = \frac{1}{2+1}\iota_\epsilon \iota_v \omega \\ & = \frac{1}{3} \iota_v \iota_\epsilon \omega \\ & = \frac{1}{3}\iota_v P_{a b} t^a \wedge \mathbf{d}t^b \\ & = \frac{1}{3} P_{a b} t^a \wedge [\iota_v,\mathbf{d}]t^b \\ & = -\frac{1}{3} P_{a b} t^a \wedge [\mathbf{d}, \iota_v]t^b \\ &= -\frac{1}{3} P_{a b} t^a \wedge (-\frac{1}{2})C^b{}_{d e} t^d \wedge t^e \\ & = +\frac{1}{6} C_{abc}t^a \wedge t^b \wedge t^c \end{aligned} \,.

Therefore in this case the Chern-Simons element from def. 12 becomes

cs =ι ϵω+π =P abt adt b+16C abct at bt c. \begin{aligned} \mathrm{cs} & = \iota_\epsilon \omega + \pi \\ & = P_{a b} t^a \wedge \mathbf{d}t^b + \frac{1}{6} C_{abc} t^a \wedge t^b \wedge t^c \end{aligned} \,.

This is indeed the familiar standard choice of Chern-Simons element on a Lie algebra. Notice that evaluated on a 𝔤\mathfrak{g}-valued form

Ω (Σ)W(𝔤):A \Omega^\bullet(\Sigma) \leftarrow \mathrm{W}(\mathfrak{g}) : A

this is

cs(A)=AF A+16A[AA]. \mathrm{cs}(A) = \langle A \wedge F_A\rangle + \frac{1}{6}\langle A \wedge [A \wedge A]\rangle \,.

If 𝔤\mathfrak{g} is a matrix Lie algebra then the Killing form is proportional to the trace of the matrix product: t a,t b=tr(t at b)\langle t_a,t_b\rangle = \mathrm{tr}(t_a t_b). In this case we have

A[AA] =A aA bA ctr(t a(t bt ct ct b)) =2A aA bA ctr(t at bt c) =2tr(AAA) \begin{aligned} \langle A \wedge [A \wedge A]\rangle &= A^a \wedge A^b \wedge A^c \,\mathrm{tr}(t_a (t_b t_c - t_c t_b)) \\ &= 2 A^a \wedge A^b \wedge A^c \,\mathrm{tr}(t_a t_b t_c ) \\ &= 2 \,\mathrm{tr}(A \wedge A \wedge A) \end{aligned}

and hence

cs(A)=tr(AF A)+13tr(AAA). \mathrm{cs}(A) = \mathrm{tr}(A \wedge F_A) + \frac{1}{3}\,\mathrm{tr}(A \wedge A \wedge A) \,.

Often this is written in terms of the de Rham differential 2-form d dRAd_{\mathrm{dR}} A instead of the curvature 2-form F A:=d dRA+12[AA]F_A := d_{\mathrm{dR}} A + \frac{1}{2}[A \wedge A]. Since the former is the image under AA of d W(𝔤)d_{\mathrm{W}(\mathfrak{g})} we can alternatively write

cs =P abt a(d Wd CE)t b+16C abct at bt c =P abt ad Wt bP abt a(12)C b cdt ct d+16C abct at bt c =P abt ad Wt b+(12+16)C abct at bt c =P abt ad Wt b+23C abct at bt c. \begin{aligned} \mathrm{cs} &= P_{a b} t^a \wedge (d_{\mathrm{W}} - d_{\mathrm{CE}})t^b + \frac{1}{6} C_{abc} t^a \wedge t^b \wedge t^c \\ &= P_{a b} t^a \wedge d_{\mathrm{W}} t^b - P_{ab} t^a \wedge (-\frac{1}{2})C^b{}_{cd}t^c \wedge t^d + \frac{1}{6} C_{abc} t^a \wedge t^b \wedge t^c \\ & = P_{ab} t^a \wedge d_{\mathrm{W}}t^b + (\frac{1}{2}+\frac{1}{6}) C_{abc} t^a \wedge t^b \wedge t^c \\ & = P_{ab} t^a \wedge d_{\mathrm{W}}t^b + \frac{2}{3}C_{abc} t^a \wedge t^b \wedge t^c \end{aligned} \,.

Hence

cs(A)=Ad dRA+23A[AA]. \mathrm{cs}(A) = \langle A \wedge d_{\mathrm{dR}} A\rangle + \frac{2}{3} \langle A \wedge [A \wedge A]\rangle \,.

n=2n=2 – The Courant σ\sigma-model

(…)

Supergravity

Lagrangians

Remark

By this discussion, what is called the cosmo-cocycle condition in the D'Auria-Fre formulation of supergravity is the condition that while the Lagrangian L_ is not quite a Chern-Simons form, we have nevertheless that its Weil algebra differential

d W(𝔤)L sugraW(𝔤)λ 2(𝔤 *[1]) d_{W(\mathfrak{g})} L_{sugra} \in W(\mathfrak{g})\otimes \lambda^2 (\mathfrak{g}^*[1])

at least biliear in the curvatures (the shifted components).

While for this case the argument of prop. \ref{TheEquationsOfMotion} does not give a closed formula for the full equations of motions, but it still implies that field configurations FF with vanishing curvature do solve the equations of motion. Hence that

F A=0 F_A = 0

is a sufficient condition for AA to be a point in the covariant phase space.

This is supposedly the reason for the terms “cosmo cocycle” for this condiiton: it ensures in supergravity that that flat Minkowski space with all fields vanishing is a (“cosmological”) solution.

References

The notion of Chern-Simons elements for L L_\infty-algebras and the associated imnfty\imnfty-Chern-Simons Lagrangians is due to

The induced construction of the ∞-Chern-Weil homomorphism with special attention to the Chern-Simons circle 3-bundle and the Chern-Simons circle 7-bundle is in

In the general context of cohesive (∞,1)-toposes \infty-Chern-Simons theory is discussed in section 4.3 of

The case of the AKSZ sigma-model is discussed in

Discussion of symplectic Lie n-algebroids is in

  • Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis (arXiv)

    On the structure of graded symplectic supermanifolds and Courant algebroids (arXiv)

Last revised on January 22, 2013 at 13:16:35. See the history of this page for a list of all contributions to it.