gerbes

To continue from the previous post

Twisted Differential Cohomology

Ulrich Bunke gave a talk introducing differential cohomology theories, and Thomas Nikolaus gave one about a twisted version of such theories (unfortunately, perhaps in the wrong order). The idea here is that cohomology can give a classification of field theories, and if we don’t want the theories to be purely topological, we would need to refine this. A cohomology theory is a (contravariant) functorial way of assigning to any space $X$, which we take to be a manifold, a $\mathbb{Z}$-graded group: that is, a tower of groups of “cocycles”, one group for each $n$, with some coboundary maps linking them. (In some cases, the groups are also rings) For example, the group of differential forms, graded by degree.

Cohomology theories satisfy some axioms – for example, the Mayer-Vietoris sequence has to apply whenever you cut a manifold into parts. Differential cohomology relaxes one axiom, the requirement that cohomology be a homotopy invariant of $X$. Given a differential cohomology theory, one can impose equivalence relations on the differential cocycles to get a theory that does satisfy this axiom – so we say the finer theory is a “differential refinement” of the coarser. So, in particular, ordinary cohomology theories are classified by spectra (this is related to the Brown representability theorem), whereas the differential ones are represented by sheaves of spectra – where the constant sheaves represent the cohomology theories which happen to be homotopy invariants.

The “twisting” part of this story can be applied to either an ordinary cohomology theory, or a differential refinement of one (though this needs similarly refined “twisting” data). The idea is that, if $R$ is a cohomology theory, it can be “twisted” over $X$ by a map $\tau: X \rightarrow Pic_R$ into the “Picard group” of $R$. This is the group of invertible $R$-modules (where an $R$-module means a module for the cohomology ring assigned to $X$) – essentially, tensoring with these modules is what defines the “twisting” of a cohomology element.

An example of all this is twisted differential K-theory. Here the groups are of isomorphism classes of certain vector bundles over $X$, and the twisting is particularly simple (the Picard group in the topological case is just $\mathbb{Z}_2$). The main result is that, while topological twists are classified by appropriate gerbes on $X$ (for K-theory, $U(1)$-gerbes), the differential ones are classified by gerbes with connection.

Fusion Categories

Scott Morrison gave a talk about Classifying Fusion Categories, the point of which was just to collect together a bunch of results constructing particular examples. The talk opens with a quote by Rutherford: “All science is either physics or stamp collecting” – that is, either about systematizing data and finding simple principles which explain it, or about collecting lots of data. This talk was unabashed stamp-collecting, on the grounds that we just don’t have a lot of data to systematically understand yet – and for that very reason I won’t try to summarize all the results, but the slides are well worth a look-over. The point is that fusion categories are very useful in constructing TQFT’s, and there are several different constructions that begin “given a fusion category $\mathcal{C}$“… and yet there aren’t all that many examples, and very few large ones, known.

Scott also makes the analogy that fusion categories are “noncommutative finite groups” – which is a little confusing, since not all finite groups are commutative anyway – but the idea is that the symmetric fusion categories are exactly the representation categories of finite groups. So general fusion categories are a non-symmetric generalization of such groups. Since classifying finite groups turned out to be difficult, and involve a laundry-list of sporadic groups, it shouldn’t be too surprising that understanding fusion categories (which, for the symmetric case, include the representation categories of all these examples) should be correspondingly tricky. Since, as he points out, we don’t have very many non-symmetric examples beyond rank 12 (analogous to knowing only finite groups with at most 12 elements), it’s likely that we don’t have a very good understanding of these categories in general yet.

There were a couple of talks – one during the workshop by Sonia Natale, and one the previous week by Sebastian Burciu, whom I also had the chance to talk with that week – about “Equivariantization” of fusion categories, and some fairly detailed descriptions of what results. The two of them have a paper on this which gives more details, which I won’t summarize – but I will say a bit about the construction.

An “equivariantization” of a category $C$ acted on by a group $G$ is supposed to be a generalization of the notion of the set of fixed points for a group acting on a set.  The category $C^G$ has objects which consist of an object $x \in C$ which is fixed by the action of $G$, together with an isomorphism $\mu_g : x \rightarrow x$ for each $g \in G$, satisfying a bunch of unsurprising conditions like being compatible with the group operation. The morphisms are maps in $C$ between the objects, which form commuting squares for each $g \in G$. Their paper, and the talks, described how this works when $C$ is a fusion category – namely, $C^G$ is also a fusion category, and one can work out its fusion rules (i.e. monoidal structure). In some cases, it’s a “group theoretical” fusion category (it looks like $Rep(H)$ for some group $H$) – or a weakened version of such a thing (it’s Morita equivalent to ).

A nice special case of this is if the group action happens to be trivial, so that every object of $C$ is a fixed point. In this case, $C^G$ is just the category of objects of $C$ equipped with a $G$-action, and the intertwining maps between these. For example, if $C = Vect$, then $C^G = Rep(G)$ (in particular, a “group-theoretical fusion category”). What’s more, this construction is functorial in $G$ itself: given a subgroup $H \subset G$, we get an adjoint pair of functors between $C^G$ and $C^H$, which in our special case are just the induced-representation and restricted-representation functors for that subgroup inclusion. That is, we have a Mackey functor here. These generalize, however, to any fusion category $C$, and to nontrivial actions of $G$ on $C$. The point of their paper, then, is to give a good characterization of the categories that come out of these constructions.

Quantizing with Higher Categories

The last talk I’d like to describe was by Urs Schreiber, called Linear Homotopy Type Theory for Quantization. Urs has been giving evolving talks on this topic for some time, and it’s quite a big subject (see the long version of the notes above if there’s any doubt). However, I always try to get a handle on these talks, because it seems to be describing the most general framework that fits the general approach I use in my own work. This particular one borrows a lot from the language of logic (the “linear” in the title alludes to linear logic).

Basically, Urs’ motivation is to describe a good mathematical setting in which to construct field theories using ingredients familiar to the physics approach to “field theory”, namely… fields. (See the description of Kevin Walker’s talk.) Also, Lagrangian functionals – that is, the notion of a physical action. Constructing TQFT from modular tensor categories, for instance, is great, but the fields and the action seem to be hiding in this picture. There are many conceptual problems with field theories – like the mathematical meaning of path integrals, for instance. Part of the approach here is to find a good setting in which to locate the moduli spaces of fields (and the spaces in which path integrals are done). Then, one has to come up with a notion of quantization that makes sense in that context.

The first claim is that the category of such spaces should form a differentially cohesive infinity-topos which we’ll call $\mathbb{H}$. The “infinity” part means we allow morphisms between field configurations of all orders (2-morphisms, 3-morphisms, etc.). The “topos” part means that all sorts of reasonable constructions can be done – for example, pullbacks. The “differentially cohesive” part captures the sort of structure that ensures we can really treat these as spaces of the suitable kind: “cohesive” means that we have a notion of connected components around (it’s implemented by having a bunch of adjoint functors between spaces and points). The “differential” part is meant to allow for the sort of structures discussed above under “differential cohomology” – really, that we can capture geometric structure, as in gauge theories, and not just topological structure.

In this case, we take $\mathbb{H}$ to have objects which are spectral-valued infinity-stacks on manifolds. This may be unfamiliar, but the main point is that it’s a kind of generalization of a space. Now, the sort of situation where quantization makes sense is: we have a space (i.e. $\mathbb{H}$-object) of field configurations to start, then a space of paths (this is WHERE “path-integrals” are defined), and a space of field configurations in the final system where we observe the result. There are maps from the space of paths to identify starting and ending points. That is, we have a span:

$A \leftarrow X \rightarrow B$

Now, in fact, these may all lie over some manifold, such as $B^n(U(1))$, the classifying space for $U(1)$ $(n-1)$-gerbes. That is, we don’t just have these “spaces”, but these spaces equipped with one of those pieces of cohomological twisting data discussed up above. That enters the quantization like an action (it’s WHAT you integrate in a path integral).

Aside: To continue the parallel, quantization is playing the role of a cohomology theory, and the action is the twist. I really need to come back and complete an old post about motives, because there’s a close analogy here. If quantization is a cohomology theory, it should come by factoring through a universal one. In the world of motives, where “space” now means something like “scheme”, the target of this universal cohomology theory is a mild variation on just the category of spans I just alluded to. Then all others come from some functor out of it.

Then the issue is what quantization looks like on this sort of scenario. The Atiyah-Singer viewpoint on TQFT isn’t completely lost here: quantization should be a functor into some monoidal category. This target needs properties which allow it to capture the basic “quantum” phenomena of superposition (i.e. some additivity property), and interference (some actual linearity over $\mathbb{C}$). The target category Urs talked about was the category of $E_{\infty}$-rings. The point is that these are just algebras that live in the world of spectra, which is where our spaces already lived. The appropriate target will depend on exactly what $\mathbb{H}$ is.

But what Urs did do was give a characterization of what the target category should be LIKE for a certain construction to work. It’s a “pull-push” construction: see the link way above on Mackey functors – restriction and induction of representations are an example . It’s what he calls a “(2-monoidal, Beck-Chevalley) Linear Homotopy-Type Theory”. Essentially, this is a list of conditions which ensure that, for the two morphisms in the span above, we have a “pull” operation for some and left and right adjoints to it (which need to be related in a nice way – the jargon here is that we must be in a Wirthmuller context), satisfying some nice relations, and that everything is functorial.

The intuition is that if we have some way of getting a “linear gadget” out of one of our configuration spaces of fields (analogous to constructing a space of functions when we do canonical quantization over, let’s say, a symplectic manifold), then we should be able to lift it (the “pull” operation) to the space of paths. Then the “push” part of the operation is where the “path integral” part comes in: many paths might contribute to the value of a function (or functor, or whatever it may be) at the end-point of those paths, because there are many ways to get from A to B, and all of them contribute in a linear way.

So, if this all seems rather abstract, that’s because the point of it is to characterize very generally what has to be available for the ideas that appear in physics notions of path-integral quantization to make sense. Many of the particulars – spectra, $E_{\infty}$-rings, infinity-stacks, and so on – which showed up in the example are in a sense just placeholders for anything with the right formal properties. So at the same time as it moves into seemingly very abstract terrain, this approach is also supposed to get out of the toy-model realm of TQFT, and really address the trouble in rigorously defining what’s meant by some of the standard practice of physics in field theory by analyzing the logical structure of what this practice is really saying. If it turns out to involve some unexpected math – well, given the underlying issues, it would have been more surprising if it didn’t.

It’s not clear to me how far along this road this program gets us, as far as dealing with questions an actual physicist would like to ask (for the most part, if the standard practice works as an algorithm to produce results, physicists seldom need to ask what it means in rigorous math language), but it does seem like an interesting question.

As usual, this write-up process has been taking a while since life does intrude into blogging for some reason.  In this case, because for a little less than a week, my wife and I have been on our honeymoon, which was delayed by our moving to Lisbon.  We went to the Azores, or rather to São Miguel, the largest of the nine islands.  We had a good time, roughly like so:

Now that we’re back, I’ll attempt to wrap up with the summaries of things discussed at the workshop on Higher Gauge Theory, TQFT, and Quantum Gravity.  In the previous post I described talks which I roughly gathered under TQFT and Higher Gauge Theory, but the latter really ramifies out in a few different ways.  As began to be clear before, higher bundles are classified by higher cohomology of manifolds, and so are gerbes – so in fact these are two slightly different ways of talking about the same thing.  I also remarked, in the summary of Konrad Waldorf’s talk, the idea that the theory of gerbes on a manifold is equivalent to ordinary gauge theory on its loop space – which is one way to make explicit the idea that categorification “raises dimension”, in this case from parallel transport of points to that of 1-dimensional loops.  Next we’ll expand on that theme, and then finally reach the “Quantum Gravity” part, and draw the connection between this and higher gauge theory toward the end.

Gerbes and Cohomology

The very first workshop speaker, in fact, was Paolo Aschieri, who has done a lot of work relating noncommutative geometry and gravity.  In this case, though, he was talking about noncommutative gerbes, and specifically referred to this work with some of the other speakers.  To be clear, this isn’t about gerbes with noncommutative group $G$, but about gerbes on noncommutative spaces.  To begin with, it’s useful to express gerbes in the usual sense in the right language.  In particular, he explain what a gerbe on a manifold $X$ is in concrete terms, giving Hitchin’s definition (viz).  A $U(1)$ gerbe can be described as “a cohomology class” but it’s more concrete to present it as:

• a collection of line bundles $L_{\alpha \beta}$ associated with double overlaps $U_{\alpha \beta} = U_{\alpha} \cap U_{\beta}$.  Note this gets an algebraic structure (multiplication $\star$ of bundles is pointwise $\otimes$, with an inverse given by the dual, $L^{-1} = L^*$, so we can require…
• $L_{\alpha \beta}^{-1} \cong L_{\beta \alpha}$, which helps define…
• transition functions $\lambda _{\alpha \beta \gamma}$ on triple overlaps $U_{\alpha \beta \gamma}$, which are sections of $L_{\alpha \beta \gamma} = L_{\alpha \beta} \star L_{\beta \gamma} \star L_{\gamma \alpha}$.  If this product is trivial, there’d be a 1-cocycle condition here, but we only insist on the 2-cocycle condition…
• $\lambda_{\beta \gamma \delta} \lambda_{\alpha \gamma \delta}^{-1} \lambda_{\alpha \beta \delta} \lambda_{\alpha \beta \gamma}^{-1} = 1$

This is a $U(1)$-gerbe on a commutative space.  The point is that one can make a similar definition for a noncommutative space.  If the space $X$ is associated with the algebra $A=C^{\infty}(X)$ of smooth functions, then a line bundle is a module for $A$, so if $A$ is noncommutative (thought of as a “space” $X$), a “bundle over $X$ is just defined to be an $A$-module.  One also has to define an appropriate “covariant derivative” operator $D$ on this module, and the $\star$-product must be defined as well, and will be noncommutative (we can think of it as a deformation of the $\star$ above).  The transition functions are sections: that is, elements of the modules in question.  his means we can describe a gerbe in terms of a big stack of modules, with a chosen algebraic structure, together with some elements.  The idea then is that gerbes can give an interpretation of cohomology of noncommutative spaces as well as commutative ones.

Mauro Spera spoke about a point of view of gerbes based on “transgressions”.  The essential point is that an $n$-gerbe on a space $X$ can be seen as the obstruction to patching together a family of  $(n-1)$-gerbes.  Thus, for instance, a $U(1)$ 0-gerbe is a $U(1)$-bundle, which is to say a complex line bundle.  As described above, a 1-gerbe can be understood as describing the obstacle to patching together a bunch of line bundles, and the obstacle is the ability to find a cocycle $\lambda$ satisfying the requisite conditions.  This obstacle is measured by the cohomology of the space.  Saying we want to patch together $(n-1)$-gerbes on the fibre.  He went on to discuss how this manifests in terms of obstructions to string structures on manifolds (already discussed at some length in the post on Hisham Sati’s school talk, so I won’t duplicate here).

A talk by Igor Bakovic, “Stacks, Gerbes and Etale Groupoids”, gave a way of looking at gerbes via stacks (see this for instance).  The organizing principle is the classification of bundles by the space maps into a classifying space – or, to get the category of principal $G$-bundles on, the category $Top(Sh(X),BG)$, where $Sh(X)$ is the category of sheaves on $X$ and $BG$ is the classifying topos of $G$-sets.  (So we have geometric morphisms between the toposes as the objects.)  Now, to get further into this, we use that $Sh(X)$ is equivalent to the category of Étale spaces over $X$ – this is a refinement of the equivalence between bundles and presheaves.  Taking stalks of a presheaf gives a bundle, and taking sections of a bundle gives a presheaf – and these operations are adjoint.

The issue at hand is how to categorify this framework to talk about 2-bundles, and the answer is there’s a 2-adjunction between the 2-category $2-Bun(X)$ of such things, and $Fib(X) = [\mathcal{O}(X)^{op},Cat]$, the 2-category of fibred categories over $X$.  (That is, instead of looking at “sheaves of sets”, we look at “sheaves of categories” here.)  The adjunction, again, involves talking stalks one way, and taking sections the other way.  One hard part of this is getting a nice definition of “stalk” for stacks (i.e. for the “sheaves of categories”), and a good part of the talk focused on explaining how to get a nice tractable definition which is (fibre-wise) equivalent to the more natural one.

Bakovic did a bunch of this work with Branislav Jurco, who was also there, and spoke about “Nonabelian Bundle 2-Gerbes“.  The paper behind that link has more details, which I’ve yet to entirely absorb, but the essential point appears to be to extend the description of “bundle gerbes” associated to crossed modules up to 2-crossed modules.  Bundles, with a structure-group $G$, are classified by the cohomology $H^1(X,G)$ with coefficients in $G$; and whereas “bundle-gerbes” with a structure-crossed-module $H \rightarrow G$ can likewise be described by cohomology $H^1(X,H \rightarrow G)$.  Notice this is a bit different from the description in terms of higher cohomology $H^2(X,G)$ for a $G$-gerbe, which can be understood as a bundle-gerbe using the shifted crossed module $G \rightarrow 1$ (when $G$ is abelian.  The goal here is to generalize this part to nonabelian groups, and also pass up to “bundle 2-gerbes” based on a 2-crossed module, or crossed complex of length 2, $L \rightarrow H \rightarrow G$ as I described previously for Joao Martins’ talk.  This would be classified in terms of cohomology valued in the 2-crossed module.  The point is that one can describe such a thing as a bundle over a fibre product, which (I think – I’m not so clear on this part) deals with the same structure of overlaps as the higher cohomology in the other way of describing things.

Finally,  a talk that’s a little harder to classify than most, but which I’ve put here with things somewhat related to string theory, was Alexander Kahle‘s on “T-Duality and Differential K-Theory”, based on work with Alessandro Valentino.  This uses the idea of the differential refinement of cohomology theories – in this case, K-theory, which is a generalized cohomology theory, which is to say that K-theory satisfies the Eilenberg-Steenrod axioms (with the dimension axiom relaxed, hence “generalized”).  Cohomology theories, including generalized ones, can have differential refinements, which pass from giving topological to geometrical information about a space.  So, while K-theory assigns to a space the Grothendieck ring of the category of vector bundles over it, the differential refinement of K-theory does the same with the category of vector bundles with connection.  This captures both local and global structures, which turns out to be necessary to describe fields in string theory – specifically, Ramond-Ramond fields.  The point of this talk was to describe what happens to these fields under T-duality.  This is a kind of duality in string theory between a theory with large strings and small strings.  The talk describes how this works, where we have a manifold with fibres at each point $M\times S^1_r$ with fibres strings of radius $r$ and $M \times S^1_{1/r}$ with radius $1/r$.  There’s a correspondence space $M \times S^1_r \times S^1_{1/r}$, which has projection maps down into the two situations.  Fields, being forms on such a fibration, can be “transferred” through this correspondence space by a “pull-back and push-forward” (with, in the middle, a wedge with a form that mixes the two directions, $exp( d \theta_r + d \theta_{1/r})$).  But to be physically the right kind of field, these “forms” actually need to be representing cohomology classes in the differential refinement of K-theory.

Quantum Gravity etc.

Now, part of the point of this workshop was to try to build, or anyway maintain, some bridges between the kind of work in geometry and topology which I’ve been describing and the world of physics.  There are some particular versions of physical theories where these ideas have come up.  I’ve already touched on string theory along the way (there weren’t many talks about it from a physicist’s point of view), so this will mostly be about a different sort of approach.

Benjamin Bahr gave a talk outlining this approach for our mathematician-heavy audience, with his talk on “Spin Foam Operators” (see also for instance this paper).  The point is that one approach to quantum gravity has a theory whose “kinematics” (the description of the state of a system at a given time) is described by “spin networks” (based on $SU(2)$ gauge theory), as described back in the pre-school post.  These span a Hilbert space, so the “dynamical” issue of such models is how to get operators between Hilbert spaces from “foams” that interpolate between such networks – that is, what kind of extra data they might need, and how to assign amplitudes to faces and edges etc. to define an operator, which (assuming a “local” theory where distant parts of the foam affect the result independently) will be of the form:

$Z(K,\rho,P) = (\prod_f A_f) \prod_v Tr_v(\otimes P_e)$

where $K$ is a particular complex (foam), $\rho$ is a way of assigning irreps to faces of the foam, and $P$ is the assignment of intertwiners to edges.  Later on, one can take a discrete version of a path integral by summing over all these $(K, \rho, P)$.  Here we have a product over faces and one over vertices, with an amplitude $A_f$ assigned (somehow – this is the issue) to faces.  The trace is over all the representation spaces assigned to the edges that are incident to a vertex (this is essentially the only consistent way to assign an amplitude to a vertex).  If we also consider spacetimes with boundary, we need some amplitudes $B_e$ at the boundary edges, as well.  A big part of the work with such models is finding such amplitudes that meet some nice conditions.

Some of these conditions are inherently necessary – to ensure the theory is invariant under gauge transformations, or (formally) changing orientations of faces.  Others are considered optional, though to me “functoriality” (that the way of deriving operators respects the gluing-together of foams) seems unavoidable – it imposes that the boundary amplitudes have to be found from the $A_f$ in one specific way.  Some other nice conditions might be: that $Z(K, \rho, P)$ depends only on the topology of $K$ (which demands that the $P$ operators be projections); that $Z$ is invariant under subdivision of the foam (which implies the amplitudes have to be $A_f = dim(\rho_f)$).

Assuming all these means the only choice is exactly which sub-projection $P_e$ is of the projection onto the gauge-invariant part of the representation space for the faces attached to edge $e$.  The rest of the talk discussed this, including some examples (models for BF-theory, the Barrett-Crane model and the more recent EPRL/FK model), and finished up by discussing issues about getting a nice continuum limit by way of “coarse graining”.

On a related subject, Bianca Dittrich spoke about “Dynamics and Diffeomorphism Symmetry in Discrete Quantum Gravity”, which explained the nature of some of the hard problems with this sort of discrete model of quantum gravity.  She began by asking what sort of models (i.e. which choices of amplitudes) in such discrete models would actually produce a nice continuum theory – since gravity, classically, is described in terms of spacetimes which are continua, and the quantum theory must look like this in some approximation.  The point is to think of these as “coarse-graining” of a very fine (perfect, in the limit) approximation to the continuum by a triangulation with a very short length-scale for the edges.  Coarse graining means discarding some of the edges to get a coarser approximation (perhaps repeatedly).  If the $Z$ happens to be triangulation-independent, then coarse graining makes no difference to the result, nor does the converse process of refining the triangulation.  So one question is:  if we expect the continuum limit to be diffeomorphism invariant (as is General Relativity), what does this say at the discrete level?  The relation between diffeomorphism invariance and triangulation invariance has been described by Hendryk Pfeiffer, and in the reverse direction by Dittrich et al.

Actually constructing the dynamics for a system like this in a nice way (“canonical dynamics with anomaly-free constraints”) is still a big problem, which Bianca suggested might be approached by this coarse-graining idea.  Now, if a theory is topological (here we get the link to TQFT), such as electromagnetism in 2D, or (linearized) gravity in 3D, coarse graining doesn’t change much.  But otherwise, changing the length scale means changing the action for the continuum limit of the theory.  This is related to renormalization: one starts with a “naive” guess at a theory, then refines it (in this case, by the coarse-graining process), which changes the action for the theory, until arriving at (or approximating to) a fixed point.  Bianca showed an example, which produces a really huge, horrible action full of very complicated terms, which seems rather dissatisfying.  What’s more, she pointed out that, unless the theory is topological, this always produces an action which is non-local – unlike the “naive” discrete theory.  That is, the action can’t be described in terms of a bunch of non-interacting contributions from the field at individual points – instead, it’s some function which couples the field values at distant points (albeit in a way that falls off exponentially as the points get further apart).

In a more specific talk, Aleksandr Mikovic discussed “Finiteness and Semiclassical Limit of EPRL-FK Spin Foam Models”, looking at a particular example of such models which is the (relatively) new-and-improved candidate for quantum gravity mentioned above.  This was a somewhat technical talk, which I didn’t entirely follow, but  roughly, the way he went at this was through the techniques of perturbative QFT.  That is, by looking at the theory in terms of an “effective action”, instead of some path integral over histories $\phi$ with action $S(\phi)$ – which looks like $\int d\phi e^{iS(\phi)}$.  Starting with some classical history $\bar{\phi}$ – a stationary point of the action $S$ – the effective action $\Gamma(\bar{\phi})$ is an integral over small fluctuations $\phi$ around it of $e^{iS(\bar{\phi} + \phi)}$.

He commented more on the distinction between the question of triangulation independence (which is crucial for using spin foams to give invariants of manifolds) and the question of whether the theory gives a good quantum theory of gravity – that’s the “semiclassical limit” part.  (In light of the above, this seems to amount to asking if “diffeomorphism invariance” really extends through to the full theory, or is only approximately true, in the limiting case).  Then the “finiteness” part has to do with the question of getting decent asymptotic behaviour for some of those weights mentioned above so as to give a nice effective action (if not necessarily triangulation independence).  So, for instance, in the Ponzano-Regge model (which gives a nice invariant for manifolds), the vertex amplitudes $A_v$ are found by the 6j-symbols of representations.  The asymptotics of the 6j symbols then becomes an issue – Alekandr noted that to get a theory with a nice effective action, those 6j-symbols need to be scaled by a certain factor.  This breaks triangulation independence (hence means we don’t have a good manifold invariant), but gives a physically nicer theory.  In the case of 3D gravity, this is not what we want, but as he said, there isn’t a good a-priori reason to think it can’t give a good theory of 4D gravity.

Now, making a connection between these sorts of models and higher gauge theory, Aristide Baratin spoke about “2-Group Representations for State Sum Models”.  This is a project Baez, Freidel, and Wise, building on work by Crane and Sheppard (see my previous post, where Derek described the geometry of the representation theory for some 2-groups).  The idea is to construct state-sum models where, at the kinematical level, edges are labelled by 2-group representations, faces by intertwiners, and tetrahedra by 2-intertwiners.  (This assumes the foam is a triangulation – there’s a certain amount of back-and-forth in this area between this, and the Poincaré dual picture where we have 4-valent vertices).  He discussed this in a couple of related cases – the Euclidean and Poincaré 2-groups, which are described by crossed modules with base groups $SO(4)$ or $SO(3,1)$ respectively, acting on the abelian group (of automorphisms of the identity) $R^4$ in the obvious way.  Then the analogy of the 6j symbols above, which are assigned to tetrahedra (or dually, vertices in a foam interpolating two kinematical states), are now 10j symbols assigned to 4-simplexes (or dually, vertices in the foam).

One nice thing about this setup is that there’s a good geometric interpretation of the kinematics – irreducible representations of these 2-groups pick out orbits of the action of the relevant $SO$ on $R^4$.  These are “mass shells” – radii of spheres in the Euclidean case, or proper length/time values that pick out hyperboloids in the Lorentzian case of $SO(3,1)$.  Assigning these to edges has an obvious geometric meaning (as a proper length of the edge), which thus has a continuous spectrum.  The areas and volumes interpreting the intertwiners and 2-intertwiners start to exhibit more of the discreteness you see in the usual formulation with representations of the $SO$ groups themselves.  Finally, Aristide pointed out that this model originally arose not from an attempt to make a quantum gravity model, but from looking at Feynman diagrams in flat space (a sort of “quantum flat space” model), which is suggestively interesting, if not really conclusively proving anything.

Finally, Laurent Freidel gave a talk, “Classical Geometry of Spin Network States” which was a way of challenging the idea that these states are exclusively about “quantum geometries”, and tried to give an account of how to interpret them as discrete, but classical.  That is, the quantization of the classical phase space $T^*(A/G)$ (the cotangent bundle of connections-mod-gauge) involves first a discretization to a spin-network phase space $\mathcal{P}_{\Gamma}$, and then a quantization to get a Hilbert space $H_{\Gamma}$, and the hard part is the first step.  The point is to see what the classical phase space is, and he describes it as a (symplectic) quotient $T^*(SU(2)^E)//SU(2)^V$, which starts by assigning $T^*(SU(2))$ to each edge, then reduced by gauge transformations.  The puzzle is to interpret the states as geometries with some discrete aspect.

The answer is that one thinks of edges as describing (dual) faces, and vertices as describing some polytopes.  For each $p$, there’s a $2(p-3)$-dimensional “shape space” of convex polytopes with $p$-faces and a given fixed area $j$.  This has a canonical symplectic structure, where lengths and interior angles at an edge are the canonically conjugate variables.  Then the whole phase space describes ways of building geometries by gluing these things (associated to vertices) together at the corresponding faces whenever the two vertices are joined by an edge.  Notice this is a bit strange, since there’s no particular reason the faces being glued will have the same shape: just the same area.  An area-1 pentagon and an area-1 square associated to the same edge could be glued just fine.  Then the classical geometry for one of these configurations is build of a bunch of flat polyhedra (i.e. with a flat metric and connection on them).  Measuring distance across a face in this geometry is a little strange.  Given two points inside adjacent cells, you measure orthogonal distance to the matched faces, and add in the distance between the points you arrive at (orthogonally) – assuming you glued the faces at the centre.  This is a rather ugly-seeming geometry, but it’s symplectically isomorphic to the phase space of spin network states – so it’s these classical geometries that spin-foam QG is a quantization of.  Maybe the ugliness should count against this model of quantum gravity – or maybe my aesthetic sense just needs work.

(Laurent also gave another talk, which was originally scheduled as one of the school talks, but ended up being a very interesting exposition of the principle of “Relativity of Localization”, which is hard to shoehorn into the themes I’ve used here, and was anyway interesting enough that I’ll devote a separate post to it.)

I recently went to California to visit Derek Wise at UC Davis – we were talking about expanding the talk he gave at Perimeter Institute into a more developed paper about ETQFT from Lie groups. Between that, the end of the Winter semester, and the beginning of the “Summer” session (in which I’m teaching linear algebra), it’s taken me a while to write up Emre Coskun’s two-part talk in our Stacks And Groupoids seminar.

Emre was explaining the theory of gerbes in terms of stacks. One way that I have often heard gerbes explained is in terms of a categorification of vector bundles – thus, the theory of “bundle gerbes”, as described by Murray in this paper here. The essential point of that point of view is that bundles can be put together by taking trivial bundles on little neighbourhoods of a space, and “gluing” them together on two-fold overlaps of those neighbourhoods – the gluing functions then have to satisfy a cocycle condition so that they agree on triple overlaps. A gerbe, on the other hand, defines line bundles (not functions) on double overlaps, and the gluing functions now live on triple overlaps. The idea is that this begins a heirarchy of concepts, each of which categorifes the previous (after “gerbe”, one just starts using terms like “2-gerbe”, “3-gerbe”, and so on). The levels of this hierarchy are supposed to be related to the various (nonabelian) cohomology groups $H^n(X,G)$ of a space $X$. I’ve mostly seen this point of view related to work by Jean-Luc Brylinski. It is a very differential-geometric sort of construction.

Emre, on the other hand, was describing another side to the theory of gerbes, which comes out of algebraic geometry, and is closely related to stacks. There’s a nice survey by Moerdijk which gives an account of gerbes from a similar point of view, though for later material, Emre said he drew on this book by Laumon and Moret-Bailly (which I can only find in the original French). As one might expect, a stack-theoretic view of gerbes thinks of them as generalizations of sheaves, rather than bundles. (The fact that there is a sheaf of sections of a bundle also generalizes to gerbes, so bundle-gerbes are a special case of this point of view).

Gerbes

So the setup is that we have some space $X$ – Emre was talking about the context of algebraic geometry, so the relevant idea of space here is scheme (which, if you’re interested, is assumed to have the etale topology – i.e. the one where covers use etale maps, the analog of local isomorphisms).  In the second talk, he generalized this to $S$-spaces: for some chosen scheme $S$.  That is, the category of “spaces” is based on the slice category $Sch/S$ of schemes equipped with maps into $S$, with the obvious morphisms.  This is a site, since there’s a notion of a cover over $S$ and so forth; an $S$-space is a sheaf (of sets) on this site.  So in particular, a scheme $X$ over $S$ determines an $S$-space, where $X : Sch/S \rightarrow Sets$ by $X(U) = Hom(U,X)$.  (That is, the usual way a space determines a representable sheaf).  There are also differential-geometric versions of gerbes.

So, whatever the right notion of space, a stack $\mathbb{F}$ over a space $X$ (in the sense of a sheaf of groupoids over $X$, which we’re assuming has the etale topology) is a gerbe if a couple of nice conditions apply:

1. There’s a cover $\{ U_i \rightarrow X \}$, such that none of the $\mathbb{F}(U_i)$ is empty.
2. Over any open $U$, all the objects $\mathbb{F}(U)$ are isomorphic (i.e. $\mathbb{F}(U)$ is connected as a category)

Notice that there doesn’t have to be a global object – that is, $\mathbb{F}(X)$ needn’t be empty – only some cover such that local objects exist – but where they do, they’re all “the same”.  These conditions can also be summarized in terms of the fibred category $\mathcal{F} \rightarrow X$.  There are two maps from $\pi, \Delta: \mathcal{F}\rightarrow \mathcal{F} \times_X \mathcal{F}$ – the projection and the diagonal.  The conditions respectively say these two maps are, locally, epi (i.e. surjective).

Emre’s first talk began by giving some examples of gerbes to motivate the rest. The first one is the “gerbe of splittings” of an Azumaya algebra. “An” Azumaya algebra $\mathcal{A}$ is actually a sheaf of algebras over some scheme $X$. The defining property is that locally it looks like the algebra of endomorphisms of a vector bundle. That is, on any neighborhood $U_i \subset X$, we have:

$\mathcal{A}(U_i) \cong End(\mathcal{E}_i)$

for some (algebraic) vector bundle $\mathcal{E}_i \rightarrow U_i$. A special case is when $X = Spec(\mathbb{R})$ is just a point, in which case an Azumaya algebra $\mathcal{A}$ is the same thing as a matrix algebra $M_n(\mathbb{R})$. So Azumaya algebras are not too complicated to describe.

The gerbe of splittings, $\mathbb{F}_{\mathcal{A}}$ for an Azumaya algebra is also not too complicated: a splitting is a way to represent an algebra as endomorphisms of a vector bundle – which in this case may only be possible locally. Over an given $U$, its objects are pairs $(E, \alpha)$, where $E$ is a vector bundle over $U$, and $\alpha : End(E) \rightarrow \mathbb{F}_{\mathcal{A}}(U)$ is an isomorphism. The morphisms are bundle isomorphisms that commute with the $\alpha$. So, roughly: if $\mathcal{A}$ is locally isomorphic to endomorphisms of vector bundles, the gerbe of splittings is the stack of all the vector bundles and isomorphisms which make this work. It’s easy to see this is a gerbe, since by definition, such bundles must exist locally, and necessarily they’re all isomorphic.

(This example – a gerbe consisting, locally, of a category of all vector bundles of a certain form – starts to suggest why one might want to think of gerbes as categorifying bundles.)

Another easily constructed gerbe in a similar spirit is found from a complex line bundle $\mathcal{L}$ over $X$ (and $n \in \mathbb{N}$). Then $\mathcal{X} \rightarrow X$ is a gerbe over $X$, where the groupoid $\mathcal{X}(U)$ over a neighborhood $U$ has, as objects, pairs $(\mathcal{M},\alpha)$ where $\alpha : \mathcal{M}^n \rightarrow \mathcal{L}$ is an isomorphism of line bundles. That is, the objects locally look like $n^{th}$ roots of $\mathcal{L}$. The gerbe is trivial (has a global object) if $\mathcal{L}$ has a root.

Cohomology

One says that a gerbe is banded by a sheaf of groups $\mathbb{G}$ on $X$ (or $\mathbb{G}$ is the band of the gerbe, or $\mathbb{F}$ is a $\mathbb{G}$-gerbe), if there are isomorphisms between the group $\mathbb{G}(U)$ and the automorphism group $Aut(u)$ for each object $u$ over $U$ (the property of a gerbe means these are all isomorphic). (These isomorphisms should also commute with the group homomorphisms induced by maps $\psi : V \rightarrow U$ of open sets.) So the band is, so to speak, the “local symmetry group over $U$” of the gerbe in a natural way.

In the case of the gerbe of splittings of $\mathcal{A}$ above, the band is $\mathbb{G}_m$, where over any given neighborhood, $\mathbb{G}_m(U) = Hom(U, G_m)$, where $G_m$ is the group of units in the base field: that is, the group $\mathbb{G}_m(U)$ consists of all the invertible sections in the structure sheaf of $X$. These get turned into bundle-automorphisms by taking a function $f$ to the automorphism that acts through multiplication by $f$. The gerbe $\mathcal{X}$ associated to a line bundle is banded by the group of $n^{th}$-roots of unity in sections in the structure sheaf.

From here, we can see how gerbes relate to cohomology. In particular, a $\mathbb{G}$-gerbe $\mathbb{F}$, we can associate a cohomology class $[F] \in H^2(X,\mathbb{G})$. This class can be thought of as “the obstruction to the existence of a global object”. So, in the case of an Azumaya algebra, it’s the obstruction to $\mathcal{A}$ being split (i.e. globally).

The way this works is, given a covering with an object $x_i$ in $\mathbb{F}(U_i)$, we take pull back this object along the morphisms corresponding to inclusions of sub-neighbourhoods, down to a triple-overlap $U_{ijk} = U_i \cup U_j \cup U_k$. Then there are isomorphisms comparing the different pullbacks: $u_{ij}^k : {x^i}_j^k \rightarrow x_i^{jk}$, and so on. (The lowered indices denote which of the $U$ we’re pulling back from).

Then we get a 2-cocycle in $\mathbb{G}(U_{ijk}$ (an isomorphism corresponding to what, for sheaves of sets, would be an identity). This is $c_{ijk} = u^i_{jk} ({u_i}^k_k)^{-1} u_{ij}^k$. The existence of this cocycle means that we’re getting an element in $H^2(X,\mathbb{G}$, which we denote $[\mathbb{F}]$. If a global object exists, then all our local objects are restrictions of a global one, the cocycle will always turn out to be the identity, so this class is trivial. A non-trivial class implies an obstruction to gluing the local objects into global ones.

Moduli Spaces

In the second talk, Emre gave some more examples of gerbes which it makes sense to think of as moduli spaces, including one which any gerbe resembles locally.

The first is the moduli space of all vector bundles $E$ over some (smooth, projective) curve $C$.  (Actually, one looks at those of some particular degree $d$ and rank $r$, and requires a condition called stability).

Actually, as discussed earlier in the seminar back in Aji’s talk, the right way to see this is that there is a “fine” moduli space – really a stack and not necessarily a space (in whichever context) – called $\mathcal{M}_C(r,d)$, and also a “coarse” moduli space called $M_C(r,d)$.  Roughly, the actual space $M_C(r,d)$ has points which are the isomorphism classes of vector bundles, while the stack remembers the whole groupoid of bundles and bundle-isomorphisms.  So there’s a map, which squashes a bundle to its isomorphism class: $\mathcal{M}_C(r,d) \rightarrow M_C(r,d)$ making the fine moduli space into a category fibred in groupoids – more than that, it’s a stack – and more than that, it’s a gerbe.  That is, there’s always a cover of $C$ such that there are some bundles locally, and (stable) bundles of a given rank and degree are always isomorphic.  In fact, this is a $\mathbb{G}_m$-gerbe, as above.

The next example is the gerbe of $G$-torsors, for a group $G$ (that is, $G$-sets which are isomorphic as $G$-sets to $G$ – the intuition is that they’re just like the group, but without a specified identity). The category $[\star / G ] = BG$ consists of $G$-torsors and their isomorphisms.  This is a gerbe over the point $\star$.  More interesting, when we’re in the context of $S$-spaces (and $S$ has a trivial action of $G$ on it), it becomes a $G$-gerbe over $S$.  Part of the point here is that any trivial gerbe (i.e. one with a section) is just such a classifying space for some group.  In particular, for the group of isomorphisms from a particular object to itself, crossed with $X$.

Since any gerbe has sections locally (that is, objects in $\mathbb{F}(U)$ for some $U$), every gerbe locally looks like one of these classifying-space gerbes.  This is the analog to the fact that any bundle locally looks like a product.