The n-Category Café

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.