The main thing happening in my end of the world is that it’s relocated from Europe back to North America. I’m taking up a teaching postdoc position in the Mathematics and Computer Science department at Mount Allison University starting this month. However, amidst all the preparations and moving, I was also recently in Edinburgh, Scotland for a workshop on Higher Gauge Theory and Higher Quantization, where I gave a talk called 2-Group Symmetries on Moduli Spaces in Higher Gauge Theory. That’s what I’d like to write about this time.

Edinburgh is a beautiful city, though since the workshop was held at Heriot-Watt University, whose campus is outside the city itself, I only got to see it on the Saturday after the workshop ended. However, John Huerta and I spent a while walking around, and as it turned out, climbing a lot: first the Scott Monument, from which I took this photo down Princes Street:

And then up a rather large hill called Arthur’s Seat, in Holyrood Park next to the Scottish Parliament.

The workshop itself had an interesting mix of participants. Urs Schreiber gave the most mathematically sophisticated talk, and mine was also quite category-theory-minded. But there were also some fairly physics-minded talks that are interesting to me as well because they show the source of these ideas. In this first post, I’ll begin with my own, and continue with David Roberts’ talk on constructing an explicit string bundle. …

### 2-Group Symmetries of Moduli Spaces

My own talk, based on work with Roger Picken, boils down to a couple of observations about the notion of symmetry, and applies them to a discrete model in higher gauge theory. It’s the kind of model you might use if you wanted to do lattice gauge theory for a BF theory, or some other higher gauge theory. But the discretization is just a convenience to avoid having to deal with infinite dimensional spaces and other issues that don’t really bear on the central point.

Part of that point was described in a previous post: it has to do with finding a higher analog for the relationship between two views of symmetry: one is “global” (I found the physics-inclined part of the audience preferred “rigid”), to do with a group action on the entire space; the other is “local”, having to do with treating the points of the space as objects of a groupoid who show how points are related to each other. (Think of trying to describe the orbit structure of just the part of a group action that relates points in a little neighborhood on a manifold, say.)

In particular, we’re interested in the symmetries of the moduli space of connections (or, depending on the context, flat connections) on a space, so the symmetries are gauge transformations. Now, here already some of the physically-inclined audience objected that these symmetries should just be eliminated by taking the quotient space of the group action. This is based on the slogan that “only gauge-invariant quantities matter”. But this slogan has some caveats: in only applies to closed manifolds, for one. When there are boundaries, it isn’t true, and to describe the boundary we need something which acts as a representation of the symmetries. Urs Schreiber pointed out a well-known example: the Chern-Simons action, a functional on a certain space of connections, is not gauge-invariant. Indeed, the boundary terms that show up due to this not-invariance explain why there is a Wess-Zumino-Witt theory associated with the boundaries when the bulk is described by Chern-Simons.

Now, I’ve described a lot of the idea of this talk in the previous post linked above, but what’s new has to do with how this applies to moduli spaces that appear in higher gauge theory based on a 2-group . The points in these space are connections on a manifold . In particular, since a 2-group is a group object in categories, the transformation groupoid (which captures global symmetries of the moduli space) will be a double category. It turns out there is another way of seeing this double category by local descriptions of the gauge transformations.

In particular, general gauge transformations in HGT are combinations of two special types, described geometrically by -valued functions, or -valued 1-forms, where is the group of objects of , and is the group of morphisms based at . If we think of connections as functors from the fundamental 2-groupoid into , these correspond to pseudonatural transformations between these functors. The main point is that there are also two special types of these, called “strict”, and “costrict”. The strict ones are just natural transformations, where the naturality square commutes strictly. The costrict ones, also called ICONs (for “identity component oplax natural transformations” – see the paper by Steve Lack linked from the nlab page above for an explanation of “costrictness”). They assign the identity morphism to each object, but the naturality square commutes only up to a specified 2-cell. Any pseudonatural transformation factors into a strict and costrict part.

The point is that taking these two types of transformation to be the horizontal and vertical morphisms of a double category, we get something that very naturally arises by the action of a big 2-group of symmetries on a category. We also find something which doesn’t happen in ordinary gauge theory: that only the strict gauge transformations arise from this global symmetry. The costrict ones must already be the morphisms in the category being acted on. This category plays the role of the moduli space in the normal 1-group situation. So moving to 2-groups reveals that in general we should distinguish between global/rigid symmetries of the moduli space, which are strict gauge transformations, and costrict ones, which do not arise from the global 2-group action and should be thought of as intrinsic to the moduli space.

### String Bundles

David Roberts gave a rather interesting talk called “Constructing Explicit String Bundles”. There are some notes for this talk here. The point is simply to give an explicit construction of a particular 2-group bundle. There is a lot of general abstract theory about 2-bundles around, and a fair amount of work that manipulates physically-motivated descriptions of things that can presumably be modelled with 2-bundles. There has been less work on giving a mathematically rigorous description of specific, concrete 2-bundles.

This one is of interest because it’s based on the String 2-group. Details are behind that link, but roughly the classifying space of (a homotopy 2-type) is fibred over the classifying space for (a 1-type). The exact map is determined by taking a pullback along a certain characteristic class (which is a map out of ). Saying “the” string 2-group is a bit of a misnomer, by the way, since such a 2-group exists for every simply connected compact Lie group . The group that’s involved here is a , the string 2-group associated to , the universal cover of the rotation group . This is the one that determines whether a given manifold can support a “string structure”. A string structure on , therefore, is a lift of a spin structure, which determines whether one can have a spin bundle over , hence consistently talk about a spin connection which gives parallel transport for spinor fields on . The string structure determines if one can consistently talk about a string-bundle over , and hence a 2-group connection giving parallel transport for strings.

In this particular example, the idea was to find, explicitly, a string bundle over Minkowski space – or its conformal compactification. In point of fact, this particular one is for $latek String(5)$, and is over 6-dimensional Minkowski space, whose compactification is . This particular is convenient because it’s possible to show abstractly that it has exactly one nontrivial class of string bundles, so exhibiting one gives a complete classification. The details of the construction are in the notes linked above. The technical details rely on the fact that we can coordinatize nicely using the projective quaternionic plane, but conceptually it relies on the fact that , and because of how the lifting works, this is also . This quotient means there’s a string bundle whose fibre is .

While this is only one string bundle, and not a particularly general situation, it’s nice to see that there’s a nice elegant presentation which gives such a bundle explicitly (by constructing cocycles valued in the crossed module associated to the string 2-group, which give its transition functions).

(Here endeth Part I of this discussion of the workshop in Edinburgh. Part II will talk about Urs Schreiber’s very nice introduction to Higher Geometric Quantization)

(This ends the first part of this update – the next will describe the physics-oriented talks, and the third will describe Urs Schreiber’s series on higher geometric quantization)

August 16, 2014 at 4:19 am

Is there a generalization from String(n) to say, a Brane(n)? Also, does higher gauge theory provide any insights on discretizing brane worldvolumes? (I’m thinking along the lines of applying noncommutative motives to the study of gauge/gravity duality.)

August 16, 2014 at 3:00 pm

Hi! I don’t know a lot about the theory of higher generalizations of beyond the string 2-group, but I do know that they exist. However, this is one of those situations in string theory where a construction only works in certain dimensions. There is a 6-group called , and a 10-group called (the off-by-one is because the important dimension is that of the worldsheet). The 6-group was introduced by Sati, Schreiber and Stasheff in this paper here, and there’s a precis over on the nLab. That’s as far as this pattern connecting branes to -group lifts of spin structures goes, however.

As far as discretization of the volume, that’s something that would come out of quantization: you find a volume operator, and it turns out to have discrete or continuous spectrum. Perhaps there’s a way to intuitively understand why it’s one or the other from the classical geometry (analogous to the way bundles over a circle have to have integral numbers of twists), but I don’t know it.

August 19, 2014 at 1:07 am

There’s also supergroup versions of these higher extensions, something that John Huerta (also at the workshop, but didn’t present) has worked on. Ultimately these arise from various Fierz identities involving spinors that encode cocycles for super Lie algebra extensions.

August 23, 2014 at 3:42 am

Don’t these Fierz identities only apply in certain finite dimensions?

August 23, 2014 at 9:44 pm

They do, but that reflects the interesting constraints on forming such n-group extensions.

August 24, 2014 at 2:53 pm

Are these constraints related to the division algebras?

August 24, 2014 at 4:41 pm

I’d like to understand the answer to this as well. What I know about the relation between the division algebras and these n-group extensions is limited to what I’ve learned from John Huerta and John Baez, as for example in here:

http://math.ucr.edu/home/baez/susy/

I don’t quite see how they connect to the constraints David is talking about. The constraints involving the division algebras invove their relation to the Clifford algebras, which connect in turn to the spin groups in various dimensions. The dimensions where division algebras are defined are , and those connections imply that the super-string, with its 2D worldsheet, should make sense in dimensions , the 2-brane with a 3D worldsheet should make sense in dimensions , and so on. I don’t see how this would be connected to a constraint on the dimensions of -branes. David?

August 24, 2014 at 4:52 pm

The k+2 comes from the Jordan algebras of 2×2 Hermitian matrices over the division algebras (i.e., spin factors). The (k+1,1) spin groups act on these algebras as determinant preserving transformation groups. One also has the split versions of the division algebras which show up directly in M-theory toroidal compactifications.

Arkani Hamed et al make use of the k=2 spin factor in writing twistor-string scattering amplitudes. I bet one can extend the twistor-string formalism in the k=4,8 cases as well. The k=8 case, when complexified, is very likely the structure behind D=11 SUGRA.

February 2, 2016 at 7:42 am

Hi Jeff,

I think I was speaking to you in Edinburgh when I claimed that String(G)-equivariant U(1)-bundle gerbes on total spaces of G-bundles are the same thing as String(G)-bundles. If so, let me point out that at least part of this has been done in a joint paper with Michael Murray, Danny Stevenson and Ray Vozzo. Or rather, since this is true from abstract oo-topos nonsense, we gave a construction of a String(G)-equivariant bundle gerbe from a trivialisation of a Chern-Simons 2-gerbe (which should be equivalent to the data of lifting a G-bundle to a String(G)-bundle). I would go so far as to claim that this is most of the way to showing that the groupoid underlying the trivialisation is the total space of the String(G)-bundle.

Ray and I can give the general version of what you wrote about above, for homogeneous bundles G –> G/H of a quite general sort, though not with cocycles, instead of just SU(3) –> S^5. If one takes the bundle gerbe viewpoint (i.e. trivialising the Chern-Simons 2-gerbe) we can also do this with explicit connections as well, but are still working towards incorporating curvings.

If it was’t you (it may have been John H), then I hope this is still interesting!