## February 12, 2008

### Construction of Cocycles for Chern-Simons 3-Bundles

#### Posted by Urs Schreiber

In the old article

A geometric construction of the first Pontryagin class
Quantum Topology, 209-220
(review)

a concrete construction of a Čech cocycle (= local transition data) for a line 3-bundle (aka 2-gerbe) classified by the first Pontrjagin class $p_1(P) \in H^4(X,\mathbb{Z})$ of a given principal $G$-bundle $P \to X$ is discussed, following

Čech cocycles for characteristic classes
Comm. Math. Phys. 178 (1996)
(pdf, review).

The point of the first article is to interpret the construction of the second in terms of the holonomy of the gerbe on $G$.

Here I want to discuss the following:

- how the fundamental 2-groupoid $\Pi_2(X_{CE(g_\mu)})$ of the smooth space obtained from the small version $g_\mu$ of the String Lie 2-algebra (section 6.4.1) as described here is another version of the strict String 2-group

$\mathbf{B} String'(G) = \Pi_2(X_{CE}(g_\mu))$

sitting inside the sequence (proposition 20)

$1 \to \mathbf{B} U(1) \to String'(G) \to G \to 1$

- how the construction described by Brylinski and McLaughlin is a special case of the construction of obstructing $n$-bundles for the attempted lift through the above sequence, using weak cokernels as described in the integral way used in the following from slide 518 on and as corresponds differentially to the construction in section 8 of $L_\infty$-connections and applications (pdf, blog, arXiv).

A) the construction of $String'(G) := \Pi_2(X_{CE(g_\mu)})$

Let $g$ be a simple Lie algebra with canonical bilinear invariant form $\langle \cdot,\cdot\rangle$ whose normalization we’ll fix such that the associated Lie algebra 3-coycle $\mu = \langle \cdot, [\cdot,\cdot]\rangle$ corresponds to an integral 3-form on the simple, simply connected compact Lie group $G$ integrating $g$. See From loop groups to 2-groups for more details.

Recall from section 6.4 that this cocycle defines a Lie 2-algebra

$g_\mu \,,$

the (small version of the) String Lie 2-algebra (section 6.4.1), which is, according to section 6.5, completly characterized by the fact that for any smooth space $Y$, flat $g_\mu$-valued differential forms on $Y$ are tuples consisting of a flat $g$-valued 1-form $A$ and a 2-form $B$ satisfying $d B = \mu(A)$:

$\Omega^\bullet_{flat}(Y,g_\m0) = \{ (A,B) \in \Omega^1(Y,g) \times \Omega^2(Y) | d B = \mu(A) \} \,.$

Of course this is at the heart of André Henriques’ integration of $g_\mu$ to a smooth Kan complex (a Lie $\infty$-groupoid) as recalled here – but as also indicated there and further expanded on in Differential forms on smooth spaces, I am thinking that we can in fact use this to integrate $g_\mu$ to a strict Lie 2-group (as done by more indirect means here, compare slide 152) by first forming the smooth space (a sheaf of sets on smooth test domains) $X_{CE(g_\mu)}$ and then forming the strict fundamental path 2-groupoid $\Pi_2(X_{CE(g_\mu)})$ of that.

For any generalized smooth space $X$ (sheaf on smooth test domains) we can form the strict fundamental path 2-groupoid $\Pi_2(X) \,,$ which is itself a groupoid internal to generalized smooth spaces, by following the construction in section 4.1 of Smooth 2-functors and differential forms (pdf, blog).

Morphisms of $\Pi_2(X)$ are smooth maps from the interval $[0,1] \to X$ with sitting instants and modulo thin homotopies $[0,1]^2 \to X$, where a homotopy is called thin if all 2-forms on $X$ pulled back along it vanish.

2-Morphisms of $\Pi_2(X)$ are smooth maps $[0,1]^2 \to X$ (not necessarily thin now, of course) modulo higher homotopies $[0,1]^3 \to X$.

As discussed in the entry on Transgression (pdf, blog), the generalized smooth space $X_{CE(g_\mu)}$ is defined by the fact that a smooth map $U \to X$ from a test domain $U$ into it is precisely a choice of flat $g_\mu$ valued forms on $U$, i.e. an element in $\Omega^\bullet_{flat}(U,g_\mu)$.

Therefore:

1-morphisms in $\Pi_2(X_{CE(g_\mu)})$ are $g$-valued 1-forms on the interval modulo reparameterization. This is the same as reparameterization classes of paths in $G$, starting at the identity. So $Mor_1(\Pi_2(X_{CE(g_\mu)})) = P G \,.$ Up to the fact that reparameterization is divided out here, this is as in From loop groups to 2-groups, only that now composition of 1-morphisms is not the pointwise multiplication in $P G$, but is concatenation of paths.

2-morphisms in $\Pi_2(X_{CE(g_\mu)})$ are flat $g$-valued 1-forms on the unit square, restricting to the given 1-forms on the boundary, together with a 2-form on the unit square, modulo a couple of relations.

By the standard lore about integration of ordinary Lie algebras along the lines discussed here we find that all paths in $G$ with the same endpoint may be connected by a 2-morphism.

We divide out all 3-dimensional homotopies:

dividing out the thin 3-dimensional homotopies amounts, again, to dividing out orientation preserving reparameterizations (automorphisms of the unit square) and making orientation-reversing reparameterizations correspond to inverses (dividing out “spikes”). For the 2-form this means, following the reasoning of Integration without integration (pdf, blog) that only its integral over the unit square is left after quotienting. So that’s just a (real, say) number.

finally, not-necessarily thin 3-dimensional homotopies identify two 2-morphism represented each by a flat $g$-valued 1-form and a 2-form on the unit square if the flat 1-forms can be extended to a flat 1-form $\tilde A \in \Omega^1_{flat}([0,1]^3,g)$ the unit cube and the 2-forms to a 2-form $\tilde B$ on $[0,1]^3$ satisfying $d \tilde B = \mu(\tilde A)$, such that the integrals of the two 2-forms differ by the integral of $\mu(\tilde A)$ over the cube: $\int_{[0,1]^3} d \tilde B = \int_{[0,1]^2} B - \int_{[0,1]^2} B' = \int_{[0,1]^3} \langle \tilde A \wedge \tilde A \wedge \tilde A \rangle \,.$

To recognize this, notice the following: on a simply connected space $U$ every flat $g$-valued 1-form $\tilde A$ may be expressed as $\tilde A = g^{-1} d g$ for $g : U \to G$ a suitable smooth function. (To construct $g$, fix any point $x_0$ and let for all $x \in U$ the value of $g(x)$ be the parallel transport of $\tilde A$ from $x_0$ to $x$ along any path connecting them.)

Since the unit cube is simply connected, we find that as a result the following characterization:

2-morphisms in $\Pi_2(X_{CE(g_\mu)})$ are triples consisting of a loop in $G$ starting at the identity (and equipped with another marked point), a surface in $G$ cobounding that loop, together with a real number, modulo the relation which identifies two such pairs if the boundary loop coindices, and if the two surfaces can be filled by a cobounding 3-ball such that the difference in the two numbers is the integral $\int_{[0,1]^3} \langle (g^{-1}dg) \wedge (g^{-1}dg) \wedge (g^{-1}dg) \rangle$ of the canonical 3-form on $G$ over this 3-ball.

You’ll recognize again the appearance of the construction of the “tautological” bundle gerbe on $G$, as also describe in From loop groups to 2-groups.

The difference here to the construction there is only in the realization of the multiplicative structure on this beast: in $\Pi_2(X_{CE(g_\mu)})$ horizontal composition is by concatenation of paths, not by the product in $G$.

Notice that $\Pi_2(X_{CE(g_\mu)})$ has just a single object (as I emphasized here).

So I write $\mathbf{B} String'(G) := \Pi_2(X_{CE(g_\mu)})$ for the strict one-object 2-groupoid defined this way, where $String'(G)$ denotes the underlying strict 2-group.

A 1) re-interpretation using $\Pi_3(X_{CE(g_\mu)})$

Notice that, while the label $\int_{[0,1]^2} B$ carried by any representative of a 2-morphism in $\Pi_2(X_{CE(g_\mu)})$ is an element in $\mathbb{R}$, its value in the equivalence class of representatives is well defined only modulo $\mathbb{Z}$: a 3-dimensional homotopy from the constant path to itself divides out the integral of $\mu(g^{-1} d g)$ over a closed 3-manifold. Since this form is integral (by assumption on the normalization of $\mu$) this is an integer. If we take $\mu$ normalized such that it yields the generator of $H^3(G,\mathbb{Z}) \simeq \mathbb{Z}$ then every integer arises this way.

We can make this more explicit by incorporating the 3-dimensional homotopies explicitly as 3-morphisms our $n$-group.

This is a general phenomenon: notice that the ordinary (1-)group $U(1)$ is equivalent, as a 2-group, to the 2-group coming from the crossed module $\mathbb{Z} \hookrightarrow \mathbb{R}$:

$U(1) \simeq (\mathbb{Z} \hookrightarrow \mathbb{R}) \,.$

Same for the shifted versions

$\mathbf{B}^{n-1} U(1) \simeq \mathbf{B}^{n-1}(\mathbb{Z} \hookrightarrow \mathbb{R}) \,.$

This means that we can regard, in particular, a String-like extension by shifted $U(1)$

$1 \to \mathbf{B}^{n-1} U(1) \to \hat G \to G \to 1$

as an extension

$1 \to \mathbf{B}^{n-1} (\mathbb{Z} \hookrightarrow \mathbb{R}) \to \hat G \to G \to 1 \,.$

From the point of view of obtaining $\hat G$ as the fundamental path $n$-groupoid of a generalized smooth space, this may actually be more natural. So if we instead compute $\Pi_3(X_{CE(g_\mu)})$ where we divide out only thin homotopies up to level three, we find, by the same arguments as before, $String''(G)$ as a strict 3-group whose 2-morphisms are labeled by loops in $G$ labeled by elements in $\mathbb{R}$, and whose 3-autmorphisms of the trivial 2-morphisms are labeled by $\mathbb{Z}$.

The two different perspectives, the 1-group $U(1)$ versus the equivalent 2-group $(\mathbb{Z} \to \mathbb{R})$ underlie the fact that $\mathbb{Z}$-$n$-gerbes are the same as $U(1)$-($n-1$)-gerbes. On p. 211 of their A geometric construction of the first Pontryagin class Brylinski and McLaughlin take the $\mathbb{Z}$-$3$-gerbe point of view, on p. 215 they take the $U(1)$-2-gerbe point of view.

B) the construction of the obstruction to the lift of a $G$-bundle through the String-extension

Recall from Obstructions for $n$-bundle lifts and from sections Obstruction theory (slide 501) and Obstructing $n$-bundles: integral picture (slide 518) how we compute the cocycles for the “lifting ($n-1$)-gerbes” or “obstructing $n$-bundles” for lifts of Čech cocycles of $G$-bundles through String-like (or other) extensions:

$1 \to B^{n-1} U(1) \to \hat G \to G \to 1$

of their structure group to a structure $n$-group $\hat G$:

we first form the $(n+1)$-group denoted

$(B^{n-1}U(1) \hookrightarrow \hat G)$

which is the weak cokernel of this inclusion. This is equivalent to $G$

$G \simeq (B^{n-1}U(1) \hookrightarrow \hat G)$

(we may need anafunctorial equivalence for this to be true, which in turn means that the following requires that we have refined the cover sufficiently)

and hence every ordinary Čech cocycle

$\array{ && (x,j) \\ & \nearrow && \searrow \\ (x,i) &&\stackrel{}{\to}&& (x,k) } \;\; \mapsto \;\; \array{ && \bullet \\ & {}^{g_{ij}(x)}\nearrow && \searrow^{g_{jk}(x)} \\ \bullet &&\stackrel{g_{ik}(x)}{\to}&& \bullet }$

may always be lifted to a $(B^{n-1}U(1) \hookrightarrow \hat G)$ $n+1$-cocycle. This just means that we can simply choose lifts $\hat g_{ij}(x)$ of all $g_{ij}(x)$: the failure of these choices to form a genuine lift of the cocyc le to $\hat G$ will always be in the image of $B^{n-1}U(1) \hookrightarrow \hat G$, hence there is a unique label in $U(1)$ of every $n+1$-simplex measuring that failure.

The obstructing $n+1$-bundle, finally, is obtained by forgetting everything except these lables of $(n+1)$-simplices, which amounts, technically, to projecting the $(B^{n-1}U(1) \hookrightarrow \hat G)$ cocycle along the canonical projection $(B^{n-1}U(1) \hookrightarrow \hat G) \to B^n U(1) \,.$

Now apply this to the String extension $B U(1) \to String'(G) \to G$ discussed above. Notice that $U(1) \simeq (\mathbb{Z} \hookrightarrow \mathbb{R} )$, turn the crank and compare with Brylinski-McLaughlin: one gets their construction.

More in detail, we obtain the following.

C) The Brylinski-McLaughlin construction in light of the above

We start with a principal $G$-bundle $P \to X$ over $X$. Choose a surjective submersion $\pi : Y \to X$ (for instance a cover of $X$ by open sets, as they do) such that there is a trivialization of the pulled back bundle $\pi^* P$.

Actually, a convenient choice of $Y$ for our particular interest here is to take $Y = P$ and use the fact that every principal bundle canonically trivializes when pulled back along its own projection map. More on that below (but this is not what Brylinski-McLaughlin consider, though they could have just as well).

From the local trivialization, we obtain a $G$-cocycle as usual: for every point in a triple intersection, $y \in Y^{[3]}$ we have an assignment: $\array{ && \pi_{2} y \\ & \nearrow && \searrow \\ \pi_1 y &&\stackrel{}{\to}&& \pi_3 y } \;\; \mapsto \;\; \array{ && \bullet \\ & {}^{\pi_{12}^* g(y)}\nearrow && \searrow^{\pi_{23}^* g(y)} \\ \bullet &&\stackrel{\pi_{13}^* g(y)}{\to}&& \bullet } \,.$

Now, to obtain the Čech coycle corresponding to the first Pontrjagin class of this bundle, Brylinski and McLaughlin ask us to first of all lift all the elements in $G$ here to paths in $G$ starting at the identity and ending at the given element in $G$, and to replace the group product along the edges of the above triangles with concatenation of these paths.

$\array{ && \pi_{2} y \\ & \nearrow && \searrow \\ \pi_1 y &&\stackrel{}{\to}&& \pi_3 y } \;\; \mapsto \;\; \array{ && \bullet \\ & {}^{\pi_{12}^* \hat g(y)}\nearrow && \searrow^{\pi_{23}^* \hat g(y)} \\ \bullet &&\stackrel{\pi_{13}^* \hat g(y)}{\to}&& \bullet } \,.$

We see now that this is nothing but the object part of an attempted lift through $String'(G) \to G \,.$

In general this will fail to be a lift of cocycles, though. As explained above, it does however always yield a lift through the weak cokernel 3-group $(U(1) \hookrightarrow String'(G)) \to G$ in that for any choice of lifts $\hat g$ as above, there will be a unique way to label the interior of every tetrahedron: namely by extending the map to $G$ to the entire tetrahedron, pulling back the 3-cocycle to that and integrating it, regarding the result modulo $\mathbb{Z}$. This is Brylinski-McLaughlin’s statement on p. 216, last sentence above section 3.

D) Relation to lifts of $L_\infty$-connections

In $L_\infty$-connections and applications (pdf, blog, arXiv) it is described how to conceive such cocycles entirely Lie algebraically by replacing smooth cocycle pseudo-functors from the Čech groupoid to the structure $n$-group by their differential version from the corresponding Lie algebroid to the corresponding Lie $n$-algebra.

For this purpose, consider in particular the choice of surjective submersion given by the $G$-bundle itself: $Y := P \,.$

By assumption, $G$ is simply connected and hence so are the fibers of $P$. But this means that the Čech groupoid whose space of morphism is $Y^{[2]} = P^{[2]} = P \times_X P$ is nothing by the fundamental vertical path groupoid of $P$ $Y^{[2]} = \Pi_1^{\mathrm{vert}}(Y) = \Pi_1^{\mathrm{vert}}(P) \,,$ the groupoid of all paths in $Y= P$ whose projection down along $p : P \to X$ is a constant path.

The Chevalley-Eilenberg algebra of the Lie algebroid corresponding to the fundamental path groupoid of a space $U$ is just the algebra of forms on $U$:

$CE(Lie(\Pi_1(U))) = \Omega^\bullet(U) \,.$

Hence the Chevalley-Eilenberg algebra of the Lie algebroid corresponding to the fundamental vertical path groupoid of a space $U$ is just the algebra of vertical forms on $U$:

$CE(Lie(\Pi_1^{\mathrm{vert}}(Y))) = \Omega^\bullet_{\mathrm{vert}}(Y)$

(definition 11, p. 23).

Hence our $G$-bundle cocycle, which is a pseudeofunctor $g : Y^{[2]} \to \mathbf{B} G$ and the obstructing Chern-Simons 3-cocycle $p(\hat g) : Y^{[2]} \to \mathbf{B}^2 U(1)$ which we constructed from the attempted lift $\hat g : Y^{[2]} \to \mathbf{B} String'(G)$ of this to a String 2-cocycle all become Lie algebroid morphism whose dual version are DGCA morphisms

$\Omega^\bullet_{\mathrm{vert}}(Y) \stackrel{Lie(g)^*}{\leftarrow} CE(g)$ and $\Omega^\bullet_{\mathrm{vert}}(Y) \stackrel{Lie(\hat g)^*}{\leftarrow} CE(string'(g))$ and $\Omega^\bullet_{\mathrm{vert}}(Y) \stackrel{Lie(p(\hat g))^*}{\leftarrow} CE(b^{2}u(1))$ respectively.

Using this translation, it becomes evident how the discussion of lifts and obstructions in section 8 is the Lie analog of the integral discussion given here.

Posted at February 12, 2008 11:23 AM UTC

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## 1 Comment & 6 Trackbacks

### Re: Construction of Cocycles for Chern-Simons 3-Bundles

I made the remark above that it is useful to look at cocycles not with respect to an open cover

$(Y = \sqcup U) \to X$

of $X$, but instead using the surjective submersion given by the total space $P$ of the principal $G$-bundle itself

$(Y = P) \to X \,.$

Since $G$ is assumed to be simply connected (I am assuming that) a morphism in the Čech groupoid

$\pi_1 y \to \pi_2 y$

is, if we take $Y = P$, precisely a homotopy class of paths in a fiber of $P$. There is the canonical flat $g$-valued 1-form $A_{vert}$ on these fibers, defined by the fact that it sends a vertical vector to the Lie algebra element generating it by the right action of $G$ on $P$.

Hence when we take $Y = P$ the lift of elements in $G$ to paths in $G$ ending at these elements which Brylinski-McLaughlin describe (and which I pointed out is a special case of the lift through weak cokernels of $n$-groups) is actually almost canonically given in that it is canonically given after we choose a representative for each homotopy class of paths in the fiber.

Once we do that the path in $G$ which we are after is just the path in the fiber, identified as a path in $G$ by identitfying the source point of the path with the identity element in the group.

Or in other words: the 1-form on the interval which defined a morphism in the String Lie 2-group $\Pi_2(X_{CE(g_\mu)})$ is then just the 1-form $A_{vert}$ on the fiber restricted to that path.

This should be compared with section 8, where this vertical 1-form on the fiber indeed represents the cocycle.

And it is a special case of a more general construction, which relates the Lie $\infty$-algebraic constructions there to the integrated cocycle constructions:

suppose you have an $n$-fold connected fiber and an $n$-functor from the fundamental $n$-groupoid of it with values in the structure $n$-group $G_n$.

Then we obtain a cocycle for $G_n$-n-bundles by picking

- one path between any two points;

- one surface between any triangle of paths such chosen paths;

- one volume between any tetrahedron of such chosen surfaces

etc, and restricting our $n$-functor to these chosen representatives.

One sees that the class of the cocycle obtained this way is independent of these choices.

This is one way to see how “a Čech cocycle is a flat transport on the fibers” while a connection for that cocycle is then an extension of that flat transport to a not-necessarily flat transport on the total space.

This is at the heart of the description of $n$-bundles in terms of flat vertical and non-flat general $L_\infty$-algebra valued forms in $L_\infty$-connections.

Posted by: Urs Schreiber on February 13, 2008 4:29 PM | Permalink | Reply to this
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