nLab universal Chern-Simons line 3-bundle



\infty-Chern-Weil theory

Differential cohomology



A Chern-Simons circle 3-bundle is the circle n-bundle with connection classified by the cocycle in degree 4 ordinary differential cohomology that is canonically associated to a GG-principal bundle with connection.

For GG a Lie group and c:GK(,4)c : \mathcal{B}G \to K(\mathbb{Z},4) a cocycle for a degree 4 characteristic class in integral cohomology and XX a smooth manifold, Chern-Weil theory provides a morphism (the refined Chern-Weil homomorphism)

c^:GBund (X)H diff 4(X) \hat c : G Bund_\nabla(X) \to H_{diff}^4(X)

from GG-principal bundles with connection \nabla to degree 4 ordinary differential cohomology. The cocycles on the right may be thought of as

By construction, the curvature 4-form of c^()\hat c(\nabla) is the curvature characteristic form F F \langle F_\nabla \wedge F_\nabla\rangle of \nabla and accordingly the 3-form connection on c^()\hat c(\nabla) is locally a Chern-Simons form CS()CS(\nabla) of \nabla.

Accordingly, the higher parallel transport induced by c^()\hat c(\nabla) over 3-dimensional manifolds ϕ:ΣX\phi : \Sigma \to X is the action functional of the quantum field theory called Chern-Simons theory. In this form it appears for instance as the gauge field called the supergravity C-field in certain supergravity theories. In particular, if (with due care) one takes \nabla to be the universal connection on the GG-universal principal bundle over a smooth version of BGB G, then c^()\hat c(\nabla) is the background gauge field for bare Chern-Simons theory.

Therefore this structure c^()\hat c(\nabla) has become known as the Chern-Simons 2-gerbe of \nabla. We may also think of it as the Chern-Simons circle 3-bundle .

At least for simply connected GG one may enhance the assignment c^()\nabla \mapsto \hat c(\nabla) to a morphism of ∞-groupoids

c^:H conn(X,BG)H diff(X,B 3U(1)), \hat \mathbf{c} : \mathbf{H}_conn(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1)) \,,

where on the left we have the groupoid of smooth GG-principal bundles with connection on XX, and on the right the 3-groupoid of circle 3-bundles with connection. The homotopy fibers of this morphism over a trivial circle 3-bundle with connection are 3-groupoids whose objects are naturally identified with pairs consisting of a connection \nabla on a GG-bundle and a trivialization of its corresponding Chern-Simons 3-bundle. This in particular implies a trivialization of the underlying cocycle in degree 4 integral cohomology and therefore defines a string structure. One calls these homotopy fibers therefore differential string structures.


In differential geometry – For Lie groups

In Cech-Deligne cohomology

In (Brylinski-McLaughlin I) there is spelled out an explicit construction of c^()\hat c(\nabla) for given \nabla in Cech-Deligne cohomology. This is a special case of the general construction presented in (Brylinski-McLaughlin II).

In this section here we review this explicit cocycle construction. In the next section we discuss a systematic way to derive this construction.

Assume that GG is a simply connected compact simple Lie group, such as the spin group, and take the characteristic class cc to be that whose transgression to GG has as image in de Rham cohomology the de Rham class of the normalized canonical Lie algebra cocycle μCE(𝔤)\mu \in CE(\mathfrak{g}).

For PXP \to X a GG-bundle with connection \nabla, there exists an open cover {U iX}\{U_i \to X\} such that we have a Cech cohomology cocycle for PP given by a smooth transition function

g i,j:U iU jG g_{i,j} : U_i \cap U_j \to G

satisfying on U iU jU kU_i \cap U_j \cap U_k the cocycle condition g ijg jk=g ikg_{i j} \cdot g_{j k} = g_{i k}.

Since GG is assumed connected and simply connected and since for every Lie group the second homotopy group is trivial we have that the first nonvanishing homotopy group of GG is the third one.

Therefore we can always find (possibly after refining the cover) a lift of this cocycle as follows:

  • on double intersections, choose smooth functions

    g^ ij:(U iU j)×Δ 1G \hat g_{i j} : (U_i \cap U_j) \times \Delta^1 \to G

    such that g^ ij(x,0)=e\hat g_{i j}(x,0 ) = e is the identity in gg, and such that such that g^ ij(x,1)=g ij(x)\hat g_{i j}(x,1) = g_{i j}(x);

  • on triple intersections, choose smooth functions

    g^ ijk:(U iU jU k)×Δ 2G \hat g_{i j k} : (U_i \cap U_j \cap U_k) \times \Delta^2 \to G

    that cobound these paths in the evident way

    g ij g^ ij g^ ijk g ijg^ jk e g^ ik g ik. \array{ && g_{i j} \\ & {}^{\mathllap{\hat g_{i j}}}\nearrow &\Downarrow^{\mathrlap{\hat g_{i j k}}}& \searrow^{g_{i j} \cdot \mathrlap{\hat g_{j k}}} \\ e &&\underset{\hat g_{i k}}{\to}&& g_{i k} }.

    (This can be done because π 1(G)=*\pi_1(G) = *.)

  • on quadruple intersection choose smooth functions

    g^ ijkl:(U iU jU kU l)×Δ 3G \hat g_{i j k l} : (U_i \cap U_j \cap U_k \cap U_l) \times \Delta^3 \to G

    such that these 3-balls fill the evident tetrahedra.

    (This can be done because π 2(G)=0\pi_2(G) = 0.)


The Cech cohomology cocycle with coefficients in B 3/\mathbf{B}^3 \mathbb{R}/\mathbb{Z} which is given by

c(g) ijkl:= Δ 3g^ ijkl *μmod c(g)_{i j k l} := \int_{\Delta^3} \hat g_{i j k l}^* \mu \;\;\;\;\; mod \mathbb{Z}

is well defined and represents the class c(P)H 4(X)c(P) \in H^4(X) in integral cohomology.

Moreover, this refines to the cocycle in Cech-Deligne cohomology that is given by

(CS(σ i *A), Δ 1CS((Pg^ ij) *A), Δ 2CS((Pg^ ijk) *A), Δ 3g^ ijkl *μmod), \left( CS(\sigma_i^* A) \,,\;\; \int_{\Delta^1} CS((P\cdot \hat g_{i j})^* A) \,,\;\; \int_{\Delta^2} CS((P\cdot \hat g_{i j k})^* A) \,,\;\; \int_{\Delta^3} \hat g_{i j k l}^* \mu \;\;\; mod \mathbb{Z} \right) \,,


  • AΩ 1(P,𝔤)A \in \Omega^1(P,\mathfrak{g}) is the incarnation of the connection \nabla as an Ehresmann connection given by a glbally defined 1-form on the total space PXP \to X of the bundle;

  • σ i:U iP\sigma_i : U_i \to P are the local sections of PXP \to X that induce the original Cech cocycle (g ij:=σ iσ j 1)(g_{i j} := \sigma_i \cdot \sigma_j^{-1});

  • Pg^ ij:Δ 1PP \cdot \hat g_{i j} : \Delta^1 \to P is given by the right action of GG on PP and analogously for the other terms;

  • CS(...)CS(...) denotes the Chern-Simons form of the given 𝔤\mathfrak{g}-valued 1-form.


First notice that this is indeed well-defined: by compactness and simplicty of GG we have π 3(G)=\pi_3(G) = \mathbb{Z}. By assumption on μΩ 3(G)\mu \in \Omega^3(G), for any map f:S 3Gf : S^3 \to G, we have S 3f *μ\int_{S^3} f^*\mu \in \mathbb{Z} \subset \mathbb{R}. This implies that c(g)c(g) is indeed a Cech cocycle.

Then the proof is effectively just the observation that the given collection of differential forms indeed does refine this to a Cech-cocycle with coefficients in the Deligne complex, and that therefore we can read off the image of the integral cohomology class [c(g)][c(g)] in de Rham cohomology from the curvature 4-form of this Deligne cocycle. That is by construction F F Ω cl 4(X)\langle F_\nabla \wedge F_\nabla \rangle \in \Omega^4_{cl}(X), which by Chern-Weil theory is indeed the image of the claimed integral class.

So the only mystery about this construction is really: where does it come from? Apart from making this clever Ansatz and checking that it works, can one somehow systematically derive this construction? This we shall try to answer the section below.

In \infty-Chern-Weil theory

The above Cech-Deligne cocycle construction of c^()\hat c(\nabla) may be understood as a special case of the general construction of Chern-Weil homomorphisms by the methods discussed at ∞-Chern-Weil theory.

We briefly recall the general approach and then spell out the details.

The basic ingredients in ∞-Chern-Weil theory that give the refinement of a characteristic class to a morphism

H conn(X,BG)H diff(X,B kU(1)) \mathbf{H}_{conn}(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X, \mathbf{B}^k U(1))

from the GG-principal bundles with connection to ordinary differential cohomology are these:

  1. for a given Lie algebra 𝔤\mathfrak{g} the realization of the corresponding Lie group as a truncation of the simplicial presheaf

    (U,[n]){Ω (U×Δ n)CE(𝔤)} (U,[n]) \mapsto \{\Omega^\bullet(U\times \Delta^n) \leftarrow CE(\mathfrak{g})\}

    (see Lie integration);

  2. the observation that, up to subtleties with the truncation, a Lie algebra cocycle

    CE(𝔤)CE(b k1):μ CE(\mathfrak{g}) \leftarrow CE(b^{k-1}\mathbb{R}) : \mu

    induces therefore an integrated cocycle BGB kU(1)\mathbf{B}G \to \mathbf{B}^k U(1);

  3. the observation that this is lifted to connections and differential refinement by

    1. thickening the simplicial presheaf for BG\mathbf{B}G to

      (U,[n]){C (U)Ω (Δ n) CE(𝔤) Ω (U)Ω (Δ n) A W(𝔤)} (U,[n]) \mapsto \left\{ \array{ C^\infty(U) \otimes \Omega^\bullet(\Delta^n) &\stackrel{}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right\}
    2. thickening the Lie algebra cocycle by its Chern-Simons element

CE(𝔤) μ CE(b k1) W(𝔤) cs W(b k1) \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) }

and then postcomposing with that. Note that the above diagram is part of a larger diagram involving the invariant polynomial \langle-\rangle for μ\mu and exhibiting the Chern-Simons element as a transgression element between these two:

CE(𝔤) μ CE(b k1) W(𝔤) cs W(b k1) inv(𝔤) inv(b k1) \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }

Also note that inv(b k1)CE(b k)inv(b^{k-1} \mathbb{R})\cong CE(b^k \mathbb{R}).

For the case at hand, let 𝔤\mathfrak{g} be a semisimple Lie algebra, :CE(b 3)W(𝔤)\langle -\rangle : CE(b^3\mathbb{R})\to W(\mathfrak{g}) its canonical Killing form invariant polynomial, μ=,[,]:CE(b 2)CE(𝔤)\mu = \langle -,[-,-]\rangle: CE(b^2\mathbb{R})\to CE(\mathfrak{g}) the corresponding Lie algebra cocycle, cs:W(b 2)W(𝔤)cs: W(b^2\mathbb{R})\to W(\mathfrak{g}) the Chern-Simons elements exhibiting the transgression between the two, GG the simply connected Lie group integrating it.

First consider the bare cocycle for the Chern-Simons circle 3-bundle as the Lie integration of the cocycle μ\mu.


Consider the simplicial presheaf

exp(𝔤):(U,[n]){C (U)Ω (Δ n)CE(𝔤)}, \exp(\mathfrak{g}) : (U,[n]) \mapsto \{C^\infty(U) \otimes \Omega^\bullet(\Delta^n) \leftarrow CE(\mathfrak{g})\} \,,

where here and in what follows differential forms ω\omega on simplices are taken to have sitting instants in that for all kk \in \mathbb{N} there exists for every kk-face of Δ n\Delta^n an open neighbourhood such that ω\omega restricted to that open neighbourhood is constant in the direction perpendicular to the boundary.


The canonical map

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

from the 3-coskeleton of exp(𝔤)\exp(\mathfrak{g}) to the delooping of the simply connected Lie group GG which is given on 1-morphisms by higher parallel transport is an equivalence ( in the model structure on simplicial presheaves [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}).


Use that a 𝔤\mathfrak{g}-valued 1-form on the interval is canonically identified with a based path in GG. Then use that for k2k \leq 2 we have π k(G)=0\pi_k(G) = 0. See Lie integration for more.


There is a commuting diagram

exp(𝔤) exp(μ) exp(b 2) Δ cosk 3exp(𝔤) B 3/, \array{ \exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^2 \mathbb{R}) \\ \downarrow && \downarrow^{\mathrlap{\int_{\Delta^\bullet}}} \\ \mathbf{cosk}_3 \exp(\mathfrak{g}) &\stackrel{}{\to}& \mathbf{B}^3 \mathbb{R}/\mathbb{Z} } \,,

where the right vertical morphism is the composite of the equivalence

Δ :exp(b 2)B 3 \int_{\Delta^\bullet} : \exp(b^2 \mathbb{R}) \stackrel{\simeq}{\to} \mathbf{B}^3 \mathbb{R}

discussed at Integration to Line n-groups with the evident quotient B 3B 3/\mathbf{B}^3 \mathbb{R} \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}, where the copy of \mathbb{Z} in \mathbb{R} is the lattice of periods of μ\mu over 3-spheres in GG.

The top morphism sends a UU-family of 3-morphisms Ω (U×Δ 3)ACE(𝔤)\Omega^\bullet(U \times \Delta^3) \stackrel{A}{\leftarrow} CE(\mathfrak{g}) – which we may think of as a UU-family of based 3-balls Σ:U×Δ 3G\Sigma : U \times \Delta^3 \to G – to the family of 3-forms

Ω (U×Δ k) vertACE(𝔤)μCE(b 2):μ *(A). \Omega^\bullet(U \times \Delta^k)_{vert} \stackrel{A}{\leftarrow} CE(\mathfrak{g}) \stackrel{\mu}{\leftarrow} CE(b^2 \mathbb{R}) : \mu^*(A) \,.

which we may think as a family of closed 3-forms

μ *(A)Ω (U×Δ k) vert \mu^*(A)\in \Omega^\bullet(U \times \Delta^k)_{vert}

The right vertical morphism sends this to the fiber integration

U Δ 3μ *(A)C (U;) U \mapsto \int_{\Delta^3} \mu^*(A) \;\;\; \in C^\infty(U;\mathbb{R})

and regards the result then modulo \mathbb{Z}. That this indeed gives a morphism down at the bottom is the statement that for a 4-morphism in cosk 3exp(𝔤)\mathbf{cosk}_3 \exp(\mathfrak{g}) – which is a 3-sphere V:S 3GV : S^3 \to G – we have that S 3V *μ *(A)=0mod\int_{S^3} V^* \mu^*(A) = 0 \;mod\; \mathbb{Z}, which is true by the fact that we take \mathbb{Z} to be precisely generated by these periods. (Alternatively we can assume μ\mu to be normalized such that it generates the image in deRham cohomology of H 3(G,)H^3(G,\mathbb{Z}) \simeq \mathbb{Z}.)

We shall by slight abuse of notation write exp(μ)\exp(\mu) also for the morphism cosk 3exp(𝔤)B 3/\mathbf{cosk}_3 \exp(\mathfrak{g}) \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}.


For {U iX}\{U_i \to X\} a cover and C(U)[CartSp op,sSet] projC(U) \in [CartSp^{op}, sSet]_{proj} the corresponding Cech nerve we have that

  • a morphism g:C(U)BGg : C(U) \to \mathbf{B}G is precisely a Cech 1-cocycle with values in GG;

  • a lift

    cosk 3exp(𝔤) g^ C(U) g BG \array{ && \mathbf{cosk}_3 \exp(\mathfrak{g}) \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow^{\mathrlap{\simeq}} \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G }

    is precisely a lift of this cocycle to a system of paths, triangles and tetrahedra in GG, as above.


Write exp(𝔤) diff\exp(\mathfrak{g})_{diff} for the simplicial presheaf

(U,[n]){Ω (U×Δ k) vert CE(𝔤) Ω (U×Δ k) W(𝔤)}. (U,[n]) \mapsto \left\{ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\leftarrow& W(\mathfrak{g}) } \right\} \,.

Its 3-coskeleton cosk 3exp(𝔤inn(𝔤))\mathbf{cosk}_3 \exp(\mathfrak{g} \to inn(\mathfrak{g})) is the coefficient for GG-principal bundles with pseudo-connection adapted to the model cosk 3exp(𝔤)\mathbf{cosk}_3 \exp(\mathfrak{g}) for BG\mathbf{B}G.


Pseudo-connections ^\hat \nabla

cosk 3exp(𝔤) diff ^ cosk 3exp(𝔤) g^ C(U) g BG \array{ && \mathbf{cosk}_3 \exp(\mathfrak{g})_{diff} \\ &{}^{\mathllap{\hat \nabla}}\nearrow& \downarrow^{\mathrlap{\simeq}} \\ && \mathbf{cosk}_3 \exp(\mathfrak{g}) \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow^{\mathrlap{\simeq}} \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G }

which are genuine \infty-connections in that their curvature components have no leg along the simplicial directions are in bijection with ordinary connections on the GG-bundle given by C(U)BGC(U) \to \mathbf{B}G.


On single patches ^\hat \nabla is a collection of 𝔤\mathfrak{g}-valued 1-forms A iΩ 1(U i,𝔤)A_i \in \Omega^1(U_i, \mathfrak{g}).

On double intersection it is a collection

A^ ij=A ij+λ ijΩ 1(U iU j,𝔤)C (Δ 1)C (U)Ω 1(Δ 1,𝔤)Ω 1(U iU j×Δ 1,𝔤) \hat A_{i j} = A_{i j} + \lambda_{i j} \in \Omega^1(U_i \cap U_j, \mathfrak{g})\otimes C^\infty(\Delta^1) \oplus C^\infty(U)\otimes \Omega^1(\Delta^1, \mathfrak{g}) \subset \Omega^1(U_i \cap U_j \times \Delta^1, \mathfrak{g})

whose restriction λ ij\lambda_{i j} to Δ 1\Delta^1 is the given path g^ ij\hat g_{i j} that is being covered. The condition that the curvature of A^ ij\hat A_{i j} has no component in the simplicial direction is the differential equation

tA ij=d U(λ ij) t+[(λ ij) t,A ij]. \frac{\partial}{\partial t} A_{i j} = d_U (\lambda_{i j})_t + [(\lambda_{i j})_t, A_{i j}] \,.

This differential equation has a unique solution for the boundary condition A ij(0)=A iA_{i j}(0) = A_i given by

A ij(t)=g^ ij(t) 1(A i+d)g^ ij(t). A_{i j}(t) = \hat g_{i j}(t)^{-1}(A_i + d)\hat g_{i j}(t) \,.

(To see this, use the formulas from parallel transport. If we assume just for notational simplicity that we are dealing with a matrix Lie algebra then we have tg^ ij=g^ ijλ\frac{\partial}{\partial t} \hat g_{i j} = \hat g_{i j} \cdot \lambda (by definition) and using that the claim follows.)

In particular this implies the forms on single patches satisfy the ordinary cocycle relation

A ij(1)=A j=g ij 1(A i+d)g ij A_{i j}(1) = A_j = g_{i j}^{-1}(A_i + d)g_{i j}

for connections.

Similarly there are differential equations on 2-simplices and 3-simplices with unique solutions.


Pasting postcomposition with the diagram

CE(𝔤) μ CE(b 2) W(𝔤) (cs,) W(b 2) \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^2 \mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(cs,\langle \rangle)}{\leftarrow}& W(b^2 \mathbb{R}) }

induces a morphism exp(𝔤) diffexp(b 2) diff\exp(\mathfrak{g})_{diff} \to \exp(b^2\mathbb{R})_{diff} and we obtain a commuting diagram

exp(𝔤) diff exp(b 2) diff cosk 3exp(𝔤) diff B 3U(1) chn,diff \array{ \exp(\mathfrak{g})_{diff} &\to& \exp(b^2\mathbb{R})_{diff} \\ \downarrow && \downarrow \\ \mathbf{cosk}_3\exp(\mathfrak{g})_{diff} &\to& \mathbf{B}^3 U(1)_{chn,diff} }

that covers the corresponding diagram we had before.

Here we are using the object B 3U(1) ch,diff\mathbf{B}^3 U(1)_{ch,diff} described in detail at circle n-bundle with connection.


The deeper reason for this construction is that the zig-zag composite

BGcosk 3exp(𝔤) diffB 3U(1) diff,chn dRB 4U(1) chn \mathbf{B}G \stackrel{\simeq}{\leftarrow} \mathbf{cosk}_3\exp(\mathfrak{g})_{diff} \to \mathbf{B}^3 U(1)_{diff,chn} \to \mathbf{\flat}_{dR}\mathbf{B}^4 U(1)_{chn}

of morphisms of simplicial presheaves models the intrinsically defined morphism

BG dRB 4U(1) \mathbf{B}G \to \mathbf{\flat}_{dR}\mathbf{B}^4 U(1)

in the (∞,1)-topos Smooth∞Grpd.


The outer composite morphism

cosk 3exp(𝔤) diff exp((cs,)) B 3U(1) diff ^ cosk 3exp(𝔤) g^ C(U) g BG \array{ && \mathbf{cosk}_3 \exp(\mathfrak{g})_{diff} &\stackrel{\exp((cs,\langle -\rangle))}{\to}& \mathbf{B}^3 U(1)_{diff} \\ &{}^{\mathllap{\hat \nabla}}\nearrow& \downarrow^{\mathrlap{\simeq}} \\ && \mathbf{cosk}_3 \exp(\mathfrak{g}) \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow^{\mathrlap{\simeq}} \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G }

is precisely the Cech-Deligne cocycle

(CS(A i), Δ 1CS(A^ ij), Δ 2CS(A^ ijk), Δ 3μ(A ijkl)). (CS(A_i), \int_{\Delta^1} CS(\hat A_{i j} ), \int_{\Delta^2} CS(\hat A_{i j k}), \int_{\Delta^3} \mu(A_{i j k l})) \,.

This is is exactly equal to the cocycle discussed above.

Notice by the way that this construction also serves as a manifest proof that this collection of data indeed does constitute a Deligne cocycle.


This is a matter of plugging the above pieces into each other. For instance, on double intersections we have that the 3-form CS(A^ ij)CS(\hat A_{i j}) is the image of the degree 3-generator on W(b 2)W(b^2 \mathbb{R}) under the composite

Ω (U×Δ k) vertA^ ijW(𝔤)(cs,)W(b 2). \Omega^\bullet(U \times \Delta^k)_{vert} \stackrel{\hat A_{i j}}{\leftarrow} W(\mathfrak{g}) \stackrel{(cs,\langle -\rangle)}{\leftarrow} W(b^2 \mathbb{R}) \,.

The remaining fiber integration is then that exhibiting the equivalence of simplicial differential forms

Δ : dRB 3U(1) diff,simp dRB 3U(1) diff,chn \int_{\Delta^\bullet} : \mathbf{\flat}_{dR} \mathbf{B}^3 U(1)_{diff,simp} \stackrel{\simeq}{\to} \mathbf{\flat}_{dR} \mathbf{B}^3 U(1)_{diff,chn}

that is described in some detail at Circle n-bundle with connection – models from ∞-Lie integration.

As a bundle 2-gerbe

We indicate (for the moment) the way the Chern-Simons 3-bundle is realized as a bundle 2-gerbe (for instance in CJMS and Waldorf CS) .

One first constructs the canonical bundle gerbe 𝒢G\mathcal{G} \to G on the Lie group and notices that (more or less implicitly by recourse to its delooping 2-gerbe on BG\mathbf{B}G) that this has a multiplicative structure .

Using this we see that for PXP \to X any GG-principal bundle and P [2]:=P× XPP×GP^{[2]} : = P \times_X P \to P \times G the principality isomorphism, the pullback of 𝒢\mathcal{G} along

f:P [2]P×Gp 2G f : P^{[2]} \to P \times G \stackrel{p_2}{\to} G

serves to provide the diagram

f *𝒢 P [2] P X \array{ f^* \mathcal{G} \\ \downarrow \\ P^{[2]} &\stackrel{\to}{\to}& P \\ && \downarrow \\ && X }

on which the pullback of the multiplicative structure on 𝒢\mathcal{G} induces the structure of a bundle 2-gerbe, in that we get morphisms of bundle gerbes

μ:π 0 *f *𝒢π 2 *f *𝒢π 2 *f *𝒢 \mu : \pi_0^{\ast} f^{\ast} \mathcal{G} \otimes \pi_2^{\ast} f^{\ast} \mathcal{G} \to \pi_2^* f^* \mathcal{G}

that are associative up to a higher coherent morphisms, etc.

In complex analytic geometry – For complex Lie groups

(…) (Brylinski 00) (…)

In arithmetic geometry – For reductive groups

For reductive algebraic group GG there is no sensible element in H 3(BG,𝔾 m)H^3(\mathbf{B}G, \mathbb{G}_m), but there is the following.

Write K 2(R)K_2(R) for the degree-2 algebraic K-theory group of a commutative ring (e.g. Isely 05, section 4) and write K 2\mathbf{K}_2 for corresponding abelian sheaf on the suitable etale site (e.g. Deligne-Brylinski 01, page 6).


H 4(BG ,)H 2(BG,K 2) H^4(B G_{\mathbb{C}}, \mathbb{Z}) \simeq H^2(\mathbf{B}G, \mathbf{K}_2)

This is (HKLV 98, theorem 4.11) also (Deligne-Brylinski 01), going back to (Bloch 80). See also (Kapranov 00, (2.1), MO discussion).

Notice that over \mathbb{C} the Beilinson regulator c 1,2c_{1,2} (e.g. Brylinski 94, theorem, 3.3) relates

H 1(X,K 2)H 2(X,𝔾 m). H^1(X, \mathbf{K}_2) \longrightarrow H^2(X, \mathbb{G}_m) \,.

The CS 3-bundle 3-connection is the extended Lagrangian for ordinary GG-Chern-Simons theory. See there for more.


For Lie groups

The Cech-Deligne cohomology realization

As cocycles in Cech-Deligne cohomology the Chern-Simons 2-gerbe has been constructed explicitly in

  • Jean-Luc Brylinski and Dennis McLaughlin, A geometric construction of the first Pontryagin class (1993) (pdf)

as a special case of the general construction in

  • Jean-Luc Brylinski and Dennis McLaughlin, Cech cocycles for characteristic classes , Communications in Mathematical Phiysics, Volume 178, Number 1, (Springer)

The 2-gerbe realization

Conceived as a genuine gerbe the Chern-Simons 2-gerbe appears in

  • Jean-Luc Brylinski and Dennis McLaughlin, The geometry of degree-4 characteristic classes and of line bundles on loop spaces II (pdf).

Among the first references to apply specifically bundle gerbe technology to this construction is

This was later refined in

Here are some slides from talks:

  • Krzysztof Gawedzki, Wess-Zumino-Witten and Chern-Simons theories for non-simply connected Lie groups (pdf)

The full Chern-Simons circle 3-connection on the full moduli stack of GG-principal connections BG connB 3U(1) conn\mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} was then constructed in

Exposition of this and further developments are in

For complex reductive groups

Discussion in complex analytic geometry of multiplicative Chern-Simons holomorphic line 2-bundles is in

The relevant Beilinson regulator is discussed also in

  • Jean-Luc Brylinski, Holomorphic gerbes and the Beilinson regulator, Astérisque 226 (1994): 145-174 (pdf)

For reductive algebraic groups

For the case of reductive algebraic groups:

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