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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)Ω^ τGPG) G_3 := (U(1) \to \hat\Omega_\tau G \to P G)

and use the fact that the 2-group

(2)String G:=Ω^ τGPG \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 τ\tau on GG.

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

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

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

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

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

Now consider this same functor as a pseudofunctor to the discrete 3-group on Σ(G)\Sigma(G): B:U [2]Σ(11G). B : U^{[2]} \to \Sigma(1 \to 1 \to G) \,. This amounts to regarding the principal (1-)bundle BB as a degenerate case of a principal 3-bundle.

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

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

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

By smoothly sending elements gGg \in G to chosen based paths in GG ending at gg, we can always lift the functor B:U [2]Σ(11BG) B : U^{[2]} \to \Sigma(1 \to 1 \to B G) to a functor B:U [2]Σ(1ΩGPG). B' : U^{[2]} \to \Sigma(1 \to \Omega G \to P G) \,. Notice that there is a unique element in ΩG\Omega G going between elements of PGP G with coinciding endpoint. This makes the lift from BB to BB' always possible.

So this still describes, up to isomorphism, a GG-bundle on XX.

But the action of PGP G on ΩG\Omega G lifts to an action on the level τ\tau central extention Ω^ τG\hat \Omega_\tau G.

Therefore we may want to further lift BB' to a pseudofunctor B:U [2]Σ(1Ω^ τGPG). 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,)H^4(X, \mathbb{Z}). If this does not vanish, we instead get a pseudofunctor B:U [2]Σ(U(1)Ω^ τGPG), B'' : U^{[2]} \to \Sigma(U(1) \to \hat \Omega_\tau G \to P G) \,, determined by that class in H 4(X,)H^4(X, \mathbb{Z}).

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

This pseudofunctor is (the local data for) a GG-Chern-Simons 2-gerbe on XX.

Notice that the 2-fold pseudoness of this functor means that it assigns elements of PGP G to points in double intersections, such that the product in PGP G is respected up to an element of Ω^ τG\hat \Omega_\tau G on triple intersections, which form a tetrahedron on quandruple intersections that is filled by an element of U(1)U(1), such that all these U(1)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 GG. (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 BB'' 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 VV of XX by open contractible sets. Pulled back to each open set in UU, the functor BB'' may be completely trivialized along (111)i(U(1)Ω^ τGPG). (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 iV_i a transition tetrahedron all of whose labels are trivial. On double intersections V iV jV_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)Ω^G τPGU(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 Id Id h i g ij h j g jk h k Id Id μ ijk Id h i g ik h k Id . \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 ih_i, h jh_j, h kh_k indicate functions from V iV jV_i\cap V_j into PGP G,

- the labels on 2-morphisms g ijg_{ij}, g jkg_{jk} and g ikg_{ik} represent functions to Ω^ τG\hat \Omega_\tau G,

- and μ ijkU(1)\mu_{ijk} \in U(1).

By construction, the μ \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 WZW G τ\mathrm{WZW}_G^\tau at level τ\tau on GG may, like any bundle gerbe, be regarded as a groupoid. The 2-group String G τ\mathrm{String}^\tau_G may also be regarded as a groupoid. And as groupoids, these are the same: WZW G τ=String G τ. \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 YGY \to G is nothing but PGGP G \to G, with projection being the end point evaluation. The line bundle over (PG) [2]PG×ΩG(P G)^{[2]} \simeq PG \times \Omega G is constructed in the same way as the central extension of ΩG\Omega G, by using the integral 3-form HH representing the class of the gerbe in deRham cohomology.

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

  • a map f:XG f : X \to G
  • a choice of sections of the pulled back bundle gerbe f *WZW G τ f^* \mathrm{WZW}^\tau_G from GG to XX.
  • 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 GG at level τ\tau) are the pullback of the canonical 1-gerbe on GG (at that level) to XX, along a given map f:XGf : 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

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

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

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

Therefore in this case, there are no transition 1-gerbes in the bulk of XX, but precisely one on its boundary. As we have seen, this data on the boundary is precisely a map of the boundary into GG, together with a pullback of the canonical gerbe on GG to M\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)Ω^GPG)G_3 = (U(1) \to \hat \Omega G \to P G)

do you havew a general theory

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

at least for n=1n=1 and 22?

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

at least for n=1n=1 and 22?

or with side conditions?

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

If you want an nn-bundle to be a total space (a category) EE together with a regular surjection EME \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 nn-bundle here, I am being sloppy and am either talking about nn-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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