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 $G$-principal bundle with connection.
For $G$ a Lie group and $c : \mathcal{B}G \to K(\mathbb{Z},4)$ a cocycle for a degree 4 characteristic class in integral cohomology and $X$ a smooth manifold, Chern-Weil theory provides a morphism (the refined Chern-Weil homomorphism)
from $G$-principal bundles with connection $\nabla$ to degree 4 ordinary differential cohomology. The cocycles on the right may be thought of as
circle 3-bundles with connection$\hat c(\nabla)$;
bundle 2-gerbes with connection.
By construction, the curvature 4-form of $\hat c(\nabla)$ is the curvature characteristic form $\langle F_\nabla \wedge F_\nabla\rangle$ of $\nabla$ and accordingly the 3-form connection on $\hat c(\nabla)$ is locally a Chern-Simons form $CS(\nabla)$ of $\nabla$.
Accordingly, the higher parallel transport induced by $\hat c(\nabla)$ over 3-dimensional manifolds $\phi : \Sigma \to X$ 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 $G$-universal principal bundle over a smooth version of $B G$, then $\hat c(\nabla)$ is the background gauge field for bare Chern-Simons theory.
Therefore this structure $\hat c(\nabla)$ 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 $G$ one may enhance the assignment $\nabla \mapsto \hat c(\nabla)$ to a morphism of ∞-groupoids
where on the left we have the groupoid of smooth $G$-principal bundles with connection on $X$, 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 $G$-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 (Brylinski-McLaughlin I) there is spelled out an explicit construction of $\hat c(\nabla)$ 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 $G$ is a simply connected compact simple Lie group, such as the spin group, and take the characteristic class $c$ to be that whose transgression to $G$ has as image in de Rham cohomology the de Rham class of the normalized canonical Lie algebra cocycle $\mu \in CE(\mathfrak{g})$.
For $P \to X$ a $G$-bundle with connection $\nabla$, there exists an open cover $\{U_i \to X\}$ such that we have a Cech cohomology cocycle for $P$ given by a smooth transition function
satisfying on $U_i \cap U_j \cap U_k$ the cocycle condition $g_{i j} \cdot g_{j k} = g_{i k}$.
Since $G$ 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 $G$ 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
such that $\hat g_{i j}(x,0 ) = e$ is the identity in $g$, and such that such that $\hat g_{i j}(x,1) = g_{i j}(x)$;
on triple intersections, choose smooth functions
that cobound these paths in the evident way
(This can be done because $\pi_1(G) = *$.)
on quadruple intersection choose smooth functions
such that these 3-balls fill the evident tetrahedra.
(This can be done because $\pi_2(G) = 0$.)
The Cech cohomology cocycle with coefficients in $\mathbf{B}^3 \mathbb{R}/\mathbb{Z}$ which is given by
is well defined and represents the class $c(P) \in H^4(X)$ in integral cohomology.
Moreover, this refines to the cocycle in Cech-Deligne cohomology that is given by
where
$A \in \Omega^1(P,\mathfrak{g})$ is the incarnation of the connection $\nabla$ as an Ehresmann connection given by a glbally defined 1-form on the total space $P \to X$ of the bundle;
$\sigma_i : U_i \to P$ are the local sections of $P \to X$ that induce the original Cech cocycle $(g_{i j} := \sigma_i \cdot \sigma_j^{-1})$;
$P \cdot \hat g_{i j} : \Delta^1 \to P$ is given by the right action of $G$ on $P$ and analogously for the other terms;
$CS(...)$ denotes the Chern-Simons form of the given $\mathfrak{g}$-valued 1-form.
First notice that this is indeed well-defined: by compactness and simplicty of $G$ we have $\pi_3(G) = \mathbb{Z}$. By assumption on $\mu \in \Omega^3(G)$, for any map $f : S^3 \to G$, we have $\int_{S^3} f^*\mu \in \mathbb{Z} \subset \mathbb{R}$. This implies that $c(g)$ 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 $[c(g)]$ in de Rham cohomology from the curvature 4-form of this Deligne cocycle. That is by construction $\langle F_\nabla \wedge F_\nabla \rangle \in \Omega^4_{cl}(X)$, 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.
The above Cech-Deligne cocycle construction of $\hat c(\nabla)$ 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
from the $G$-principal bundles with connection to ordinary differential cohomology are these:
for a given Lie algebra $\mathfrak{g}$ the realization of the corresponding Lie group as a truncation of the simplicial presheaf
(see Lie integration);
the observation that, up to subtleties with the truncation, a Lie algebra cocycle
induces therefore an integrated cocycle $\mathbf{B}G \to \mathbf{B}^k U(1)$;
the observation that this is lifted to connections and differential refinement by
thickening the simplicial presheaf for $\mathbf{B}G$ to
thickening the Lie algebra cocycle by its Chern-Simons element
and then postcomposing with that. Note that the above diagram is part of a larger diagram involving the invariant polynomial $\langle-\rangle$ for $\mu$ and exhibiting the Chern-Simons element as a transgression element between these two:
Also note that $inv(b^{k-1} \mathbb{R})\cong CE(b^k \mathbb{R})$.
For the case at hand, let $\mathfrak{g}$ be a semisimple Lie algebra, $\langle -\rangle : CE(b^3\mathbb{R})\to W(\mathfrak{g})$ its canonical Killing form invariant polynomial, $\mu = \langle -,[-,-]\rangle: CE(b^2\mathbb{R})\to CE(\mathfrak{g})$ the corresponding Lie algebra cocycle, $cs: W(b^2\mathbb{R})\to W(\mathfrak{g})$ the Chern-Simons elements exhibiting the transgression between the two, $G$ 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 $\mu$.
Consider the simplicial presheaf
where here and in what follows differential forms $\omega$ on simplices are taken to have sitting instants in that for all $k \in \mathbb{N}$ there exists for every $k$-face of $\Delta^n$ an open neighbourhood such that $\omega$ restricted to that open neighbourhood is constant in the direction perpendicular to the boundary.
The canonical map
from the 3-coskeleton of $\exp(\mathfrak{g})$ to the delooping of the simply connected Lie group $G$ which is given on 1-morphisms by higher parallel transport is an equivalence ( in the model structure on simplicial presheaves $[CartSp^{op}, sSet]_{proj}$).
Use that a $\mathfrak{g}$-valued 1-form on the interval is canonically identified with a based path in $G$. Then use that for $k \leq 2$ we have $\pi_k(G) = 0$. See Lie integration for more.
There is a commuting diagram
where the right vertical morphism is the composite of the equivalence
discussed at Integration to Line n-groups with the evident quotient $\mathbf{B}^3 \mathbb{R} \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}$, where the copy of $\mathbb{Z}$ in $\mathbb{R}$ is the lattice of periods of $\mu$ over 3-spheres in $G$.
The top morphism sends a $U$-family of 3-morphisms $\Omega^\bullet(U \times \Delta^3) \stackrel{A}{\leftarrow} CE(\mathfrak{g})$ – which we may think of as a $U$-family of based 3-balls $\Sigma : U \times \Delta^3 \to G$ – to the family of 3-forms
which we may think as a family of closed 3-forms
The right vertical morphism sends this to the fiber integration
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 $\mathbf{cosk}_3 \exp(\mathfrak{g})$ – which is a 3-sphere $V : S^3 \to G$ – we have that $\int_{S^3} V^* \mu^*(A) = 0 \;mod\; \mathbb{Z}$, which is true by the fact that we take $\mathbb{Z}$ to be precisely generated by these periods. (Alternatively we can assume $\mu$ to be normalized such that it generates the image in deRham cohomology of $H^3(G,\mathbb{Z}) \simeq \mathbb{Z}$.)
We shall by slight abuse of notation write $\exp(\mu)$ also for the morphism $\mathbf{cosk}_3 \exp(\mathfrak{g}) \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}$.
For $\{U_i \to X\}$ a cover and $C(U) \in [CartSp^{op}, sSet]_{proj}$ the corresponding Cech nerve we have that
a morphism $g : C(U) \to \mathbf{B}G$ is precisely a Cech 1-cocycle with values in $G$;
a lift
is precisely a lift of this cocycle to a system of paths, triangles and tetrahedra in $G$, as above.
Write $\exp(\mathfrak{g})_{diff}$ for the simplicial presheaf
Its 3-coskeleton $\mathbf{cosk}_3 \exp(\mathfrak{g} \to inn(\mathfrak{g}))$ is the coefficient for $G$-principal bundles with pseudo-connection adapted to the model $\mathbf{cosk}_3 \exp(\mathfrak{g})$ for $\mathbf{B}G$.
Pseudo-connections $\hat \nabla$
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 $G$-bundle given by $C(U) \to \mathbf{B}G$.
On single patches $\hat \nabla$ is a collection of $\mathfrak{g}$-valued 1-forms $A_i \in \Omega^1(U_i, \mathfrak{g})$.
On double intersection it is a collection
whose restriction $\lambda_{i j}$ to $\Delta^1$ is the given path $\hat g_{i j}$ that is being covered. The condition that the curvature of $\hat A_{i j}$ has no component in the simplicial direction is the differential equation
This differential equation has a unique solution for the boundary condition $A_{i j}(0) = A_i$ given by
(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 $\frac{\partial}{\partial t} \hat g_{i j} = \hat g_{i j} \cdot \lambda$ (by definition) and using that the claim follows.)
In particular this implies the forms on single patches satisfy the ordinary cocycle relation
for connections.
Similarly there are differential equations on 2-simplices and 3-simplices with unique solutions.
Pasting postcomposition with the diagram
induces a morphism $\exp(\mathfrak{g})_{diff} \to \exp(b^2\mathbb{R})_{diff}$ and we obtain a commuting diagram
that covers the corresponding diagram we had before.
Here we are using the object $\mathbf{B}^3 U(1)_{ch,diff}$ described in detail at circle n-bundle with connection.
The deeper reason for this construction is that the zig-zag composite
of morphisms of simplicial presheaves models the intrinsically defined morphism
in the (∞,1)-topos Smooth∞Grpd.
The outer composite morphism
is precisely the Cech-Deligne cocycle
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 $CS(\hat A_{i j})$ is the image of the degree 3-generator on $W(b^2 \mathbb{R})$ under the composite
The remaining fiber integration is then that exhibiting the equivalence of simplicial differential forms
that is described in some detail at Circle n-bundle with connection – models from ∞-Lie integration.
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 $\mathcal{G} \to G$ on the Lie group and notices that (more or less implicitly by recourse to its delooping 2-gerbe on $\mathbf{B}G$) that this has a multiplicative structure .
Using this we see that for $P \to X$ any $G$-principal bundle and $P^{[2]} : = P \times_X P \to P \times G$ the principality isomorphism, the pullback of $\mathcal{G}$ along
serves to provide the diagram
on which the pullback of the multiplicative structure on $\mathcal{G}$ induces the structure of a bundle 2-gerbe, in that we get morphisms of bundle gerbes
that are associative up to a higher coherent morphisms, etc.
(…) (Brylinski 00) (…)
For reductive algebraic group $G$ there is no sensible element in $H^3(\mathbf{B}G, \mathbb{G}_m)$, but there is the following.
Write $K_2(R)$ for the degree-2 algebraic K-theory group of a commutative ring (e.g. Isely 05, section 4) and write $\mathbf{K}_2$ for corresponding abelian sheaf on the suitable etale site (e.g. Deligne-Brylinski 01, page 6).
Then
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 $c_{1,2}$ (e.g. Brylinski 94, theorem, 3.3) relates
The higher parallel transport of a Chern-Simons circle 3-bundle is the action functional for Chern-Simons theory.
For the case that $G = O$ is the orthogonal group and $X \to \mathbf{B}O$ the classifying map of the tangent bundle of $X$, a trivialization of the corresponding Chern-Simons 3-bundle is a string structure on $X$. A trivialization of the Chern-Simons 3-bundle with connection is a differential string structure on $X$.
For products of torus groups and the cup product class the same construction yields the T-duality 2-group
universal Chern-Simons n-bundle
Chern-Simons circle 3-bundle
The CS 3-bundle 3-connection is the extended Lagrangian for ordinary $G$-Chern-Simons theory. See there for more.
As cocycles in Cech-Deligne cohomology the Chern-Simons 2-gerbe has been constructed explicitly in
as a special case of the general construction in
Conceived as a genuine gerbe the Chern-Simons 2-gerbe appears in
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:
The full Chern-Simons circle 3-connection on the full moduli stack of $G$-principal connections $\mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$ was then constructed in
Exposition of this and further developments are in
Discussion in complex analytic geometry of multiplicative Chern-Simons holomorphic line 2-bundles is in
Jean-Luc Brylinski, around theorem 5.4.10 (p. 226-227) of Loop spaces and characteristic classes, Birkhäuser
Jean-Luc Brylinski, Gerbes on complex reductive Lie groups (arXiv:math/0002158)
The relevant Beilinson regulator is discussed also in
For the case of reductive algebraic groups:
Spencer Bloch, The dilogarithm and extensions of Lie algebras, Algebraic K-Theory Evanston 1980 Lecture Notes in Mathematics Volume 854, 1981, pp 1-23 (publisher)
Hélène Esnault, Bruno Kahn, Marc Levine, Eckart Viehweg, The Arason invariant and mod 2 algebraic cycles, J. Amer. Math. Soc. 11 (1998), 73-118 (pdf,publisher page)
Mikhail Kapranov, The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups (arXiv.math/0001005)
Pierre Deligne, Jean-Luc Brylinski, Central extensions of reductive groups by $K_2$, Publications Mathématiques de l’IHÉS 2001 (web, pdf)
Last revised on February 4, 2021 at 09:17:31. See the history of this page for a list of all contributions to it.