The n-Category Café

Chern-Simons 2-Gerbe and its Structure 3-Group

Let $X$ be a $d$-dimensional manifold. (Later we want $d=3$, but at the moment $d$ can be arbitrary.)

Let $B\to X$ be a principal $G$-bundle on $X$, for $G$ a compact, simple and simply connection Lie group. (For the time being I consider everything without connection.)

Let $U \to X$ be a good covering by open contractible sets of $X$ and let $U^{[2]} \stackrel{\to}{\to} U$ be the corresponding Lie groupoid.

Then, up to isomorphism, the bundle $B$ is a strict functor $$B : U^{[2]} \to \Sigma(G) \,.$$

Now consider this same functor as a pseudofunctor to the discrete 3-group on $\Sigma(G)$: $$B : U^{[2]} \to \Sigma(1 \to 1 \to G) \,.$$ This amounts to regarding the principal (1-)bundle $B$ as a degenerate case of a principal 3-bundle.

Denote by $P G$ the group of piecewise smooth parameterized paths in $G$, based at the neutral element.

Denote by $\Omega G$ the group of piecewise smooth parameterized loops in $G$, based at the neutral element.

There is an obvious action of $P G$ on $\Omega G$ by conjugation, and an obvious homomorphism from loops to paths, which makes $$\Omega G \to P G$$ a crossed module of groups, hence a strict 2-group.

By smoothly sending elements $g \in G$ to chosen based paths in $G$ ending at $g$, we can always lift the functor $$B : U^{[2]} \to \Sigma(1 \to 1 \to B G)$$ to a functor $$B' : U^{[2]} \to \Sigma(1 \to \Omega G \to P G) \,.$$ Notice that there is a unique element in $\Omega G$ going between elements of $P G$ with coinciding endpoint. This makes the lift from $B$ to $B'$ always possible.

So this still describes, up to isomorphism, a $G$-bundle on $X$.

But the action of $P G$ on $\Omega G$ lifts to an action on the level $\tau$ central extention $\hat \Omega_\tau G$.

Therefore we may want to further lift $B'$ to a pseudofunctor $$B'' : U^{[2]} \to \Sigma(1 \to \hat \Omega_\tau G \to P G) \,.$$

This lift, however, is obstructed by the Pontryagin class in $H^4(X, \mathbb{Z})$. If this does not vanish, we instead get a pseudofunctor $$B'' : U^{[2]} \to \Sigma(U(1) \to \hat \Omega_\tau G \to P G) \,,$$ determined by that class in $H^4(X, \mathbb{Z})$.

The 3-group $U(1) \to \hat \Omega_\tau G \to P G$ may be thought of as the largest strict 3-group inside the automorphism 3-group $\mathrm{AUT}(\mathrm{String}_G)$ (following the calculation here).

This pseudofunctor is (the local data for) a $G$-Chern-Simons 2-gerbe on $X$.

Notice that the 2-fold pseudoness of this functor means that it assigns elements of $P G$ to points in double intersections, such that the product in $P G$ is respected up to an element of $\hat \Omega_\tau G$ on triple intersections, which form a tetrahedron on quandruple intersections that is filled by an element of $U(1)$, such that all these $U(1)$-elements make a 4-simplex commute on quintuple intersections.

Because if we did equip this with a connection, its curvature 4-form would have to be the deRham representative of the Pontryagin class, hence locally be the Chern-Simons 3-form of the connection on $G$. (Compare for instance section 6 of Danny Stevenson’s notes here).

Transition1-Gerbes for the CS 2-Gerbe.

We can compute the nature of transition 1-gerbes for $B''$ along the lines of the computation of transition bundles for nonabelian gerbes given in section 3.2 of this text.

In order to do so, we choose another good covering $V$ of $X$ by open contractible sets. Pulled back to each open set in $U$, the functor $B''$ may be completely trivialized along $$(1 \to 1 \to 1) \stackrel{i}{\to} (U(1) \to \hat \Omega_\tau G \to P G ) \,.$$

This way we obtain on each patch $V_i$ a transition tetrahedron all of whose labels are trivial. On double intersections $V_i \cap V_j$, these transition tetrahedra are now related by cylinders with triangular base, expressing a morphism between two trivial pseudofunctors with values in $U(1) \to \hat \Omega G_\tau \to P G$.

Think of two trivial transition tetrahedra, a large and a small one, concentrically sitting inside each other. From each vertex of the small tetrahedron draw a line to the corresponding vertex of the large one. This partitions the volume in between the two tetrahedra by those triangular cylinders.

I’ll indicate a triangular cylinder by cutting its sides open like this $$\array{ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet \\ \;\;\downarrow h_i &\stackrel{g_{ij}}{\Leftarrow}& \;\;\downarrow h_j &\stackrel{g_{jk}}{\Leftarrow}& \;\;\downarrow h_k \\ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet } \; \stackrel{\mu_{ijk}}{\rightarrow} \; \array{ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet \\ \;\;\downarrow h_i &\stackrel{g_{ik}}{\Leftarrow}& \;\;\downarrow h_k \\ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet } \,.$$ (Here the bottom and top triangle of the cyclinder are suppressed, since in the present context they are labeled by identity morphisms only.)

Since this is supposed to define a pseudonatural transformation of 3-functors (compare page 6 of these notes):

- the labels on 1-morphisms $h_i$, $h_j$, $h_k$ indicate functions from $V_i\cap V_j$ into $P G$,

- the labels on 2-morphisms $g_{ij}$, $g_{jk}$ and $g_{ik}$ represent functions to $\hat \Omega_\tau G$,

- and $\mu_{ijk} \in U(1)$.

By construction, the $\mu_{\cdots}$ will make a tretrahedron commute (meaning that they do consistently interpolate between the small and the large trivial tetrahedron mentioned before).

Now we need a simple but important

Fact. The tautological bundle gerbe $\mathrm{WZW}_G^\tau$ at level $\tau$ on $G$ may, like any bundle gerbe, be regarded as a groupoid. The 2-group $\mathrm{String}^\tau_G$ may also be regarded as a groupoid. And as groupoids, these are the same: $$\mathrm{WZW}^\tau_G = \mathrm{String}^\tau_G \,.$$

This is manifest from the very construction of both these groupoids.

The construction of the tautological bundle gerbe is described for instance in Michael Murray’s original article. The surjective submersion $Y \to G$ is nothing but $P G \to G$, with projection being the end point evaluation. The line bundle over $(P G)^{[2]} \simeq PG \times \Omega G$ is constructed in the same way as the central extension of $\Omega G$, by using the integral 3-form $H$ representing the class of the gerbe in deRham cohomology.

Given this identification of the 2-group $\mathrm{String}^\tau_G$ with the tautological bundle gerbe over $G$, we find that the maps involved in the above triangular cylinder are nothing but

• a map $$f : X \to G$$
• a choice of sections of the pulled back bundle gerbe $$f^* \mathrm{WZW}^\tau_G$$ from $G$ to $X$.
• a labeling of triangular cylinders by the resulting abelian 3-cocycle.

In other words, we find

Consequence: The transition 1-gerbes of a Chern-Simons 2-gerbe (on $G$ at level $\tau$) are the pullback of the canonical 1-gerbe on $G$ (at that level) to $X$, along a given map $f : X \to G$.

A different perspective on the relation between Chern-Simons 2-gerbes and WZW 1-gerbes can be found in

Alan L. Carey, Stuart Johnson, Michael K. Murray, Danny Stevenson, Bai-Ling Wang
Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories
math/0410013.

My discussion here emphasizes the 3-group relevant for Chern-Simons 2-gerbes, and the role played by the String 2-group.

Finally, lect $X$ be a 3-manifold with boundary. Then the class in $H^4(X,\mathbb{Z})$ controlling the above construction of the Chern-Simons 2-gerbe necessarily vanishes, and hence $B''$ may be trivialized globally, on all of $X$.

Moreover, we would choose trivializations of $B''$ restricted to the boundary, following a general logic # of how to choose sections of $n$-bundles representing states of $n$-particles coupled to them.

Therefore in this case, there are no transition 1-gerbes in the bulk of $X$, but precisely one on its boundary. As we have seen, this data on the boundary is precisely a map of the boundary into $G$, together with a pullback of the canonical gerbe on $G$ to $\partial M$. This way the Wess-Zumino-Witten term arises on the boundary of Chern-Simons theory.