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Chern-Simons element

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

\infty-Chern-Weil theory

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A Chern-Simons element on an L-∞ algebroid (named after Shiing-shen Chern and James Simons who considered this for semisimple Lie algebras) is an element of its Weil algebra that exhibits a transgression between an ∞-Lie algebroid cocycle and an invariant polynomial.

It is construct that arises in the presentation of the ∞-Chern-Weil homomorphism by an ∞-anafunctor of simplicial presheaves.

Definition

We discuss ∞-Lie algebras and ∞-Lie algebroids 𝔞\mathfrak{a} of finite type in terms of their Chevalley-Eilenberg algebras CE(𝔞)CE(\mathfrak{a}). For \infty-Lie algebras these are objects in the category dgAlg of dg-algebras (over a given ground field). For \infty-Lie algebroids these are dg-algebras equipped with a lift of the degree-0 algebra to an algebra over a given Fermat theory TT and such that the differential is a TT-derivation in this degree. (See ∞-Lie algebroid for details). We shall write in the following dgAlgdgAlg also for the category of dg-algebras with this extra structure and leave the Fermat theory TT implicit.

Definition

For 𝔤\mathfrak{g} an ∞-Lie algebra or more generally ∞-Lie algebroid, μCE(𝔤)\mu \in CE(\mathfrak{g}) a ∞-Lie algebra cocycle (a closed element of the Chevalley-Eilenberg algebra) and W(𝔤)\langle - \rangle \in W(\mathfrak{g}) an invariant polynomial, a Chern-Simons element exhibiting the transgression between the two is an element

csW(𝔤) cs \in W(\mathfrak{g})

such that

  1. we have d W(𝔤)cs=d_{W(\mathfrak{g})} cs = \langle -\rangle

  2. and cs CE(𝔤)=μcs|_{CE(\mathfrak{g})} = \mu

where the restriction is along the canonical morphism W(𝔤)CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}).

Remark

Notice that a degree-nn ∞-Lie algebroid cocycle μ\mu is equivalently a morphism

CE(𝔤)CE(b n1):μ CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathbb{R}) : \mu

and an invariant polynomial of degree n+1n+1 is equivalently a morphism

int(𝔤)inv(b n1)=CE(b n): int(\mathfrak{g}) \leftarrow inv(b^{n-1}\mathbb{R}) = CE(b^n \mathbb{R}) : \langle - \rangle

in dgAlg.

Observation

A Chern-Simons element cscs an 𝔞\mathfrak{a} witnessing the transgression of \langle - \rangle to μ\mu is equvivalently a morphism

W(𝔤)W(b n1):cs W(\mathfrak{g}) \leftarrow W(b^{n-1} \mathbb{R}) : cs

such that we have a commuting diagram in dgAlg

(1)CE(𝔤) μ CE(b n1) cocycle i * W(𝔤) cs W(b n1) ChernSimonselement inv(𝔤) inv(b n1) invariantpolynomial, \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1} \mathbb{R}) &&& cocycle \\ \uparrow && \uparrow^{\mathrlap{i^*}} \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{n-1} \mathbb{R}) &&& Chern-Simons\;element \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& inv(b^{n-1} \mathbb{R}) &&& invariant polynomial } \,,

where the vertical morphisms are the canonical ones.

Remark

If we think of

  • W(𝔤)W(\mathfrak{g}) as differential forms on the total space of the universal GG-bundles;

  • CE(𝔤)CE(\mathfrak{g}) as differential forms on the fiber

  • inv(𝔤)inv(\mathfrak{g}) as differential forms on the base space

then the abov expresses the classical notion of transgression of forms from the fiber to the base of a fibe bundle (for instance Borel, section 9).

Properties

Uniqueness

Observation

For a given transgressive cocycle μ\mu and transgressing invariant polynomial \langle - \rangle the set of Chern-Simons elements witnessing the transgression is a torsor (based over the point and) over the additive group

{ωW(𝔤)d W(𝔤)ω=0,i *ω=0} \{\omega \in W(\mathfrak{g}) | d_{W(\mathfrak{g}) \omega = 0, i^* \omega = 0}\}

of Chern-Simons elements for vanishing cocycle and vanishing invariant polynomial.

Canonical \infty-Chern-Simons elements

Since the Weil algebra of an L-∞ algebra has trivial cohomolgy in positive degree, every invariant polynomial ,,\langle -,\cdots, -\rangle has a Chern-Simons element and there is a standard formula for it.

Proposition

Let 𝔤\mathfrak{g} be an L-∞ algebra with kk-ary brackets [,,] k:𝔤 k𝔤[-,\cdots, -]_k : \mathfrak{g}^{\otimes k} \to \mathfrak{g} and equipped with a quadratic invariant polynomial ,\langle -,-\rangle.

A Chern-Simons element for ,\langle-,-\rangle is given by the formula

cs(A)=A,d dRA+ k=1 2(k+1)!A,[A,,A] k, cs(A) = \langle A, d_{dR} A\rangle + \sum_{k = 1}^\infty \frac{2}{(k+1)!} \langle A, [A,\cdots,A]_k\rangle \,,

where A:W(𝔤)Ω (Σ)A : W(\mathfrak{g}) \to \Omega^\bullet(\Sigma) is any 𝔤\mathfrak{g}-valued form

Proof

There is a canonical contracting homotopy

τ:W(𝔤)W(𝔤) \tau : W(\mathfrak{g}) \to W(\mathfrak{g})

satisfying [d W,τ]=Id[d_W, \tau] = Id

and the above element is

cs=τ,. cs = \tau \langle -,-\rangle \,.

To see this, let {t a}\{t_a\} be a basis and {t a}\{t^a\} the dual basis. Then the differential of the Chevalley-Eilenberg algebra can be written

d CE(𝔤)t a= k=1 1k![t a 1,t a 2,,t a k] at a 1t a 2t a k, d_{CE(\mathfrak{g})} t^a = - \sum_{k = 1}^\infty \frac{1}{k!} [t_{a_1}, t_{a_2}, \cdots, t_{a_k}]^a \,\, t^{a_1} \wedge t^{a_2}\wedge \cdots t^{a_k} \,,

where

[,,,]:𝔤 k𝔤 [-,-, \cdots, -] : \mathfrak{g}^{\otimes_k} \to \mathfrak{g}

is the corresponding kk-ary bracket.

Write

P ab:=t a,t b, P_{a b} := \langle t_a , t_b\rangle \,,

for the components of the invariant polynomial in this basis.

Then the claim is that

cs=2P abt ad W(𝔤)t b+ k=1 C ab 1,,b kt at b 1t b k, cs = 2 P_{a b} t^a \wedge d_{W(\mathfrak{g})} t^b + \sum_{k = 1}^\infty C_{a b_1, \cdots, b_k} \,\, t^a \wedge t^{b_1} \wedge \cdots \wedge t^{b_k} \,,

where the coefficients are

C ab 1,,b k:=1(k+1)!(P ab[t b 1,,t b k] b) C_{a b_1, \cdots, b_k} := \frac{1}{(k+1)!} (P_{a b} [t_{b_1}, \cdots, t_{b_k}]^b)

Write F(𝔤)F(\mathfrak{g}) for the free dg-algebra on the graded vector space 𝔤 *\mathfrak{g}^*. In terms of the above basis this is generated from {t a,dt a}\{t^a, \mathbf{d}t^a\}. As discussed at Weil algebra, there is a dg-algebra isomorphism

F(𝔤)W(𝔤) F(\mathfrak{g}) \stackrel{\simeq}{\to} W(\mathfrak{g})

given by sending t at at^a \mapsto t^a and dt ad CEt a+r a\mathbf{d}t^a \mapsto d_{CE} t^a + r^a.

Let h:F(𝔤)F(𝔤)h : F(\mathfrak{g}) \to F(\mathfrak{g}) be the derivation which on generators is defined by

h:t a0 h : t^a \mapsto 0
h:dt at a. h : \mathbf{d}t^a \mapsto t^a \,.

Notice that this is not the homotopy that exhibits the triviality of Id F(𝔤 *)Id_{F(\mathfrak{g}^*)}, rather that homotopy is 1Lh\frac{1}{L} h, where LL is the word length operator for element in F(𝔤 *)F(\mathfrak{g}^*) in terms of the generators {t a,dt a}\{t^a , \mathbf{d}t^a\}.

Therefore the homotopy τ\tau is the composite top morphism in the diagram

W(𝔤) τ W(𝔤) F(𝔤) 1Lh F(𝔤). \array{ W(\mathfrak{g}) &\stackrel{\tau}{\to}& W(\mathfrak{g}) \\ \downarrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}} \\ F(\mathfrak{g}) &\stackrel{\frac{1}{L} h}{\to}& F(\mathfrak{g}) } \,.

Unwinding this, we find

cs :=τ(P abr ar b) =τ(d W(𝔤)t a+ k=1 [t a 1,,t a k] at a 1t a k)(d W(𝔤)t b+ k=1 [t b 1,,t b k] bt b 1t b k) =P abt ad W(𝔤)t b+ k=1 2k!(k+1)P ab[t b 1,t b k] bt b 1t b k. \begin{aligned} cs & := \tau \left( P_{a b} r^{a} \wedge r^b \right) \\ & = \tau \left( d_{W(\mathfrak{g})} t^a + \sum_{k = 1}^\infty [t_{a_1}, \cdots, t_{a_k}]^a t^{a_1} \wedge \dots t^{a_k} \right) \wedge \left( d_{W(\mathfrak{g})} t^b + \sum_{k = 1}^\infty [t_{b_1}, \cdots, t_{b_k}]^b t^{b_1} \wedge \dots t^{b_k} \right) \\ & = P_{a b} t^a \wedge d_{W(\mathfrak{g})} t^b + \sum_{k = 1}^\infty \frac{2}{k! (k+1) } P_{a b} [t_{b_1}, \cdots t_{b_k}]^b t^{b_1} \wedge \cdots \wedge t^{b_k} \end{aligned} \,.
Example

We consider the ordinary Chern-Simons element as an example of this formula: let 𝔤\mathfrak{g} be a semisimple Lie algebra and ,\langle -,-\rangle the Killing form invariant polynomial. Then the above computation gives

cs =τ(P abr ar b) =P ab(d Wt a+12C a a 1a 2t a 1t a 2)(d Wt b+12C b b 1b 2t b 1t b 2) =P abt ad Wt b+22!3P abt aC b 1b 2 bt b 1t b 2 =P abt ad Wt b+13C abct at bt c. \begin{aligned} cs & = \tau \left( P_{a b} r^a \wedge r^b \right) \\ & = P_{a b} \left(d_W t^a + \frac{1}{2}C^a{}_{a_1 a_2}t^{a_1} \wedge t^{a_2} \right) \wedge \left(d_W t^b + \frac{1}{2}C^b{}_{b_1 b_2}t^{b_1} \wedge t^{b_2} \right) \\ & = P_{a b} t^a \wedge d_W t^b + \frac{2}{2! 3} P_{a b} t^a \wedge C^b_{b_1 b_2} t^{b_1} \wedge t^{b_2} \\ & = P_{a b} t^a \wedge d_W t^b + \frac{1}{3} C_{a b c} t^a \wedge t^{b} \wedge t^{c} \end{aligned} \,.

Origin and relation to other concepts

We discuss the general abstract structures of which Chern-Simons elements are presentations and how they are related to other structures.

The term Chern-Simons element alludes to the term Chern-Simons form and Chern-Simons theory. In the following we explain the relation.

As presentations for the \infty-Chern-Weil homomorphism

We explain here briefly how Chern-Simons elements provide a presentation of a generalization of the Chern-Weil homomorphism – the ∞-Chern-Weil homomorphism in cohesive (∞,1)-topos theory – in the sense in which (∞,1)-toposes have presentations by a model structure on simplicial presheaves.

To warm up, we start with considering a traditional setup of Lie groupoid theory. Recall that for GG a Lie group, we may form its delooping Lie groupoid dnoted *//G*//G or BG\mathbf{B}G. Then with XX any smooth manifold, we have that the groupoid of morphisms of Lie groupoids XBGX \to \mathbf{B}G is equivalent to that of GG-principal bundles on XX:

SmoothGrpd(X,BG)GBund(X). SmoothGrpd(X, \mathbf{B}G) \simeq G Bund(X) \,.

Here we are thinking of Lie groupoids as differentiable stacks, hence as objects in the (2,1)-topos

SmoothGrpd:=Sh (2,1)(SmoothMfd) SmoothGrpd := Sh_{(2,1)}(SmoothMfd)

of stacks/(2,1)-sheaves on the site SmoothMfd (equivalently on its small dense subsite CartSp of Cartesian spaces). (This is discussed in detail at principal bundle ).

There is a differential refinement of the Lie groupoid BG\mathbf{B}G, to the smooth groupoid

BG conn:=SmoothGrpd(P 1(),BG), \mathbf{B}G_{conn} := SmoothGrpd(\mathbf{P}_1(-), \mathbf{B}G) \,,

where P 1(X)\mathbf{P}_1(X) is the path groupoid of XX. This is the (2,1)-sheaf given by the (2,1)-sheafification of the assignment that sends a smooth manifold UU to the groupoid of Lie algebra-valued 1-forms on UU.

There is a corresponding natural equivalence

SmoothGrpd(X,BG conn)GBund conn(X) SmoothGrpd(X, \mathbf{B}G_{conn}) \simeq G Bund_{conn}(X)

of morphisms into BG conn\mathbf{B}G_{conn} with the groupoid of GG-principal bundles with with connection on XX. (This is described in detail at connection on a bundle ).

In particular if G=U(1)G = U(1) is the circle group, a morphism XBU(1) connX\to \mathbf{B}U(1)_{conn} is a circle bundle with connection. This

This allows already to consider a simple case of a characteristic class and its refinement to a differential characteristic class: Let UU be the unitary group. There is a canonical morphism of Lie groupoids c 1:BUBU(1)\mathbf{c}_1 : \mathbf{B}U \to \mathbf{B}U(1) given by the determinant. This – or rather its image in cohomology

c 1:SmoothGrpd(,BU)SmoothGrpd(,BU(1)) \mathbf{c}_1 : SmoothGrpd(- ,\mathbf{B}U) \to SmoothGrpd(-, \mathbf{B} U(1))

is a smooth representative of the characteristic class called the first Chern class . Its differential refinement is the evident morphism

c^ 1:BU connBU(1) conn \hat \mathbf{c}_1 : \mathbf{B}U_{conn} \to \mathbf{B}U(1)_{conn}

that sends a 𝔲\mathfrak{u}-valued differential form to the trace of its Lie algebra value. Postcomposition with this is the refined Chern-Weil homomorphism

SmoothGrpd(X,BU) conn c^ 1 SmoothGrpd(X,BU(1) conn) UBund (X) U(1)Bund (X) \array{ SmoothGrpd(X, \mathbf{B}U)_{conn} &\stackrel{\hat \mathbf{c}_1}{\to}& SmoothGrpd(X, \mathbf{B}U(1)_{conn}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ U Bund_\nabla(X) &\to& U(1) Bund_\nabla(X) }

with values in circle bundles with connection, hence in degree-2 ordinary differential cohomology.

It is this kind of construction on Lie groupoids that we now want to generalize to a notion of smooth ∞-groupoids, to see that Chern-Simons elements are a means to constructi morphisms akind to the differential first Chern-class c^ 1\hat \mathbf{c}_1.

A general abstract context for higher geometry equipped with differential cohomology is a cohesive (∞,1)-topos H\mathbf{H} of ∞-groupoids equipped with cohesive structure , such as smooth cohesive structure.

An example for such is the ∞-stack-analog of the stack-(2,1)-topos over SmoothMfd: the ∞-stack (∞,1)-topos Smooth∞Grpd :=Sh^ (,1)(SmoothMfd):= \hat Sh_{(\infty,1)}(SmoothMfd).

In that context we have for instance all the higher deloopings of U(1)U(1): the circle Lie (n+1)-groups

B nU(1)SmoothGrpd. \mathbf{B}^n U(1) \in Smooth\infty Grpd \,.

This is such that the evident generalizations of the above classification statements hold: we have that morphisms XB nU(1)X \to \mathbf{B}^n U(1) form an n-groupoid

SmoothGrpd(X,B nU(1))U(1)(n1)Bund(X) Smooth\infty Grpd(X, \mathbf{B}^n U(1)) \simeq U(1) (n-1)Bund(X)

equivalent to that of circle n-bundles/(n1)(n-1)-bundle gerbes on XX.

If here X=BGX = \mathbf{B}G is again the delooping of a Lie group, this means that now also the higher characteristic classes are represented by morphisms

BGB nU(1). \mathbf{B}G \to \mathbf{B}^n U(1) \,.

For instance for G=SpinG = Spin the spin group, the first fractional Pontryagin class has a smooth incarnation given by a morphism of the form

12p 1:BSpinB 3U(1) \frac{1}{2}\mathbf{p}_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1)

corresponding under the above equivalence to the ordinary Chern-Simons circle 3-bundle on BG\mathbf{B}G.

Every cohesive (∞,1)-topos comes canonically and essentially uniquely equipped with all the intrinsic structure that we need for the discussion of a refinement of this to differential characteristic classes:

There is an endo-(∞,1)-adjunction

(Π):SmoothGrpdSmoothGrpd (\mathbf{\Pi} \dashv \mathbf{\flat}) : Smooth\infty Grpd \to Smooth \infty Grpd

where

A morphism Π(X)B nU(1)\mathbf{\Pi}(X) \to \mathbf{B}^n U(1) encodes the flat higher parallel transport of a flat circle n-bundle with connection, and we have that the n-groupoid of morphisms

SmoothGrpd(Π(X),B nU(1))U(1)nBund flat(X) Smooth \infty Grpd(\mathbf{\Pi}(X), \mathbf{B}^n U(1)) \simeq U(1) n Bund_{\nabla_{flat}}(X)

is that of flat circle n-bundles with connection/ (n-1)-bundle gerbes with connection.

We observe that a trivial circle nn-bundle with connection is equivalently just a globally defined differential n-form. Therefore if we define the modified (∞,1)-adjunction

(Π dR dR):*/SmoothGrpdSmoothGrpd (\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR}) : */Smooth\infty Grpd \stackrel{\leftarrow}{\to} Smooth\infty Grpd

by forming the (∞,1)-pullback

dRB nU(1):=* B nU(1)B nU(1), \mathbf{\flat}_{dR}\mathbf{B}^n U(1) := * \prod_{\mathbf{B}^n U(1)} \mathbf{\flat} \mathbf{B}^n U(1) \,,

which is the coefficient object for trivial principal \infty-bundles equipped with flat \infty-connection, one finds (discussed in detail here) that morphisms X dRB nU(1)X \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1) correspond to trivial circle bundle with connection, hence to cocycles in de Rham cohomology of XX;

π 0SmoothGrpd(X, dRB nU(1))={H dR n(X) n2 Ω cl 1(X) n=1. \pi_0 Smooth \infty Grpd(X, \mathbf{\flat}_{dR} \mathbf{B}^n U(1)) = \left\{ \array{ H_{dR}^n(X) & n \geq 2 \\ \Omega^1_{cl}(X) & n = 1 } \right. \,.

This now allows us to construct differential refinements:

one can show (detailed discussion is here) that there are canonical cocycles

curv:B nU(1) dRB n+1 curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}

in the degree (n+1)(n+1)-de Rham cohomology of B nU(1)\mathbf{B}^n U(1): these are the universal curvature characteristic forms on B nU(1)\mathbf{B}^n U(1).

Then for c:BGB nU(1)\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1) any smooth characteristic class, the corresponding (unrefined) differential characteristic class is simply the composite

c dR:BGcB nU(1)curv dRB n+1U(1). \mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1) \,.

The (unrefined) ∞-Chern-Weil homomorphism is postcomposition with this morphism:

(c dR) *:H 1(X,G)H dR n+1(X). (\mathbf{c}_{dR})_* : H^1(X,G) \to H_{dR}^{n+1}(X) \,.

This is finally where the Chern-Simons elements come in:

Chern-Simons elements are a means to present the composite morphism c dR\mathbf{c}_{dR} of smooth ∞-groupoids by an ∞-anafunctor between smooth Kan complexes.

This presentation we describe in the next section.

(In fact a bit more is true: the serve to present the refinement of c dR\mathbf{c}_{dR} to a morphism c^\hat \mathbf{c} with values in ordinary differential cohomology. This we come to further below.)

Chern-Simons forms

We explain now how Chern-Simons elements arise as a presentation of a differential characteristic class c dR\mathbf{c}_{dR} by a span of simplicial presheaves.

At the heart of the presentation of differenial characteristic classes by morphisms of simplicial presheaves is a differential refinement of the Lie integration of L-∞ algebras and ∞-Lie algebroid: for 𝔤\mathfrak{g} an ordinary Lie algebra, one finds that the 3-coskeleton of the simplicial presheaf that assigns flat vertical 𝔤\mathfrak{g}-Lie algebra valued 1-forms

exp(𝔤):(U,[k]){Ω si,vert (U×Δ k) A vert CE(𝔤)} \exp(\mathfrak{g}) : (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{si, vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) } \right\}

is the delooping of the simply connected Lie group GG integrating 𝔤\mathfrak{g}

cosk 3exp(𝔤)BG. \mathbf{cosk}_3 \exp(\mathfrak{g}) \stackrel{\simeq}{\to} \mathbf{B}G \,.

Similarly the Lie integration of the line Lie n-algebra b n1b^{n-1}\mathbb{R}

exp(b n1):(U,[k]){Ω si,vert (U×Δ k) CE(b n1)} \exp(b^{n-1}\mathbb{R}) : (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n-1}\mathbb{R}) } \right\}

is the nn-fold delooping of \mathbb{R}:

exp(b n1)B n. \exp(b^{n-1}\mathbb{R}) \stackrel{\simeq}{\to} \mathbf{B}^n \mathbb{R} \,.

Moreover, for μ:𝔤b n1\mu : \mathfrak{g} \to b^{n-1}\mathbb{R} a degree-nn cocycle in Lie algebra cohomology, simple postcomoposition gives its image under Lie integration

exp(μ):exp(𝔤)exp(b n1). \exp(\mu) : \exp(\mathfrak{g}) \to \exp(b^{n-1}\mathbb{R}) \,.

Under coskeletization on the left this carves out the periods of μ\mu as a lattice in \mathbb{R}, which typically is the integers, so that this descends to degree nn-cocycle in Lie group cohomology with coefficients in U(1)/U(1) \simeq \mathbb{R}/\mathbb{Z}

exp(μ):BGB n/. \exp(\mu) : \mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\mathbb{Z} \,.

(See Lie group cohomology and smooth ∞-groupoid for discussion of the refined notion of Lie group cohomology arising here.)

The differential refinement of these construction is based on the following fact (discussed in detail here)

  1. the object B nU(1)\mathbf{B}^n U(1) \in Smooth∞Grpd is equivalently presented by a quotient of the presheaf of Kan complexes given by

    B nU(1) diff:(USmoothMfd,[k]Δ){Ω si,vert (U×Δ k) CE(b n1) Ω si (U×Δ k) ω W(b n1)}, \mathbf{B}^n U(1)_{diff} : (U \in SmoothMfd, [k] \in \Delta) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{\omega}{\leftarrow}& W(b^{n-1} \mathbb{R}) } \right\} \,,

    where on the right we have the set of horizontal morphisms in dgAlg that make a commuting diagram with the canonical vertical morphisms as indicated.

    B nU(1) diff B nU(1). \array{ \mathbf{B}^n U(1)_{diff} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n U(1) } \,.

    (We may think of a morphism of simplicial presheaves XB nU(1) diffX \to \mathbf{B}^n U(1)_{diff} as a circle n-bundle/(n1)(n-1)-bundle gerbe equipped with a pseudo-connection . )

    Notice that the bottom morphism here encodes precisely a degree-nn differential form ω\omega on U×Δ kU \times \Delta^k,

  2. The morphism curv:B nU(1) dRB n+1U(1)curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1) is presented on this by the ∞-anafunctor

    B nU(1) diff dRB n+1U(1) sim B nU(1) \array{ \mathbf{B}^n U(1)_{diff} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{sim} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n U(1) }

    by the map that sends such a form ω\omega to its curvature dωd \omega. If the pseudo-connections that we are dealing with are genuine connections the curvature is a basic form down on UU and this means diagrammatically that it forms the pasting composite

    Ω si,vert (U×Δ k) CE(b n1) Ω si (U×Δ k) ω W(b n1) Ω (U) dω inv(b n1) \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{\omega}{\leftarrow}& W(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{d \omega}{\leftarrow}& inv(b^{n-1} \mathbb{R}) }

    and then picks out the bottom horizontal morphism.

Therefore our task of presenting c dR\mathbf{c}_{dR} amounts to computing the composition of ∞-anafunctors

B nU(1) diff dRB n+1U(1) sim BG exp(μ) B nU(1) \array{ && \mathbf{B}^n U(1)_{diff} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{sim} \\ && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G &\stackrel{\exp(\mu)}{\to}& \mathbf{B}^n U(1) }

To do se we need to complete componentwise to commuting diagrams. To this end we first complete the assignment of exp(𝔤)\exp(\mathfrak{g}) to a diagram

(2)(U,[k]){Ω (U×Δ k) vert A vert CE(𝔤) transitionfunction Ω (U×Δ k) A W(𝔤) connection Ω (U) F A inv(𝔤) curvaturecharacteristicform}. (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &&& transition\;function \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &&& connection \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A \rangle}{\leftarrow}& inv(\mathfrak{g}) &&& curvature\;characteristic\;form } \right\} \,.

Here in the middle row an unrestricted 𝔤\mathfrak{g}-∞-Lie algebra valued differential form appears, which is the local 𝔤\mathfrak{g}-∞-connection. And in the lower row all its curvature characteristic forms appear, obtained by evaluating the curvature F AF_A in the invariant polynomials on 𝔤\mathfrak{g}.

This is such that a choice of Chern-Simons element witnessing the transgression of an invariant polynomial to μ\mu allows to refine exp(μ)\exp(\mu) to

(3)exp(μ) conn{Ω (U×Δ k) vert A vert CE(𝔤) μ CE(b n1) characteristicclass Ω (U×Δ k) A W(𝔤) cs W(b n1) ChernSimonsform Ω (U) inv(𝔤) inv(b n1) curvaturecharacteristicform}. \cdots \stackrel{\exp(\mu)_{conn}}{\mapsto} \left\{ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) &&& characteristic\;class \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{n-1}\mathbb{R}) &&& Chern-Simons\;form \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle -\rangle}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& inv(b^{n-1}\mathbb{R}) &&& curvature\;characteristic\;form } \right\} \,.

Here now the middle row is the evaluationn of the connection form inside the Chern-Simons element. This is the corresponding Chern-Simons form

Ω (U)AW(𝔤)csW(b n1):CS(A) \Omega^\bullet(U) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{cs}{\leftarrow} W(b^{n-1}\mathbb{R}) : CS(A)

of the 𝔤\mathfrak{g}-connection evaluated in the given Chern-Simons element. Its curvature is the curvature characteristic form F A\langle F_A \rangle appearing in the bottom line of the diagram, which is obtained by evaluating the 𝔤\mathfrak{g}-valued curvature in the given invariant polynomial.

A more comprehensive account of this is at Chern-Weil homomorphism in Smooth∞Grpd.

Chern-Simons action functionals

By the above construction, every Chern-Simons element csW(𝔞)cs \in W(\mathfrak{a}) of degree dd on an ∞-Lie algebroid 𝔞\mathfrak{a} induces an action functional on the space of ∞-Lie algebroid valued forms on 𝔞\mathfrak{a} over a dd-dimensional smooth manifold Σ\Sigma

S cs:Ω(Σ,𝔞) S_{cs} : \Omega(\Sigma, \mathfrak{a}) \to \mathbb{R}

given by

(A,B,C,) ΣCS(A,B,C,). (A,B,C, \cdots) \mapsto \int_\Sigma CS(A,B,C, \cdots) \,.

This generalizes the action functional of ordinary Chern-Simons theory to general Chern-Simons elements. In the examples below is a list of various quantum field theories that arise as generalized Chern-Simons theories this way.

For more details see infinity-Chern-Simons theory.

Examples

On semisimple Lie algebra– Standard Chern-Simons action functional

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\} denotes the corresponding degree-shifted basis of 𝔤 *[1]\mathfrak{g}^*[1].

Notice that in terms of this the differential of the CE-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 familiar form 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.

Higher CS-forms on semisimple Lie algebras

For μCE(𝔤)\mu \in CE(\mathfrak{g}) any higher order cocycle, CS μ(A)CS_\mu(A) is the corresponding higher order Chern-Simons form.

Action functional for Chern-Simons (super-)gravity

Higher Chern-Simons elements on the Poincare Lie algebra 𝔤=𝔦𝔰𝔬(d,1)\mathfrak{g} = \mathfrak{iso}(d,1) or the super Poincare Lie algebra 𝔤=𝔰𝔦𝔰𝔬(d,1)\mathfrak{g} = \mathfrak{siso}(d,1) yield action functionals for gravity and supergravity.

(…)

See (Zanelli).

Fractional secondary Pontryagin classes

For instance for μ 7\mu_7 the 7-cocycle on a semisimple Lie algebra, CS μ 7(A)CS_{\mu_7}(A) is the corresponding Chern-Simons 7-form, corresponding to the second Pontryagin class.

Notice that this we may also think of as a 7-cocycle on the corresponding string Lie 2-algebra. As such it is the one that classifies the extension to the fivebrane Lie 6-algebra. The corresponding Chern-Simons 7-form appears as the local conneciton data in the Chern-Simons circle 7-bundle with connection that obstructions the lift from a differential string structure to a differential fivebrane structure.

On strict Lie 2-algebras – BF-theory action functional

Proposition

Let 𝔤=(𝔤 2𝔤) 1\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g})_1 be a strict Lie 2-algebra.

Then

  • 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}, restricting to the trivial ∞-Lie algebra cocycle;

  • 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 ∞-Lie 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):σt ad W(𝔤 1)+ a iσb i. d_{W(\mathfrak{g})} : \sigma t^a \mapsto d_{W(\mathfrak{g}_1)} + \partial^a{}_i \sigma b^i \,.

Therefore with 𝔤 1=P a 1a nσr a 1σt a n\langle -\rangle_{\mathfrak{g}_1} = P_{a_1 \cdots a_n} \sigma r^{a_1} \wedge \cdots \sigma t^{a_n} we have

d W(𝔤) 𝔤 1=nP a 1a n a 1 iσb iσt a n. d_{W(\mathfrak{g})} \langle - \rangle_{\mathfrak{g}_1} = n P_{a_1 \cdots a_n}\partial^{a_1}{}_i \sigma b^{i} \wedge \cdots \sigma 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)(A,B) 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} \,.

On symplectic \infty-Lie algebroids – The AKSZ Lagrangian

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

The corresponding Chern-Simons elements of ω\omega are the integrands for the action functionals of various TQFT sigma-models.

With {,}:CE(𝔞)CE(𝔞)CE(𝔞)\{-,-\} : CE(\mathfrak{a}) \otimes CE(\mathfrak{a}) \to CE(\mathfrak{a}) the graded Poisson bracket induced by ω\omega we have (see Roytenberg) that there exists a ∞-Lie algebra cocycle μCE(𝔞)\mu \in CE(\mathfrak{a}) such that

d CE(𝔞)={μ,}. d_{CE(\mathfrak{a})} = \{\mu, -\} \,.

So in particular μ\mu being a cocycle means that

d CE(𝔞)μ={μ,μ}=0. d_{CE(\mathfrak{a})} \mu = \{\mu, \mu\} = 0 \,.
Proposition

The cocycle μ\mu is in transgression with the invariant polynomial n2ω\frac{n}{2}\omega via the Chern-Simons element

cs =12ι ϵωμ, \begin{aligned} cs &= \frac{1}{2 }\iota_{\epsilon} \omega - \mu \end{aligned} \,,

where ϵ\epsilon is the Euler vector field (Roytenberg).

Here d\mathbf{d} is the shift derivation in the Weil algebra, in that d W(𝔞)=d CE(𝔞)+dd_{W(\mathfrak{a})} = d_{CE(\mathfrak{a})} + \mathbf{d}.

Proof

To safe typing signs, we write as if all functions were even graded. By standard reasoning the computation holds true then also for arbitrary grading.

Observe that

  1. on unshifted generators we have

    (dx b)ω ab{x a,}=d (\mathbf{d}x^b) \omega_{a b} \{x^a , -\} = \mathbf{d}
  2. we have graded commutators

    • [d,ι v]=N[\mathbf{d}, \iota_v] = N (the degree operator)

    and

    • [d CE(𝔞),ι v]=d CE(𝔞)d 1[d_{CE(\mathfrak{a})}, \iota_v] = -d_{CE(\mathfrak{a})}\mathbf{d}^{-1}.

    (as one checks on generators).

Therefore

d W(𝔞)12ι vω =[d W(𝔞),ι v]12ω =(nd CE(𝔞)d 1)12ω =n2ωω ab{μ,x a}dx b =12ω+dμ, \begin{aligned} d_{W(\mathfrak{a})} \frac{1}{2}\iota_{v} \omega &= [d_{W(\mathfrak{a})}, \iota_{v}] \frac{1}{2}\omega \\ & = (n - d_{CE(\mathfrak{a})} \mathbf{d}^{-1} ) \frac{1}{2}\omega \\ & = \frac{n}{2}\omega - \omega_{a b} \{\mu, x^a\} \mathbf{d}x^b \\ &= \frac{1}{2} \omega + \mathbf{d}\mu \end{aligned} \,,

where in the first line we used that by definition of invariant polynomial d W(𝔞)ω=0d_{W(\mathfrak{a})} \omega = 0. Similarly, using that by definition d CE(𝔞)μ=0d_{CE(\mathfrak{a})} \mu = 0 we have

d W(𝔞)μ=dμ. d_{W(\mathfrak{a})} \mu = \mathbf{d}\mu \,.

So in total we have

d W(𝔞)(12ι ϵωμ)=12ω. d_{W(\mathfrak{a})} (\frac{1}{2} \iota_\epsilon \omega - \mu) = \frac{1}{2}\omega \,.
Remark

In local coordinates where ω=ω abdx adx b\omega = \omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b we have

cs=nω abx adx b+μ. cs = n \omega_{a b} x^a \wedge \mathbf{d}x^b + \mu \,.

The Chern-Simons action functional corresponding to this Chern-Simons element on 𝔞\mathfrak{a} is that considered in AKSZ theory.

Below we spell out some low-dimensional cases explicitly.

Higher phase space —Hamiltonian and Lagrangian mechanics

The symplectic Lie nn-algebroid (𝔓,ω)(\mathfrak{P}, \omega) may be thought of as an n-symplectic manifold that models the phase space of a physical system.

This means for (𝔤,)=(𝔓,ω)(\mathfrak{g},\langle-\rangle) = (\mathfrak{P}, \omega) a symplectic Lie nn-algebroid, the general diagram (1) exhibiting the transgression between cocycles and invariant polynomials via Chern-Simons elements may be labeled in terms of Hamiltonian mechanics, Lagrangian mechanics and symplectic geometry as follows

(4)CE(𝔓) H CE(b n) Hamiltonian W(𝔓) L W(b n) Lagrangian inv(𝔓) ω inv(b n) symplecticstructure \array{ CE(\mathfrak{P}) &\stackrel{H}{\leftarrow}& CE(b^{n}\mathbb{R}) &&& Hamiltonian \\ \uparrow && \uparrow \\ W(\mathfrak{P}) &\stackrel{L}{\leftarrow}& W(b^n \mathbb{R}) &&& Lagrangian \\ \uparrow && \uparrow \\ inv(\mathfrak{P}) &\stackrel{\omega}{\leftarrow}& inv(b^n \mathbb{R}) &&& symplectic\;structure }

See Hamiltonian, Lagrangian, symplectic structure.

On a symplectic manifold – 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”).

On a Poisson Lie algebroid – The Poisson σ\sigma-model

Proposition

Let 𝔞=𝔓(X,π)\mathfrak{a} = \mathfrak{P}(X,\pi) by a Poisson Lie algebroid. This comes with the canonical invariant polynomial ω=d idx i\omega = \mathbf{d} \partial_i \wedge \mathbf{d} x^i.

The corresponding ∞-Lie algebroid cocycle is

μ ω=π \mu_{\omega} = \pi

and a Chern-Simons element for this is

cs ω=π ij i j+ idx i. cs_\omega = \pi^{i j} \partial_i \wedge \partial_j + \partial_i \wedge \mathbf{d}x^i \,.

For Σ\Sigma a 2-dimensional smooth manifold the corresponding action functional on ∞-Lie algebroid-valued forms S:Ω (X,𝔓(X,π))S : \Omega^\bullet(X, \mathfrak{P}(X,\pi)) \to \mathbb{R} is the actional functional of the Poisson sigma-model

S:(X,η) Σ(ηdX+π(ηη). S : (X, \eta) \mapsto \int_\Sigma (\eta \wedge d X + \pi(\eta \wedge \eta) \,.
Proof

We compute in a local coordinte patch:

d W(𝔓(X,π))(π ij i j+ idx i)= dx k( kπ ij) i j +2π ij(d i) j ( iπ jk) j kdx i +(d i)(dx i) +()()2π ij id j = (d i)(dx i). \begin{aligned} d_{W(\mathfrak{P}(X,\pi))} (\pi^{i j} \partial_i \wedge \partial_j + \partial_i \wedge \mathbf{d} x^i) = & \mathbf{d}x^k (\partial_k \pi^{i j}) \partial_i \wedge \partial_j \\ & + 2 \pi^{i j} (\mathbf{d}\partial_i) \wedge \partial_j \\ & - (\partial_i \pi^{j k}) \partial_j \wedge \partial_k \wedge \mathbf{d}x^i \\ & + (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i) \\ & + (-)(-) 2\pi^{i j} \partial_i \wedge \mathbf{d}\partial_j \\ = & (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i) \end{aligned} \,.

On higher extensions of the super Poincare Lie algebra – supergravity

See D'Auria-Fre formulation of supergravity for the moment.

References

A classical reference on transgression of differential forms from the fiber to the base of a fiber bundle is section 9 of.

  • Armand Borel, Topology of Lie groups and characteristic classes Bull. Amer. Math. Soc. Volume 61, Number 5 (1955), 397-432. (EUCLID)

The general definition of Chern-Simons element on \infty-Lie algebras and \infty-Lie algebroids is definition 21 in

The examples of the BF-theory invariant polynomials and Chern-Simons elements are in prop. 18 and def. 26 and the BF-action functional itself is extracted below proposition 28.

Dedicated discussion of \infty-Chern-Simons theory is at

A comprehensive account is in

A survey of higher Chern-Simons elements and their action functionals as applied to gravity and supergravity is in

  • Jorge Zanelli, Lecture notes on Chern-Simons (super-)gravities arXiv:0502193

Symplectic Lie nn-algebroids are discussed 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)

A talk about the historical origins of the standard Chern-Simons forms see

  • Jim Simons, Origin of Chern-Simons talk at Simons Center for Geometry and Physics (2011) (video)

Revised on September 11, 2011 19:25:13 by Urs Schreiber (82.113.99.29)