Continuing from the previous post, there are a few more lecture series from the school to talk about.
Higher Gauge Theory
The next was John Huerta’s series on Higher Gauge Theory from the point of view of 2-groups. John set this in the context of “categorification”, a slightly vague program of replacing set-based mathematical ideas with category-based mathematical ideas. The general reason for this is to get an extra layer of “maps between things”, or “relations between relations”, etc. which tend to be expressed by natural transformations. There are various ways to go about this, but one is internalization: given some sort of structure, the relevant example in this case being “groups”, one has a category , and can define a 2-group as a “category internal to “. So a 2-group has a group of objects, a group of morphisms, and all the usual maps (source and target for morphisms, composition, etc.) which all have to be group homomorphisms. It should be said that this all produces a “strict 2-group”, since the objects necessarily form a group here. In particular, satisfies group axioms “on the nose” – which is the only way to satisfy them for a group made of the elements of a set, but for a group made of the elements of a category, one might require only that it commute up to isomorphism. A weak 2-group might then be described as a “weak model” of the theory of groups in , but this whole approach is much less well-understood than the strict version as one goes to general n-groups.
Now, as mentioned in the previous post, there is a 1-1 correspondence between 2-groups and crossed modules (up to equivalence): given a crossed module , there’s a 2-group whose objects are and whose morphisms are ; given a 2-group with objects , there’s a crossed module . (The action acts on a morphism in such as way as to act by multiplication on its source and target). Then, for instance, the Peiffer identity for crossed modules (see previous post) is a consequence of the fact that composition of morphisms is supposed to be a group homomorphism.
Looking at internal categories in [your favourite setting here] isn’t the only way to do categorification, but it does produce some interesting examples. Baez-Crans 2-vector spaces are defined this way (in ), and built using these are Lie 2-algebras. Looking for a way to integrate Lie 2-algebras up to Lie 2-groups (which are internal categories in Lie groups) brings us back to the current main point. This is the use of 2-groups to do higher gauge theory. This requires the use of “2-bundles”. To explain these, we can say first of all that a “2-space” is an internal category in (whether that be manifolds, or topological spaces, or what-have-you), and that a (locally trivial) 2-bundle should have a total 2-space , a base 2-space , and a (functorial) projection map , such that there’s some open cover of by neighborhoods where locally the bundle “looks like” , where is the fibre of the bundle. In the bundle setting, “looks like” means “is isomorphic to” by means of isomorphisms . With 2-bundles, it’s interpreted as “is equivalent to” in the categorical sense, likewise by maps .
Actually making this precise is a lot of work when is a general 2-space – even defining open covers and setting up all the machinery properly is quite hard. This has been done, by Toby Bartels in his thesis, but to keep things simple, John restricted his talk to the case where is just an ordinary manifold (thought of as a 2-space which has only identity morphisms). Then a key point is that there’s an analog to how (principal) -bundles (where as a -set) are classified up to isomorphism by the first Cech cohomology of the manifold, . This works because one can define transition functions on double overlaps , by . Then these will automatically satisfy the 1-cocycle condidion ( on the triple overlap ) which means they represent a cohomology class .
A comparable thing can be said for the “transition functors” for a 2-bundle – they’re defined superficially just as above, except that being functors, we can now say there’s a natural isomorphism , and it’s these , defined on triple overlaps, which satisfy a 2-cocycle condition on 4-fold intersections (essentially, the two ways to compose them to collapse into agree). That is, we have and which fit together nicely. In particular, we have an element of the second Cech cohomology of : “principal -bundles are classified by second Cech cohomology of “. This sort of thing ties in to an ongoing theme of the later talks, the relationship between gerbes and higher cohomology – a 2-bundle corresponds to a “gerbe”, or rather a “1-gerbe”. (The consistent terminology would have called a bundle a “0-gerbe”, but as usual, terminology got settled before the general pattern was understood).
Finally, having defined bundles, one usually defines connections, and so we do the same with 2-bundles. A connection on a bundle gives a parallel transport operation for paths in , telling how to identify the fibres at points along by means of a functor , thinking of as a category with one object, and where is the path groupoid whose objects are points in and whose morphisms are paths (up to “thin” homotopy). At least, it does so once we trivialize the bundle around , anyway, to think of it as locally – in general we need to get the transition functions involved when we pass into some other local neighborhood. A connection on a 2-bundle is similar, but tells how to parallel transport fibres not only along paths, but along homotopies of paths, by means of , where is seen as a 2-category with one object, and now has 2-morphisms which are (essentially) homotopies of paths.
Just as connections can be described by 1-forms valued in , which give by integrating, a similar story exists for 2-connections: now we need a 1-form valued in and a 2-form valued in . These need to satisfy some relations, essentially that the curvature of has to be controlled by . Moreover, that is related to the -field of string theory, as I mentioned in the post on the pre-school… But really, this is telling us about the Lie 2-algebra associated to , and how to integrate it up to the group!
Infinite Dimensional Lie Theory and Higher Gauge Theory
This series of talks by Christoph Wockel returns us to the question of “integrating up” to a Lie group from a Lie algebra , which is seen as the tangent space of at the identity. This is a well-understood, well-behaved phenomenon when the Lie algebras happen to be finite dimensional. Indeed the classification theorem for the classical Lie groups can be got at in just this way: a combinatorial way to characterize Lie algebras using Dynkin diagrams (which describe the structure of some weight lattice), followed by a correspondence between Lie algebras and Lie groups. But when the Lie algebras are infinite dimensional, this just doesn’t have to work. It may be impossible to integrate a Lie algebra up to a full Lie group: instead, one can only get a little neighborhood of the identity. The point of such infinite-dimensional groups, and ultimately their representation theory, is to deal with string groups that have to do with motions of extended objects. Christoph Wockel was describing a result which says that, going to 2-groups, this problem can be overcome. (See the relevant paper here.)
The first lecture in the series presented some background on a setting for infinite dimensional manifolds. There are various approaches, a popular one being Frechet manifolds, but in this context, the somewhat weaker notion of locally convex spaces is sufficient. These are “locally modelled” by (infinite dimensional) locally convex vector spaces, the way finite dimensonal manifolds are locally modelled by Euclidean space. Being locally convex is enough to allow them to support a lot of differential calculus: one can find straight-line paths, locally, to define a notion of directional derivative in the direction of a general vector. Using this, one can build up definitions of differentiable and smooth functions, derivatives, and integrals, just by looking at the restrictions to all such directions. Then there’s a fundamental theorem of calculus, a chain rule, and so on.
At this point, one has plenty of differential calculus, and it becomes interesting to bring in Lie theory. A Lie group is defined as a group object in the category of manifolds and smooth maps, just as in the finite-dimensional case. Some infinite-dimensional Lie groups of interest would include: , the group of diffeomorphisms of some compact manifold ; and the group of smooth functions from into some (finite-dimensional) Lie group (perhaps just ), with the usual pointwise multiplication. These are certainly groups, and one handy fact about such groups is that, if they have a manifold structure near the identity, on some subset that generates as a group in a nice way, you can extend the manifold structure to the whole group. And indeed, that happens in these examples.
Well, next we’d like to know if we can, given an infinite dimensional Lie algebra , “integrate up” to a Lie group – that is, find a Lie group for which is the “infinitesimal” version of . One way this arises is from central extensions. A central extension of Lie group by is an exact sequence where (the image of) is in the centre of . The point here is that extends . This setup makes is a principal -bundle over .
Now, finding central extensions of Lie algebras is comparatively easy, and given a central extension of Lie groups, one always falls out of the induced maps. There will be an exact sequence of Lie algebras, and now the special condition is that there must exist a continuous section of the second map. The question is to go the other way: given one of these, get back to an extension of Lie groups. The problem of finding extensions of by , in particular as a problem of finding a bundle with connection having specified curvature, which brings us back to gauge theory. One type of extension is the universal cover of , which appears as , so that the fibre is .
In general, whether an extension can exist comes down to a question about a cocycle: that is, if there’s a function which is locally smooth (i.e. in some neighborhood in ), and is a cocyle (so that ), by the same sorts of arguments we’ve already seen a bit of. For this reason, central extensions are classified by the cohomology group . The cocycle enables a “twisting” of the multiplication associated to a nontrivial loop in , and is used to construct (by specifying how multiplication on lifts to ). Given a 2-cocycle at the Lie algebra level (easier to do), one would like to lift that up the Lie group. It turns out this is possible if the period homomorphism – which takes a chain (with ) to the integral of the original cocycle on it, – lands in a discrete subgroup of . A popular example of this is when is just , and the discrete subgroup is (or, similarly, and respectively). This business of requiring a cocycle to be integral in this way is sometimes called a “prequantization” problem.
So suppose we wanted to make the “2-connected cover” as a central extension: since will be abelian, this is conceivable. If the dimension of is finite, this is trivial (since in finite dimensions), which is why we need some theory of infinite-dimensional manifolds. Moreover, though, this may not work in the context of groups: the in the extension above needs to be a “twisting” of associativity, not multiplication, being lifted from . Such twistings come from the THIRD cohomology of (see here, e.g.), and describe the structure of 2-groups (or crossed modules, whichever you like). In fact, the solution (go read the paper for more if you like) to define a notion of central extension for 2-groups (essentially the same as the usual definition, but with maps of 2-groups, or crossed modules, everywhere). Since a group is a trivial kind of 2-group (with only trivial automorphisms of any element), the usual notion of central extension turns out to be a special case. Then by thinking of and as crossed modules, one can find a central extension which is like the 2-connected cover we wanted – though it doesn’t work as an extension of groups because we think of as the base group of the crossed module, and as the second group in the tower.
The pattern of moving to higher group-like structures, higher cohomology, and obstructions to various constructions ran all through the workshop, and carried on in the next school session…
Higher Spin Structures in String Theory
Hisham Sati gave just one school-lecture in addition to his workshop talk, but it was packed with a lot of material. This is essentially about cohomology and the structures on manifolds to which cohomology groups describe the obstructions. The background part of the lecture referenced this book by Fridrich, and the newer parts were describing some of Sati’s own work, in particular a couple of papers with Schreiber and Stasheff (also see this one).
The basic point here is that, for physical reasons, we’re often interested in putting some sort of structure on a manifold, which is really best described in terms of a bundle. For instance, a connection or spin connection on spacetime lets us transport vectors or spinors, respectively, along paths, which in turn lets us define derivatives. These two structures really belong on vector bundles or spinor bundles. Now, if these bundles are trivial, then one can make the connections on them trivial as well by gauge transformation. So having nontrivial bundles really makes this all more interesting. However, this isn’t always possible, and so one wants to the obstruction to being able to do it. This is typically a class in one of the cohomology groups of the manifold – a characteristic class. There are various examples: Chern classes, Pontrjagin classes, Steifel-Whitney classes, and so on, each of which comes in various degrees . Each one corresponds to a different coefficient group for the cohomology groups – in these examples, the groups and which are the limits of the unitary and orthogonal groups (such as )
The point is that these classes are obstructions to building certain structures on the manifold – which amounts to finding sections of a bundle. So for instance, the first Steifel-Whitney classes, of a bundle are related to orientations, coming from cohomology with coefficients in . Orientations for the manifold can be described in terms of its tangent bundle, which is an -bundle (tangent spaces carry an action of the rotation group). Consider , where we have actually . The group has two elements, and there are two types of line bundle on the circle : ones with a nowhere-zero section, like the trivial bundle; and ones without, like the Moebius strip. The circle is orientable, because its tangent bundle is of the first sort.
Generally, an orientation can be put on if the tangent bundle, as a map , can be lifted to a map – that is, it’s “secretly” an -bundle – the special orthogonal group respects orientation, which is what the determinant measures. Its two values, , are what’s behind the two classes of bundles. (In short, this story relates to the exact sequence ; in just the same way we have big groups , , and so forth.)
So spin structures have a story much like the above, but where the exact sequence plays a role – the spin groups are the universal covers (which are all double-sheeted covers) of the special rotation groups. A spin structure on some bundle , let’s say represented by is thus, again, a lifting to . The obstruction to doing this (the thing which must be zero for the lifting to exist) is the second Stiefel-Whitney class, . Hisham Sati also explained the example of “generalized” spin structures in these terms. But the main theme is an analogous, but much more general, story for other cohomology groups as obstructions to liftings of some sort of structures on manifolds. These may be bundles, for the lower-degree cohomology, or they may be gerbes or n-bundles, for higher-degree, but the setup is roughly the same.
The title’s term “higher spin structures” comes from the fact that we’ve so far had a tower of classifying spaces (or groups), , and so on. Then the problem of putting various sorts of structures on has been turned into the problem of lifting a map up this tower. At each point, the obstruction to lifting is some cohomology class with coefficients in the groups (, , etc.) So when are these structures interesting?
This turns out to bring up another theme, which is that of special dimensions – it’s just not true that the same phenomena happen in every dimension. In this case, this has to do with the homotopy groups – of and its cousins. So it turns out that the homotopy group (which is the same as as long as is bigger than ) follows a pattern, where if , and if . The fact that this pattern repeats mod-8 is one form of the (real) Bott Periodicity theorem. These homotopy groups reflect that, wherever there’s nontrivial homotopy in some dimension, there’s an obstruction to contracting maps into from such a sphere.
All of this plays into the question of what kinds of nontrivial structures can be put on orthogonal bundles on manifolds of various dimensions. In the dimensions where these homotopy groups are non-trivial, there’s an obstruction to the lifting, and therefore some interesting structure one can put on which may or may not exist. Hisham Sati spoke of “killing” various homotopy groups – meaning, as far as I can tell, imposing conditions which get past these obstructions. In string theory, his application of interest, one talks of “anomaly cancellation” – an anomaly being the obstruction to making these structures. The first part of the punchline is that, since these are related to nontrivial cohomology groups, we can think of them in terms of defining structures on n-bundles or gerbes. These structures are, essentially, connections – they tell us how to parallel-transport objects of various dimensions. It turns out that the homotopy group is related to parallel transport along -dimensional surfaces in , which can be thought of as the world-sheets of -dimensional “particles” (or rather, “branes”).
So, for instance, the fact that is nontrivial means there’s an obstruction to a lifting in the form of a class in , which has to do with spin structure – as above. “Cancelling” this “anomaly” means that for a theory involving such a spin structure to be well-defined, then this characteristic class for must be zero. The fact that is nontrivial means there’s an obstruction to a lifting in the form of a class in . This has to do with “string bundles”, where the string group is a higher analog of in exactly the sense we’ve just described. If such a lifting exists, then there’s a “string-structure” on which is compatible with the spin structure we lifted (and with the orientation a level below that). Similarly, being nontrivial, by way of an obstruction in , means there’s an interesting notion of “five-brane” structure, and a group, and so on. Personally, I think of these as giving a geometric interpretation for what the higher cohomology groups actually mean.
A slight refinement of the above, and actually more directly related to “cancellation” of the anomalies, is that these structures can be defined in a “twisted” way: given a cocycle in the appropriate cohomology group, we can ask that a lifting exist, not on the nose, but as a diagram commuting only up to a higher cell, which is exactly given by the cocycle. I mentioned, in the previous section, a situation where the cocycle gives an associator, so that instead of being exactly associative, a structure has a “twisted” associativity. This is similar, except we’re twisting the condition that makes a spin structure (or higher spin structure) well-defined. So if has the wrong characteristic class, we can only define one of these twisted structures at that level.
This theme of higher cohomology and gerbes, and their geometric interpretation, was another one that turned up throughout the talks in the workshop…
And speaking of that: coming up soon, some descriptions of the actual workshop.