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 \cite{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)

Revised on January 22, 2013 13:16:35 by Urs Schreiber (89.204.138.238)