## October 29, 2006

### WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe

#### Posted by Urs Schreiber

As a kind of comment to M. Hopkins’ lecture on Chern-Simons theory (I, II, III) I want to describe how the Wess-Zumino-Witten 1-gerbe arises as the transition 1-gerbe of the Chern-Simons 2-gerbe (analogous to how a 1-gerbe itself has transition 0-gerbes, i.e. transition bundles).

I’ll advertise a point of view # where we consider the Chern-Simons 2-gerbe as a 3-bundle with structure 3-group

(1)$G_3 := (U(1) \to \hat\Omega_\tau G \to P G)$

and use the fact that the 2-group

(2)$\mathrm{String}_G := \hat\Omega_\tau G \to P G$

is #, as a groupoid with monoidal structure, nothing but the tautological bundle gerbe representation of the canonical “WZW” gerbe at level $\tau$ on $G$.

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.

Posted at October 29, 2006 3:02 PM UTC

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### Re: WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe

You write:

we consider the Chern-Simons 2-gerbe as a 3-bundle with structure 3-group $G_3 = (U(1) \to \hat \Omega G \to P G)$

do you havew a general theory

an $n$-gerbe is an $(n+1)$-bundle?

at least for $n=1$ and $2$?

or with side conditions?

Posted by: jim stasheff on November 20, 2006 3:34 PM | Permalink | Reply to this

### Re: WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe

do you havew a general theory

an $n$-gerbe is an $(n+1)$-bundle?

at least for $n=1$ and $2$?

or with side conditions?

In principle yes. It depends a little on what precisely you want to understand under an “$n$-bundle”.

If you want an $n$-bundle to be a total space (a category) $E$ together with a regular surjection $E \to M$ such that there is a typical fiber, etc. then I know the following:

it is not so difficult to show that principal 2-bundles of this sort, if locally trivializable, give rise to nonabelian cohomology, i.e. nonablian cocycles of local transition data as found for (nonabelian) gerbes. This has been done for instance by Toby Bartels and by Igor Bakovic.

Both have announced proofs that there is in fact an equivalence between such “total space 2-bundles” and the respective nonabelian cocycles. But showing that one can reconstruct a total space from the cocycle is harder, as you know best, and I haven’t yet seen the proof of these equivalences. (Maybe soon one is available.)

So as far as a 2-bundle is supposed to be a total 2-space, etc, I know that every such 2-bundle is also a gerbe, but not the other way around.

But for all my recent work, I use a slightly different definition of 2-bundle. I found that for what I want to do I don’t need a total space and shouldn’t bother. For me, a 2-bundle with connection is, as you know, a suitable 2-functor. We have talked about that a while ago here.

In terms of these 2-functors, I think I can prove (together with Konrad Waldorff) that suitable 2-functors are equivalent to their categories of local cocycle data. (Essentially amounts to showing that they form a 2-stack.)

So whenever I say $n$-bundle here, I am being sloppy and am either talking about $n$-bundles with connection, then in terms of their parallel transport functors, or I just mean the cocycle data (the local transition data) itself. In the above entry I am only talking about cocycle data.

Posted by: urs on November 20, 2006 3:51 PM | Permalink | Reply to this
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