### Why Higher Geometric Quantization

The largest single presentation was a pair of talks on “The Motivation for Higher Geometric Quantum Field Theory” by Urs Schreiber, running to about two and a half hours, based on these notes. This was probably the clearest introduction I’ve seen so far to the motivation for the program he’s been developing for several years. Broadly, the idea is to develop a higher-categorical analog of geometric quantization (GQ for short).

One guiding idea behind this is that we should really be interested in quantization over (higher) stacks, rather than merely spaces. This leads inexorably to a higher-categorical version of GQ itself. The starting point, though, is that the defining features of stacks capture two crucial principles from physics: the gauge principle, and locality. The gauge principle means that we need to keep track not just of connections, but gauge transformations, which form respectively the objects and morphisms of a groupoid. “Locality” means that these groupoids of configurations of a physical field on spacetime is determined by its local configuration on regions as small as you like (together with information about how to glue together the data on small regions into larger regions).

Some particularly simple cases can be described globally: a scalar field gives the space of all scalar functions, namely maps into $\mathbb{C}$; sigma models generalise this to the space of maps $\Sigma \rightarrow M$ for some other target space. These are determined by their values pointwise, so of course are local.

More generally, physicists think of a field theory as given by a fibre bundle $V \rightarrow \Sigma$ (the previous examples being described by trivial bundles $\pi : M \times \Sigma \rightarrow \Sigma$), where the fields are sections of the bundle. Lagrangian physics is then described by a form on the jet bundle of $V$, i.e. the bundle whose fibre over $p \in \Sigma$ consists of the space describing the possible first $k$ derivatives of a section over that point.

More generally, a field theory gives a procedure $F$ for taking some space with structure – say a (pseudo-)Riemannian manifold $\Sigma$ – and produce a moduli space $X = F(\Sigma)$ of fields. The Sigma models happen to be representable functors: $F(\Sigma) = Maps(\Sigma,M)$ for some $M$, the representing object. A prestack is just any functor taking $\Sigma$ to a moduli space of fields. A stack is one which has a “descent condition”, which amounts to the condition of locality: knowing values on small neighbourhoods and how to glue them together determines values on larger neighborhoods.

The Yoneda lemma says that, for reasonable notions of “space”, the category $\mathbf{Spc}$ from which we picked target spaces $M$ embeds into the category of stacks over $\mathbf{Spc}$ (Riemannian manifolds, for instance) and that the embedding is faithful – so we should just think of this as a generalization of space. However, it’s a generalization we need, because gauge theories determine non-representable stacks. What’s more, the “space” of sections of one of these fibred stacks is also a stack, and this is what plays the role of the moduli space for gauge theory! For higher gauge theories, we will need higher stacks.

All of the above is the classical situation: the next issue is how to quantize such a theory. It involves a generalization of Geometric Quantization (GQ for short). Now a physicist who actually uses GQ will find this perspective weird, but it flows from just the same logic as the usual method.

In ordinary GQ, you have some classical system described by a phase space, a manifold $X$ equipped with a pre-symplectic 2-form $\omega \in \Omega^2(X)$. Intuitively, $\omega$ describes how the space, locally, can be split into conjugate variables. In the phase space for a particle in $n$-space, these “position” and “momentum” variables, and $\omega = \sum_x dx^i \wedge dp^i$; many other systems have analogous conjugate variables. But what really matters is the form $\omega$ itself, or rather its cohomology class.

Then one wants to build a Hilbert space describing the quantum analog of the system, but in fact, you need a little more than $(X,\omega)$ to do this. The Hilbert space is a space of sections of some bundle whose sections look like copies of the complex numbers, called the “prequantum line bundle“. It needs to be equipped with a connection, whose curvature is a 2-form in the class of $\omega$: in general, . (If $\omega$ is not symplectic, i.e. is degenerate, this implies there’s some symmetry on $X$, in which case the line bundle had better be equivariant so that physically equivalent situations correspond to the same state). The easy case is the trivial bundle, so that we get a space of functions, like $L^2(X)$ (for some measure compatible with $\omega$). In general, though, this function-space picture only makes sense locally in $X$: this is why the choice of prequantum line bundle is important to the interpretation of the quantized theory.

Since the crucial geometric thing here is a bundle over the moduli space, when the space is a stack, and in the context of higher gauge theory, it’s natural to seek analogous constructions using higher bundles. This would involve, instead of a (pre-)symplectic 2-form $\omega$, an $(n+1)$-form called a (pre-)$n$-plectic form (for an introductory look at this, see Chris Rogers’ paper on the case $n=2$ over manifolds). This will give a higher analog of the Hilbert space.

Now, maps between Hilbert spaces in QG come from Lagrangian correspondences – these might be maps of moduli spaces, but in general they consist of a “space of trajectories” equipped with maps into a space of incoming and outgoing configurations. This is a span of pre-symplectic spaces (equipped with pre-quantum line bundles) that satisfies some nice geometric conditions which make it possible to push a section of said line bundle through the correspondence. Since each prequantum line bundle can be seen as maps out of the configuration space into a classifying space (for $U(1)$, or in general an $n$-group of phases), we get a square. The action functional is a cell that fills this square (see the end of 2.1.3 in Urs’ notes). This is a diagrammatic way to describe the usual GQ construction: the advantage is that it can then be repeated in the more general setting without much change.

This much is about as far as Urs got in his talk, but the notes go further, talking about how to extend this to infinity-stacks, and how the Dold-Kan correspondence tells us nicer descriptions of what we get when linearizing – since quantization puts us into an Abelian category.

I enjoyed these talks, although they were long and Urs came out looking pretty exhausted, because while I’ve seen several others on this program, this was the first time I’ve seen it discussed from the beginning, with a lot of motivation. This was presumably because we had a physically-minded part of the audience, whereas I’ve mostly seen these for mathematicians, and usually they come in somewhere in the middle and being more time-limited miss out some of the details and the motivation. The end result made it quite a natural development. Overall, very helpful!

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 $\mathcal{G}$. The points in these space are connections on a manifold $M$. 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 $G$-valued functions, or $Lie(H)$-valued 1-forms, where $G$ is the group of objects of $\mathcal{G}$, and $H$ is the group of morphisms based at $1_G$. If we think of connections as functors from the fundamental 2-groupoid $\Pi_2(M)$ into $\mathcal{G}$, 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 $String(G)$ (a homotopy 2-type) is fibred over the classifying space for $G$ (a 1-type). The exact map is determined by taking a pullback along a certain characteristic class (which is a map out of $BG$). 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 $G$. The group that’s involved here is a $String(n)$, the string 2-group associated to $Spin(n)$, the universal cover of the rotation group $SO(n)$. This is the one that determines whether a given manifold can support a “string structure”. A string structure on $M$, therefore, is a lift of a spin structure, which determines whether one can have a spin bundle over $M$, hence consistently talk about a spin connection which gives parallel transport for spinor fields on $M$. The string structure determines if one can consistently talk about a string-bundle over $M$, 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 $M = S^5 \times S^1$. This particular $M$ 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 $M$ nicely using the projective quaternionic plane, but conceptually it relies on the fact that $S^5 \cong SU(3)/SU(2)$, and because of how the lifting works, this is also $String(SU(3))/String(SU(2))$. This quotient means there’s a string bundle $String(SU(3)) \rightarrow S^5$ whose fibre is $String(SU(2))$.

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)

So it’s been a while since I last posted – the end of 2013 ended up being busy with a couple of visits to Jamie Vicary in Oxford, and Roger Picken in Lisbon. In the aftermath of the two trips, I did manage to get a major revision of this paper submitted to a journal, and put this one out in public. A couple of others will be coming down the pipeline this year as well.

I’m hoping to get back to a post about motives which I planned earlier, but for the moment, I’d like to write a little about the second paper, with Roger Picken.

### Global and Local Symmetry

The upshot is that it’s about categorifying the concept of symmetry. More specifically, it’s about finding the analog in the world of categories for the interplay between global and local symmetry which occurs in the world of set-based structures (sets, topological spaces, vector spaces, etc.) This distinction is discussed in a nice way by Alan Weinstein in this article from the Notices of the AMS from

The global symmetry of an object $X$ in some category $\mathbf{C}$ can be described in terms of its group of automorphisms: all the ways the object can be transformed which leave it “the same”. This fits our understanding of “symmetry” when the morphisms can really be interpreted as transformations of some sort. So let’s suppose the object is a set with some structure, and the morphisms are set-maps that preserve the structure: for example, the objects could be sets of vertices and edges of a graph, so that morphisms are maps of the underlying data that preserve incidence relations. So a symmetry of an object is a way of transforming it into itself – and an invertible one at that – and these automorphisms naturally form a group $Aut(X)$. More generally, we can talk about an action of a group $G$ on an object $X$, which is a map $\phi : G \rightarrow Aut(X)$.

“Local symmetry” is different, and it makes most sense in a context where the object $X$ is a set – or at least, where it makes sense to talk about elements of $X$, so that $X$ has an underlying set of some sort.

Actually, being a set-with-structure, in a lingo I associate with Jim Dolan, means that the forgetful functor $U : \mathbf{C} \rightarrow \mathbf{Sets}$ is faithful: you can tell morphisms in $\mathbf{C}$ (in particular, automorphisms of $X$) apart by looking at what they do to the underlying set. The intuition is that the morphisms of $\mathbf{C}$ are exactly set maps which preserve the structure which $U$ forgets about – or, conversely, that the structure on objects of $\mathbf{C}$ is exactly that which is forgotten by $U$. Certainly, knowing only this information determines $\mathbf{C}$ up to equivalence. In any case, suppose we have an object like this: then knowing about the symmetries of $X$ amounts to knowing about a certain group action, namely the action of $Aut(X)$, on the underlying set $U(X)$.

From this point of view, symmetry is about group actions on sets. The way we represent local symmetry (following Weinstein’s discussion, above) is to encode it as a groupoid – a category whose morphisms are all invertible. There is a level-slip happening here, since $X$ is now no longer seen as an object inside a category: it is the collection of all the objects of a groupoid. What makes this a representation of “local” symmetry is that each morphism now represents, not just a transformation of the whole object $X$, but a relationship under some specific symmetry between one element of $X$ and another. If there is an isomorphism between $x \in X$ and $y \in X$, then $x$ and $y$ are “symmetric” points under some transformation. As Weinstein’s article illustrates nicely, though, there is no assumption that the given transformation actually extends to the entire object $X$: it may be that only part of $X$ has, for example, a reflection symmetry, but the symmetry doesn’t extend globally.

### Transformation Groupoid

The “interplay” I alluded to above, between the global and local pictures of symmetry, is to build a “transformation groupoid” (or “action groupoid“) associated to a group $G$ acting on a set $X$. The result is called $X // G$ for short. Its morphisms consist of pairs such that  $(g,x) : x \rightarrow (g \rhd x)$ is a morphism taking $x$ to its image under the action of $g \in G$. The “local” symmetry view of $X // G$ treats each of these symmetry relations between points as a distinct bit of data, but coming from a global symmetry – that is, a group action – means that the set of morphisms comes from the product $G \times X$.

Indeed, the “target” map in $X // G$ from morphisms to objects is exactly a map $G \times X \rightarrow X$. It is not hard to show that this map is an action in another standard sense. Namely, if we have a real action $\phi : G \rightarrow Hom(X,X)$, then this map is just $\hat{\phi} : G \times X \rightarrow X$, which moves one of the arguments to the left side. If $\phi$ was a functor, then $\hat{\phi}$ satisfies the “action” condition, namely that the following square commutes:

(Here, $m$ is the multiplication in $G$, and this is the familiar associativity-type axiom for a group action: acting by a product of two elements in $G$ is the same as acting by each one successively.

So the starting point for the paper with Roger Picken was to categorify this. It’s useful, before doing that, to stop and think for a moment about what makes this possible.

First, as stated, this assumed that $X$ either is a set, or has an underlying set by way of some faithful forgetful functor: that is, every morphism in $Aut(X)$ corresponds to a unique set map from the elements of $X$ to itself. We needed this to describe the groupoid $X // G$, whose objects are exactly the elements of $X$. The diagram above suggests a different way to think about this. The action diagram lives in the category $\mathbf{Set}$: we are thinking of $G$ as a set together with some structure maps. $X$ and the morphism $\hat{\phi}$ must be in the same category, $\mathbf{Set}$, for this characterization to make sense.

So in fact, what matters is that the category $X$ lived in was closed: that is, it is enriched in itself, so that for any objects $X,Y$, there is an object $Hom(X,Y)$, the internal hom. In this case, it’s $G = Hom(X,X)$ which appears in the diagram. Such an internal hom is supposed to be a dual to $\mathbf{Set}$‘s monoidal product (which happens to be the Cartesian product $\times$): this is exactly what lets us talk about $\hat{\phi}$.

So really, this construction of a transformation groupoid will work for any closed monoidal category $\mathbf{C}$, producing a groupoid in $\mathbf{C}$. It may be easier to understand in cases like $\mathbf{C}=\mathbf{Top}$, the category of topological spaces, where there is indeed a faithful underlying set functor. But although talking explicitly about elements of $X$ was useful for intuitively seeing how $X//G$ relates global and local symmetries, it played no particular role in the construction.

### Categorify Everything

In the circles I run in, a popular hobby is to “categorify everything“: there are different versions, but what we mean here is to turn ideas expressed in the world of sets into ideas in the world of categories. (Technical aside: all the categories here are assumed to be small). In principle, this is harder than just reproducing all of the above in any old closed monoidal category: the “world” of categories is $\mathbf{Cat}$, which is a closed monoidal 2-category, which is a more complicated notion. This means that doing all the above “strictly” is a special case: all the equalities (like the commutativity of the action square) might in principle be replaced by (natural) isomorphisms, and a good categorification involves picking these to have good properties.

(In our paper, we left this to an appendix, because the strict special case is already interesting, and in any case there are “strictification” results, such as the fact that weak 2-groups are all equivalent to strict 2-groups, which mean that the weak case isn’t as much more general as it looks. For higher $n$-categories, this will fail – which is why we include the appendix to suggest how the pattern might continue).

Why is this interesting to us? Bumping up the “categorical level” appeals for different reasons, but the ones matter most to me have to do with taking low-dimensional (or -codimensional) structures, and finding analogous ones at higher (co)dimension. In our case, the starting point had to do with looking at the symmetries of “higher gauge theories” – which can be used to describe the transport of higher-dimensional surfaces in a background geometry, the way gauge theories can describe the transport of point particles. But I won’t ask you to understand that example right now, as long as you can accept that “what are the global/local symmetries of a category like?” is a possibly interesting question.

So let’s categorify the discussion about symmetry above… To begin with, we can just take our (closed monoidal) category to be $\mathbf{Cat}$, and follow the same construction above. So our first ingredient is a 2-group $\mathcal{G}$. As with groups, we can think of a 2-group either as a 2-category with just one object $\star$, or as a 1-category with some structure – a group object in $\mathbf{Cat}$, which we’ll call $C(\mathcal{G})$ if it comes from a given 2-group. (In our paper, we keep these distinct by using the term “categorical group” for the second. The group axioms amount to saying that we have a monoidal category $(\mathcal{G}, \otimes, I)$. Its objects are the morphisms of the 2-group, and the composition becomes the monoidal product $\otimes$.)

(In fact, we often use a third equivalent definition, that of crossed modules of groups, but to avoid getting into that machinery here, I’ll be changing our notation a little.)

### 2-Group Actions

So, again, there are two ways to talk about an action of a 2-group on some category $\mathbf{C}$. One is to define an action as a 2-functor $\Phi : \mathcal{G} \rightarrow \mathbf{Cat}$. The object being acted on, $\mathbf{C} \in \mathbf{Cat}$, is the unique object $\Phi(\star)$ – so that the 2-functor amounts to a monoidal functor from the categorical group $C(\mathcal{G})$ into $Aut(\mathbf{C})$. Notice that here we’re taking advantage of the fact that $\mathbf{Cat}$ is closed, so that the hom-“sets” are actually categories, and the automorphisms of $\mathbf{C}$ – invertible functors from $\mathbf{C}$ to itself – form the objects of a monoidal category, and in fact a categorical group. What’s new, though, is that there are also 2-morphisms – natural transformations between these functors.

To begin with, then, we show that there is a map $\hat{\Phi} : \mathcal{G} \times \mathbf{C} \rightarrow \mathbf{C}$, which corresponds to the 2-functor $\Phi$, and satisfies an action axiom like the square above, with $\otimes$ playing the role of group multiplication. (Again, remember that we’re only talking about the version where this square commutes strictly here – in an appendix of the paper, we talk about the weak version of all this.) This is an intuitive generalization of the situation for groups, but it is slightly more complicated.

The action $\Phi$ directly gives three maps. First, functors $\Phi(\gamma) : \mathbf{C} \rightarrow \mathbf{C}$ for each 2-group morphism $\gamma$ – each of which consists of a function between objects of $\mathbf{C}$, together with a function between morphisms of $\mathbf{C}$. Second, natural transformations $\Phi(\eta) : \Phi(\gamma) \rightarrow \Phi(\gamma ')$ for 2-morphisms $\eta : \gamma \rightarrow \gamma'$ in the 2-group – each of which consists of a function from objects to morphisms of $\mathbf{C}$.

On the other hand, $\hat{\Phi} : \mathcal{G} \times \mathbf{C} \rightarrow \mathbf{C}$ is just a functor: it gives two maps, one taking pairs of objects to objects, the other doing the same for morphisms. Clearly, the map $(\gamma,x) \mapsto x'$ is just given by $x' = \Phi(\gamma)(x)$. The map taking pairs of morphisms $(\eta,f) : (\gamma,x) \rightarrow (\gamma ', y)$ to morphisms of $\mathbf{C}$ is less intuitively obvious. Since I already claimed $\Phi$ and $\hat{\Phi}$ are equivalent, it should be no surprise that we ought to be able to reconstruct the other two parts of $\Phi$ from it as special cases. These are morphism-maps for the functors, (which give $\Phi(\gamma)(f)$ or $\Phi(\gamma ')(f)$), and the natural transformation maps (which give $\Phi(\eta)(x)$ or $\Phi(\eta)(y)$). In fact, there are only two sensible ways to combine these four bits of information, and the fact that $\Phi(\eta)$ is natural means precisely that they’re the same, so:

$\hat{\Phi}(\eta,f) = \Phi(\eta)(y) \circ \Phi(\gamma)(f) = \Phi(\gamma ')(f) \circ \Phi(\eta)(x)$

Given the above, though, it’s not so hard to see that a 2-group action really involves two group actions: of the objects of $\mathcal{G}$ on the objects of $\mathbf{C}$, and of the morphisms of $\mathcal{G}$ on objects of $\mathbf{C}$. They fit together nicely because objects can be identified with their identity morphisms: furthermore, $\Phi$ being a functor gives an action of $\mathcal{G}$-objects on $\mathbf{C}$-morphisms which fits in between them nicely.

But what of the transformation groupoid? What is the analog of the transformation groupoid, if we repeat its construction in $\mathbf{Cat}$?

### The Transformation Double Category of a 2-Group Action

The answer is that a category (such as a groupoid) internal to $\mathbf{Cat}$ is a double category. The compact way to describe it is as a “category in $\mathbf{Cat}$“, with a category of objects and a category of morphisms, each of which of course has objects and morphisms of its own. For the transformation double category, following the same construction as for sets, the object-category is just $\mathbf{C}$, and the morphism-category is $\mathcal{G} \times \mathbf{C}$, and the target functor is just the action map $\hat{\Phi}$. (The other structure maps that make this into a category in $\mathbf{Cat}$ can similarly be worked out by following your nose).

This is fine, but the internal description tends to obscure an underlying symmetry in the idea of double categories, in which morphisms in the object-category and objects in the morphism-category can switch roles, and get a different description of “the same” double category, denoted the “transpose”.

A different approach considers these as two different types of morphism, “horizontal” and “vertical”: they are the morphisms of horizontal and vertical categories, built on the same set of objects (the objects of the object-category). The morphisms of the morphism-category are then called “squares”. This makes a convenient way to draw diagrams in the double category. Here’s a version of a diagram from our paper with the notation I’ve used here, showing what a square corresponding to a morphism $(\chi,f) \in \mathcal{G} \times \mathbf{C}$ looks like:

The square (with the boxed label) has the dashed arrows at the top and bottom for its source and target horizontal morphisms (its images under the source and target functors: the argument above about naturality means they’re well-defined). The vertical arrows connecting them are the source and target vertical morphisms (its images under the source and target maps in the morphism-category).

### Horizontal and Vertical Slices of $\mathbf{C} // \mathcal{G}$

So by construction, the horizontal category of these squares is just the object-category $\mathbf{C}$.  For the same reason, the squares and vertical morphisms, make up the category $\mathcal{G} \times \mathbf{C}$.

On the other hand, the vertical category has the same objects as $\mathbf{C}$, but different morphisms: it’s not hard to see that the vertical category is just the transformation groupoid for the action of the group of $\mathbf{G}$-objects on the set of $\mathbf{C}$-objects, $Ob(\mathbf{C}) // Ob(\mathcal{G})$. Meanwhile, the horizontal morphisms and squares make up the transformation groupoid $Mor(\mathbf{C}) // Mor(\mathcal{G})$. These are the object-category and morphism-category of the transpose of the double-category we started with.

We can take this further: if squares aren’t hip enough for you – or if you’re someone who’s happy with 2-categories but finds double categories unfamiliar – the horizontal and vertical categories can be extended to make horizontal and vertical bicategories. They have the same objects and morphisms, but we add new 2-cells which correspond to squares where the boundaries have identity morphisms in the direction we’re not interested in. These two turn out to feel quite different in style.

First, the horizontal bicategory extends $\mathbf{C}$ by adding 2-morphisms to it, corresponding to morphisms of $\mathcal{G}$: roughly, it makes the morphisms of $\mathbf{C}$ into the objects of a new transformation groupoid, based on the action of the group of automorphisms of the identity in $\mathcal{G}$ (which ensures the square has identity edges on the sides.) This last point is the only constraint, and it’s not a very strong one since $Aut(1_G)$ and $G$ essentially determine the entire 2-group: the constraint only relates to the structure of $\mathcal{G}$.

The constraint for the vertical bicategory is different in flavour because it depends more on the action $\Phi$. Here we are extending a transformation groupoid, $Ob(\mathbf{C}) // Ob(\mathcal{G})$. But, for some actions, many morphisms in $\mathcal{G}$ might just not show up at all. For 1-morphisms $(\gamma, x)$, the only 2-morphisms which can appear are those taking $\gamma$ to some $\gamma '$ which has the same effect on $x$ as $\gamma$. So, for example, this will look very different if $\Phi$ is free (so only automorphisms show up), or a trivial action (so that all morphisms appear).

In the paper, we look at these in the special case of an adjoint action of a 2-group, so you can look there if you’d like a more concrete example of this difference.

### Speculative Remarks

The starting point for this was a project (which I talked about a year ago) to do with higher gauge theory – see the last part of the linked post for more detail. The point is that, in gauge theory, one deals with connections on bundles, and morphisms between them called gauge transformations. If one builds a groupoid out of these in a natural way, it turns out to result from the action of a big symmetry group of all gauge transformations on the moduli space of connections.

In higher gauge theory, one deals with connections on gerbes (or higher gerbes – a bundle is essentially a “0-gerbe”). There are now also (2-)morphisms between gauge transformations (and, in higher cases, this continues further), which Roger Picken and I have been calling “gauge modifications”. If we try to repeat the situation for gauge theory, we can construct a 2-groupoid out of these, which expresses this local symmetry. The thing which is different for gerbes (and will continue to get even more different if we move to $n$-gerbes and the corresponding $(n+1)$-groupoids) is that this is not the same type of object as a transformation double category.

Now, in our next paper (which this one was written to make possible) we show that the 2-groupoid is actually very intimately related to the transformation double category: that is, the local picture of symmetry for a higher gauge theory is, just as in the lower-dimensional situation, intimately related to a global symmetry of an entire moduli 2-space, i.e. a category. The reason this wasn’t obvious at first is that the moduli space which includes only connections is just the space of objects of this category: the point is that there are really two special kinds of gauge transformations. One should be thought of as the morphisms in the moduli 2-space, and the other as part of the symmetries of that 2-space. The intuition that comes from ordinary gauge theory overlooks this, because the phenomenon doesn’t occur there.

Physically-motivated theories are starting to use these higher-categorical concepts more and more, and symmetry is a crucial idea in physics. What I’ve sketched here is presumably only the start of a pattern in which “symmetry” extends to higher-categorical entities. When we get to 3-groups, our simplifying assumptions that use “strictification” results won’t even be available any more, so we would expect still further new phenomena to show up – but it seems plausible that the tight relation between global and local symmetry will still exist, but in a way that is more subtle, and refines the standard understanding we have of symmetry today.

Since the last post, I’ve been busily attending some conferences, as well as moving to my new job at the University of Hamburg, in the Graduiertenkolleg 1670, “Mathematics Inspired by String Theory and Quantum Field Theory”.  The week before I started, I was already here in Hamburg, at the conference they were organizing “New Perspectives in Topological Quantum Field Theory“.  But since I last posted, I was also at the 20th Oporto Meeting on Geometry, Topology, and Physics, as well as the third Higher Structures in China workshop, at Jilin University in Changchun.  Right now, I’d like to say a few things about some of the highlights of that workshop.

Higher Structures in China III

So last year I had a bunch of discussions I had with Chenchang Zhu and Weiwei Pan, who at the time were both in Göttingen, about my work with Jamie Vicary, which I wrote about last time when the paper was posted to the arXiv.  In that, we showed how the Baez-Dolan groupoidification of the Heisenberg algebra can be seen as a representation of Khovanov’s categorification.  Chenchang and Weiwei and I had been talking about how these ideas might extend to other examples, in particular to give nice groupoidifications of categorified Lie algebras and quantum groups.

That is still under development, but I was invited to give a couple of talks on the subject at the workshop.  It was a long trip: from Lisbon, the farthest-west of the main cities of (continental) Eurasia all the way to one of the furthest-East.   (Not quite the furthest, but Changchun is in the northeast of China, just a few hours north of Korea, and it took just about exactly 24 hours including stopovers to get there).  It was a long way to go for a three day workshop, but as there were also three days of a big excursion to Changbai Mountain, just on the border with North Korea, for hiking and general touring around.  So that was a sort of holiday, with 11 other mathematicians.  Here is me with Dany Majard, in a national park along the way to the mountains:

Here’s me with Alex Hoffnung, on Changbai Mountain (in the background is China):

And finally, here’s me a little to the left of the previous picture, where you can see into the volcanic crater.  The lake at the bottom is cut out of the picture, but you can see the crater rim, of which this particular part is in North Korea, as seen from China:

Well, that was fun!

Anyway, the format of the workshop involved some talks from foreigners and some from locals, with a fairly big local audience including a good many graduate students from Jilin University.  So they got a chance to see some new work being done elsewhere – mostly in categorification of one kind or another.  We got a chance to see a little of what’s being done in China, although not as much as we might have. I gather that not much is being done yet that fit the theme of the workshop, which was part of the reason to organize the workshop, and especially for having a session aimed specially at the graduate students.

### Categorified Algebra

This is a sort of broad term, but certainly would include my own talk.  The essential point is to show how the groupoidification of the Heisenberg algebra is a representation of Khovanov’s categorification of the same algebra, in a particular 2-category.  The emphasis here is on the fact that it’s a representation in a 2-category whose objects are groupoids, but whose morphisms aren’t just functors, but spans of functors – that is, composites of functors and co-functors.  This is a pretty conservative weakening of “representations on categories” – but it lets one build really simple combinatorial examples.  I’ve discussed this general subject in recent posts, so I won’t elaborate too much.  The lecture notes are here, if you like, though – they have more detail than my previous post, but are less technical than the paper with Jamie Vicary.

Aaron Lauda gave a nice introduction to the program of categorifying quantum groups, mainly through the example of the special case $U_q(sl_2)$, somewhat along the same lines as in his introductory paper on the subject.  The story which gives the motivation is nice: one has knot invariants such as the Jones polynomial, based on representations of groups and quantum groups.  The Jones polynomial can be categorified to give Khovanov homology (which assigns a complex to a knot, whose graded Euler characteristic is the Jones polynomial) – but also assigns maps of complexes to cobordisms of knots.  One then wants to categorify the representation theory behind it – to describe actions of, for instance, quantum $sl_2$ on categories.  This starting point is nice, because it can work by just mimicking the construction of $sl_2$ and $U_q(sl_2)$ representations in terms of weight spaces: one gets categories $V_{-N}, \dots, V_N$ which correspond to the “weight spaces” (usually just vector spaces), and the $E$ and $F$ operators give functors between them, and so forth.

Finding examples of categories and functors with this structure, and satisfying the right relations, gives “categorified representations” of the algebra – the monoidal categories of diagrams which are the “categorifications of the algebra” then are seen as the abstraction of exactly which relations these are supposed to satisfy.  One such example involves flag varieties.  A flag, as one might eventually guess from the name, is a nested collection of subspaces in some $n$-dimensional space.  A simple example is the Grassmannian $Gr(1,V)$, which is the space of all 1-dimensional subspaces of $V$ (i.e. the projective space $P(V)$), which is of course an algebraic variety.  Likewise, $Gr(k,V)$, the space of all $k$-dimensional subspaces of $V$ is a variety.  The flag variety $Fl(k,k+1,V)$ consists of all pairs $W_k \subset W_{k+1}$, of a $k$-dimensional subspace of $V$, inside a $(k+1)$-dimensional subspace (the case $k=2$ calls to mind the reason for the name: a plane intersecting a given line resembles a flag stuck to a flagpole).  This collection is again a variety.  One can go all the way up to the variety of “complete flags”, $Fl(1,2,\dots,n,V)$ (where $V$ is $n$-dimenisonal), any point of which picks out a subspace of each dimension, each inside the next.

The way this relates to representations is by way of geometric representation theory. One can see those flag varieties of the form $Fl(k,k+1,V)$ as relating the Grassmanians: there are projections $Fl(k,k+1,V) \rightarrow Gr(k,V)$ and $Fl(k,k+1,V) \rightarrow Gr(k+1,V)$, which act by just ignoring one or the other of the two subspaces of a flag.  This pair of maps, by way of pulling-back and pushing-forward functions, gives maps between the cohomology rings of these spaces.  So one gets a sequence $H_0, H_1, \dots, H_n$, and maps between the adjacent ones.  This becomes a representation of the Lie algebra.  Categorifying this, one replaces the cohomology rings with derived categories of sheaves on the flag varieties – then the same sort of “pull-push” operation through (derived categories of sheaves on) the flag varieties defines functors between those categories.  So one gets a categorified representation.

Heather Russell‘s talk, based on this paper with Aaron Lauda, built on the idea that categorified algebras were motivated by Khovanov homology.  The point is that there are really two different kinds of Khovanov homology – the usual kind, and an Odd Khovanov Homology, which is mainly different in that the role played in Khovanov homology by a symmetric algebra is instead played by an exterior (antisymmetric) algebra.  The two look the same over a field of characteristic 2, but otherwise different.  The idea is then that there should be “odd” versions of various structures that show up in the categorifications of $U_q(sl_2)$ (and other algebras) mentioned above.

One example is the fact that, in the “even” form of those categorifications, there is a natural action of the Nil Hecke algebra on composites of the generators.  This is an algebra which can be seen to act on the space of polynomials in $n$ commuting variables, $\mathbb{C}[x_1,\dots,x_n]$, generated by the multiplication operators $x_i$, and the “divided difference operators” based on the swapping of two adjacent variables.  The Hecke algebra is defined in terms of “swap” generators, which satisfy some $q$-deformed variation of the relations that define the symmetric group (and hence its group algebra).   The Nil Hecke algebra is so called since the “swap” (i.e. the divided difference) is nilpotent: the square of the swap is zero.  The way this acts on the objects of the diagrammatic category is reflected by morphisms drawn as crossings of strands, which are then formally forced to satisfy the relations of the Nil Hecke algebra.

The ODD Nil Hecke algebra, on the other hand, is an analogue of this, but the $x_i$ are anti-commuting, and one has different relations satisfied by the generators (they differ by a sign, because of the anti-commutation).  This sort of “oddification” is then supposed to happen all over.  The main point of the talk was to to describe the “odd” version of the categorified representation defined using flag varieties.  Then the odd Nil Hecke algebra acts on that, analogously to the even case above.

Marco Mackaay gave a couple of talks about the $sl_3$ web algebra, describing the results of this paper with Weiwei Pan and Daniel Tubbenhauer.  This is the analog of the above, for $U_q(sl_3)$, describing a diagram calculus which accounts for representations of the quantum group.  The “web algebra” was introduced by Greg Kuperberg – it’s an algebra built from diagrams which can now include some trivalent vertices, along with rules imposing relations on these.  When categorifying, one gets a calculus of “foams” between such diagrams.  Since this is obviously fairly diagram-heavy, I won’t try here to reproduce what’s in the paper – but an important part of is the correspondence between webs and Young Tableaux, since these are labels in the representation theory of the quantum group – so there is some interesting combinatorics here as well.

### Algebraic Structures

Some of the talks were about structures in algebra in a more conventional sense.

Jiang-Hua Lu: On a class of iterated Poisson polynomial algebras.  The starting point of this talk was to look at Poisson brackets on certain spaces and see that they can be found in terms of “semiclassical limits” of some associative product.  That is, the associative product of two elements gives a power series in some parameter $h$ (which one should think of as something like Planck’s constant in a quantum setting).  The “classical” limit is the constant term of the power series, and the “semiclassical” limit is the first-order term.  This gives a Poisson bracket (or rather, the commutator of the associative product does).  In the examples, the spaces where these things are defined are all spaces of polynomials (which makes a lot of explicit computer-driven calculations more convenient). The talk gives a way of constructing a big class of Poisson brackets (having some nice properties: they are “iterated Poisson brackets”) coming from quantum groups as semiclassical limits.  The construction uses words in the generating reflections for the Weyl group of a Lie group $G$.

Li Guo: Successors and Duplicators of Operads – first described a whole range of different algebra-like structures which have come up in various settings, from physics and dynamical systems, through quantum field theory, to Hopf algebras, combinatorics, and so on.  Each of them is some sort of set (or vector space, etc.) with some number of operations satisfying some conditions – in some cases, lots of operations, and even more conditions.  In the slides you can find several examples – pre-Lie and post-Lie algebras, dendriform algebras, quadri- and octo-algebras, etc. etc.  Taken as a big pile of definitions of complicated structures, this seems like a terrible mess.  The point of the talk is to point out that it’s less messy than it appears: first, each definition of an algebra-like structure comes from an operad, which is a formal way of summing up a collection of operations with various “arities” (number of inputs), and relations that have to hold.  The second point is that there are some operations, “successor” and “duplicator”, which take one operad and give another, and that many of these complicated structures can be generated from simple structures by just these two operations.  The “successor” operation for an operad introduces a new product related to old ones – for example, the way one can get a Lie bracket from an associative product by taking the commutator.  The “duplicator” operation takes existing products and introduces two new products, whose sum is the previous one, and which satisfy various nice relations.  Combining these two operations in various ways to various starting points yields up a plethora of apparently complicated structures.

Dany Majard gave a talk about algebraic structures which are related to double groupoids, namely double categories where all the morphisms are invertible.  The first part just defined double categories: graphically, one has horizontal and vertical 1-morphisms, and square 2-morphsims, which compose in both directions.  Then there are several special degenerate cases, in the same way that categories have as degenerate cases (a) sets, seen as categories with only identity morphisms, and (b) monoids, seen as one-object categories.  Double categories have ordinary categories (and hence monoids and sets) as degenerate cases.  Other degenerate cases are 2-categories (horizontal and vertical morphisms are the same thing), and therefore their own special cases, monoidal categories and symmetric monoids.  There is also the special degenerate case of a double monoid (and the extra-special case of a double group).  (The slides have nice pictures showing how they’re all degenerate cases).  Dany then talked about some structure of double group(oids) – and gave a list of properties for double groupoids, (such as being “slim” – having at most one 2-cell per boundary configuration – as well as two others) which ensure that they’re equivalent to the semidirect product of an abelian group with the “bicrossed product”  $H \bowtie K$ of two groups $H$ and $K$ (each of which has to act on the other for this to make sense).  He gave the example of the Poincare double group, which breaks down as a triple bicrossed product by the Iwasawa decomposition:

$Poinc = (SO(3) \bowtie (SO(1; 1) \bowtie N)) \ltimes \mathbb{R}_4$

($N$ is certain group of matrices).  So there’s a unique double group which corresponds to it – it has squares labelled by $\mathbb{R}_4$, and the horizontial and vertical morphisms by elements of $SO(3)$ and $N$ respectively.  Dany finished by explaining that there are higher-dimensional analogs of all this – $n$-tuple categories can be defined recursively by internalization (“internal categories in $(n-1)$-tuple-Cat”).  There are somewhat more sophisticated versions of the same kind of structure, and finally leading up to a special class of $n$-tuple groups.  The analogous theorem says that a special class of them is just the same as the semidirect product of an abelian group with an $n$-fold iterated bicrossed product of groups.

Also in this category, Alex Hoffnung talked about deformation of formal group laws (based on this paper with various collaborators).  FGL’s are are structures with an algebraic operation which satisfies axioms similar to a group, but which can be expressed in terms of power series.  (So, in particular they have an underlying ring, for this to make sense).  In particular, the talk was about formal group algebras – essentially, parametrized deformations of group algebras – and in particular for Hecke Algebras.  Unfortunately, my notes on this talk are mangled, so I’ll just refer to the paper.

### Physics

I’m using the subject-header “physics” to refer to those talks which are most directly inspired by physical ideas, though in fact the talks themselves were mathematical in nature.

Fei Han gave a series of overview talks intorducing “Equivariant Cohomology via Gauged Supersymmetric Field Theory”, explaining the Stolz-Teichner program.  There is more, using tools from differential geometry and cohomology to dig into these theories, but for now a summary will do.  Essentially, the point is that one can look at “fields” as sections of various bundles on manifolds, and these fields are related to cohomology theories.  For instance, the usual cohomology of a space $X$ is a quotient of the space of closed forms (so the $k^{th}$ cohomology, $H^{k}(X) = \Omega^{k}$, is a quotient of the space of closed $k$-forms – the quotient being that forms differing by a coboundary are considered the same).  There’s a similar construction for the $K$-theory $K(X)$, which can be modelled as a quotient of the space of vector bundles over $X$.  Fei Han mentioned topological modular forms, modelled by a quotient of the space of “Fredholm bundles” – bundles of Banach spaces with a Fredholm operator around.

The first two of these examples are known to be related to certain supersymmetric topological quantum field theories.  Now, a TFT is a functor into some kind of vector spaces from a category of $(n-1)$-dimensional manifolds and $n$-dimensional cobordisms

$Z : d-Bord \rightarrow Vect$

Intuitively, it gives a vector space of possible fields on the given space and a linear map on a given spacetime.  A supersymmetric field theory is likewise a functor, but one changes the category of “spacetimes” to have both bosonic and fermionic dimension.  A normal smooth manifold is a ringed space $(M,\mathcal{O})$, since it comes equipped with a sheaf of rings (each open set has an associated ring of smooth functions, and these glue together nicely).  Supersymmetric theories work with manifolds which change this sheaf – so a $d|\delta$-dimensional space has the sheaf of rings where one introduces some new antisymmetric coordinate functions $\theta_i$, the “fermionic dimensions”:

$\mathcal{O}(U) = C^{\infty}(U) \otimes \bigwedge^{\ast}[\theta_1,\dots,\theta_{\delta}]$

Then a supersymmetric TFT is a functor:

$E : (d|\delta)-Bord \rightarrow STV$

(where $STV$ is the category of supersymmetric topological vector spaces – defined similarly).  The connection to cohomology theories is that the classes of such field theories, up to a notion of equivalence called “concordance”, are classified by various cohomology theories.  Ordinary cohomology corresponds then to $0|1$-dimensional extended TFT (that is, with 0 bosonic and 1 fermionic dimension), and $K$-theory to a $1|1$-dimensional extended TFT.  The Stoltz-Teichner Conjecture is that the third example (topological modular forms) is related in the same way to a $2_1$-dimensional extended TFT – so these are the start of a series of cohomology theories related to various-dimension TFT’s.

Last but not least, Chris Rogers spoke about his ideas on “Higher Geometric Quantization”, on which he’s written a number of papers.  This is intended as a sort of categorification of the usual ways of quantizing symplectic manifolds.  I am still trying to catch up on some of the geometry This is rooted in some ideas that have been discussed by Brylinski, for example.  Roughly, the message here is that “categorification” of a space can be thought of as a way of acting on the loop space of a space.  The point is that, if points in a space are objects and paths are morphisms, then a loop space $L(X)$ shifts things by one categorical level: its points are loops in $X$, and its paths are therefore certain 2-morphisms of $X$.  In particular, there is a parallel to the fact that a bundle with connection on a loop space can be thought of as a gerbe on the base space.  Intuitively, one can “parallel transport” things along a path in the loop space, which is a surface given by a path of loops in the original space.  The local description of this situation says that a 1-form (which can give transport along a curve, by integration) on the loop space is associated with a 2-form (giving transport along a surface) on the original space.

Then the idea is that geometric quantization of loop spaces is a sort of higher version of quantization of the original space. This “higher” version is associated with a form of higher degree than the symplectic (2-)form used in geometric quantization of $X$.   The general notion of n-plectic geometry, where the usual symplectic geometry is the case $n=1$, involves a $(n+1)$-form analogous to the usual symplectic form.  Now, there’s a lot more to say here than I properly understand, much less can summarize in a couple of paragraphs.  But the main theorem of the talk gives a relation between n-plectic manifolds (i.e. ones endowed with the right kind of form) and Lie n-algebras built from the complex of forms on the manifold.  An important example (a theorem of Chris’ and John Baez) is that one has a natural example of a 2-plectic manifold in any compact simple Lie group $G$ together with a 3-form naturally constructed from its Maurer-Cartan form.

At any rate, this workshop had a great proportion of interesting talks, and overall, including the chance to see a little more of China, was a great experience!

This blog has been on hiatus for a while, as I’ve been doing various other things, including spending some time in Hamburg getting set up for the move there. Another of these things has been working with Jamie Vicary on our project on the groupoidified Quantum Harmonic Oscillator (QHO for short). We’ve now put the first of two papers on the arXiv – this one is a relatively nonrigorous look at how this relates to categorification of the Heisenberg Algebra. Since John Baez is a high-speed blogging machine, he’s already beaten me to an overview of what the paper says, and there’s been some interesting discussion already. So I’ll try to say some different things about what it means, and let you take a look over there, or read the paper, for details.

I’ve given some talks about this project, but as we’ve been writing it up, it’s expanded considerably, including a lot of category-theoretic details which are going to be in the second paper in this series. But the basic point of this current paper is essentially visual and, in my opinion, fairly simple. The groupoidification of the QHO has a nice visual description, since it is all about the combinatorics of finite sets. This was described originally by Baez and Dolan, and in more detail in my very first paper. The other visual part here is the relation to Khovanov’s categorification of the Heisenberg algebra using a graphical calculus. (I wrote about this back when I first became aware of it.)

As a Representation

The scenario here actually has some common features with my last post. First, we have a monoidal category with duals, let’s say $C$ presented in terms of some generators and relations. Then, we find some concrete model of this abstractly-presented monoidal category with duals in a specific setting, namely $Span(Gpd)$.

Calling this “concrete” just refers to the fact that the objects in $Span(Gpd)$ have some particular structure in terms of underlying sets and so on. By a “model” I just mean a functor $C \rightarrow Span(Gpd)$ (“model” and “representation” mean essentially the same thing in this context). In fact, for this to make sense, I think of $C$ as a 2-category with one object. Then a model is just some particular choices: a groupoid to represent the unique object, spans of groupoids to represent the generating morphisms, spans of spans to represent the generating 2-morphisms, all chosen so that the defining relations hold.

In my previous post, $C$ was a category of cobordisms, but in this case, it’s essentially Khovanov’s monoidal category $H'$ whose objects are (oriented) dots and whose morphisms are certain classes of diagrams. The nice fact about the particular model we get is that the reasons these relations hold are easy to see in terms of a the combinatorics of sets. This is why our title describes what we got as “a combinatorial representation” Khovanov’s category $H'$ of diagrams, for which the ring of isomorphism classes of objects is the integral form of the algebra. This uses that $Span(Gpd)$ is not just a monoidal category: it can be a monoidal 2-category. What’s more, the monoidal category $H'$ “is” also a 2-category – with one object. The objects of $H'$ are really the morphisms of this 2-category.

So $H'$ is in some sense a universal theory (because it’s defined freely in terms of generators and relations) of what a categorification of the Heisenberg algebra must look like. Baez-Dolan groupoidification of the QHO then turns out to be a representation or model of it. In fact, the model is faithful, so that we can even say that it provides a combinatorial interpretation of that category.

The Combinatorial Model

Between the links above, you can find a good summary of the situation, so I’ll be a bit cursory. The model is described in terms of structures on finite sets. This is why our title calls this a “combinatorial representation” of Khovanov’s categorification.

This means that the one object of $H$ (as a 2-category) is taken to the groupoid $FinSet_0$ of finite sets and bijections (which we just called $S$ in the paper for brevity). This is the “Fock space” object. For simplicity, we can take an equivalent groupoid, which has just one $n$-element set for each $n$.

Now, a groupoid represents a system, whose possible configurations are the objects and whose symmetries are the morphisms. In this case, the possible configurations are the different numbers of “quanta”, and the symmetries (all set-bijections) show that all the quanta are interchangeable. I imagine a box containing some number of ping-pong balls.

A span of groupoids represents a process. It has a groupoid whose objects are histories (and morphisms are symmetries of histories). This groupoid has a pair of maps: to the system the process starts in, and to the system it ends in. In our model, the most important processes (which generate everything else) are the creation and annihilation operators, $a^{\dagger}$ and $a$ – and their categorified equivalents, $A$ and $A^{\dagger}$. The spans that represent them are very simple: they are processes which put a new ball into the box, or take one out, respectively. (Algebraically, they’re just a way to organize all the inclusions of symmetric groups $S_n \subset S_{n+1}$.)

The “canonical commutation relation“, which we write without subtraction thus:

$A A^{\dagger} = A^{\dagger} A + 1$

is already understood in the Baez-Dolan story: it says that there is one more way to remove a ball from a box after putting a new one into it (one more history for the process $A A^{\dagger}$) than to remove a ball and then add a new one (histories for $a^{\dagger} a$). This is fairly obvious: in the first instance, you have one more to choose from when removing the ball.

But the original Baez-Dolan story has no interesting 2-morphisms (the actual diagrams which are the 1-morphisms in $H$), whereas these are absolutely the whole point of a categorification in the sense Khovanov gets one, since the 1-morphisms of $H'$ determine what the isomorphism classes of objects even are.

So this means that we need to figure out what the 2-morphisms in $Span(Gpd)$ need to be – first in general, and second in our particular representation of $H$.

In general, a 2-morphism in $Span(Gpd)$ is a span of span-maps. You’ll find other people who take it to be a span-map. This would be a functor between the groupoids of histories: roughly, a map which assigns a history in the source span to a history in the target span (and likewise for symmetries), in a way that respects how they’re histories. But we don’t want just a map: we want a process which has histories of its own. We want to describe a “movie of processes” which change one process into another. These can have many histories of their own.

In fact, they’re not too complicated. Here’s one of Khovanov’s relation in $H'$ which forms part of how the commutation relation is expressed (shuffled to get rid of negatives, which we constantly need to do in the combinatorial model since we have no negative sets):

We read an upward arrow as “add a ball to the box”, and a downward arrow as “remove a ball”, and read right-to-left.  Both processes begin and end with“add then remove”. The right-hand side just leaves this process alone: it’s the identity.

The left-hand side shows a process-movie whose histories have two different cases. Suppose we begin with a history for which we add $x$ and then remove $y$. The first case is that $x = y$: we remove the same ball we put in. This amounts to doing nothing, so the first part of the movie eliminates all the adding and removing. The second part puts the add-remove pair back in.

The second case ensures that $x \neq y$, since it takes the initial history to the history (of a different process!) in which we remove $y$ and then add $x$ (impossible if $y = x$, since we can’t remove this ball before adding it). This in turn is taken to the history (of the original process!) where we add $x$ and then remove $y$; so this relates every history to itself, except for the case that $x = y$. Overall the sum of these relations give the identity on histories, which is the right hand side.

This picture includes several of the new 2-morphisms that we need to add to the Baez-Dolan picture: swapping the order of two generators, and adding or removing a pair of add/remove operations. Finding spans of spans which accomplish this (and showing they satisfy the right relations) is all that’s needed to finish up the combinatorial model.  So, for instance, the span of spans which adds a “remove-then-add” pair is this one:

If this isn’t clear, well, it’s explained in more detail in the paper.  (Do notice, though, that this is a diagram in groupoids: we need to specify that there are identity 2-cells in the span, rather than some other 2-cells.)

So this is basically how the combinatorial model works.

But in fact this description is (as often happens) chronologically backwards: what actually happened was that we had worked out what the 2-morphisms should be for different reasons. While trying to to understand what kind of structure this produced, we realized (thanks to Marco Mackaay) that the result was related to $H$, which in turn shed more light on the 2-morphisms we’d found.

So far so good. But what makes it possible to represent the kind of monoidal category we’re talking about in this setting is adjointness. This is another way of saying what I meant up at the top by saying we start with a monoidal category with duals.  This means morphisms each have a partner – a dual, or adjoint – going in the opposite direction.  The representations of the raising and lowering operators of the Heisenberg algebra on the Hilbert space for the QHO are linear adjoints. Their categorifications also need to be adjoints in the sense of adjoint 1-morphisms in a 2-category.

This is an abstraction of what it means for two functors $F$ and $G$ to be adjoint. In particular, it means there have to be certain 2-cells such as the unit $\eta : Id \Rightarrow G \circ F$ and counit $\epsilon : F \circ G \Rightarrow Id$ satisfying some nice relations. In fact, this only makes $F$ a left adjoint and $G$ a right adjoint – in this situation, we also have another pair which makes $F$ a right adjoint and $G$ a left one. That is, they should be “ambidextrous adjoints”, or “ambiadjoints” for short. This is crucial if they’re going to represent any graphical calculus of the kind that’s involved here (see the first part of this paper by Aaron Lauda, for instance).

So one of the theorems in the longer paper will show concretely that any 1-morphism in $Span(Gpd)$ has an ambiadjoint – which happens to look like the same span, but thought of as going in the reverse direction. This is somewhat like how the adjoint of a real linear map, expressed as a matrix relative to well-chosen bases, is just the transpose of the same matrix. In particular, $A$ and $A^{\dagger}$ are adjoints in just this way. The span-of-span-maps I showed above is exactly the unit for one side of this ambi-adjunction – but it is just a special case of something that will work for any span and its adjoint.

Finally, there’s something a little funny here. Since the morphisms of $Span(Gpd)$ aren’t functors or maps, this combinatorial model is not exactly what people often mean by a “categorified representation”. That would be an action on a category in terms of functors and natural transformations. We do talk about how to get one of these on a 2-vector space out of our groupoidal representation toward the end.

In particular, this amounts to a functor into $2Vect$ – the objects of $2Vect$ being categories of a particular kind, and the morphisms being functors that preserve all the structure of those categories. As it turns out, the thing about this setting which is good for this purpose is that all those functors have ambiadjoints. The “2-linearization” that takes $Span(Gpd)$ into $2Vect$ is a 2-functor, and this means that all the 2-cells and equations that make two morphisms ambiadjoints carry over. In $2Vect$, it’s very easy for this to happen, since all those ambiadjoints are already present. So getting representations of categorified algebras that are made using these monoidal categories of diagrams on 2-vector spaces is fairly natural – and it agrees with the usual intuition about what “representation” means.

Anything I start to say about this is in danger of ballooning, but since we’re already some 40 pages into the second paper, I’ll save the elaboration for that…

So I’ve been travelling a lot in the last month, spending more than half of it outside Portugal. I was in Ottawa, Canada for a Fields Institute workshop, “Categorical Methods in Representation Theory“. Then a little later I was in Erlangen, Germany for one called “Categorical and Representation-Theoretic Methods in Quantum Geometry and CFT“. Despite the similar-sounding titles, these were on fairly different themes, though Marco Mackaay was at both, talking about categorifying the $q$-Schur algebra by diagrams.  I’ll describe the meetings, but for now I’ll start with the first.  Next post will be a summary of the second.

The Ottawa meeting was organized by Alistair Savage, and Alex Hoffnung (like me, a former student of John Baez). Alistair gave a talk here at IST over the summer about a $q$-deformation of Khovanov’s categorification of the Heisenberg Algebra I discussed in an earlier entry. A lot of the discussion at the workshop was based on the Khovanov-Lauda program, which began with categorifying quantum version of the classical Lie groups, and is now making lots of progress in the categorification of algebras, representation theory, and so on.

The point of this program is to describe “categorifications” of particular algebras. This means finding monoidal categories with the property that when you take the Grothendieck ring (the ring of isomorphism classes, with a multiplication given by the monoidal structure), you get back the integral form of some algebra. (And then recover the original by taking the tensor over $\mathbb{Z}$ with $\mathbb{C}$). The key thing is how to represent the algebra by generators and relations. Since free monoidal categories with various sorts of structures can be presented as categories of string diagrams, it shouldn’t be surprising that the categories used tend to have objects that are sequences (i.e. monoidal products) of dots with various sorts of labelling data, and morphisms which are string diagrams that carry those labels on strands (actually, usually they’re linear combinations of such diagrams, so everything is enriched in vector spaces). Then one imposes relations on the “free” data given this way, by saying that the diagrams are considered the same morphism if they agree up to some local moves. The whole problem then is to find the right generators (labelling data) and relations (local moves). The result will be a categorification of a given presentation of the algebra you want.

So for instance, I was interested in Sabin Cautis and Anthony Licata‘s talks connected with this paper, “Heisenberg Categorification And Hilbert Schemes”. This is connected with a generalization of Khovanov’s categorification linked above, to include a variety of other algebras which are given a similar name. The point is that there’s such a “Heisenberg algebra” associated to different subgroups $\Gamma \subset SL(2,\mathbf{k})$, which in turn are classified by Dynkin diagrams. The vertices of these Dynkin diagrams correspond to some generators of the Heisenberg algebra, and one can modify Khovanov’s categorification by having strands in the diagram calculus be labelled by these vertices. Rules for local moves involving strands with different labels will be governed by the edges of the Dynkin diagram. Their paper goes on to describe how to represent these categorifications on certain categories of Hilbert schemes.

Along the same lines, Aaron Lauda gave a talk on the categorification of the NilHecke algebra. This is defined as a subalgebra of endomorphisms of $P_a = \mathbb{Z}[x_1,\dots,x_a]$, generated by multiplications (by the $x_i$) and the divided difference operators $\partial_i$. There are different from the usual derivative operators: in place of the differences between values of a single variable, they measure how a function behaves under the operation $s_i$ which switches variables $x_i$ and $x_{i+1}$ (that is, the reflection in the hyperplane where $x_i = x_{i+1}$). The point is that just like differentiation, this operator – together with multiplication – generates an algebra in $End(\mathbb{Z}[x_1,\dots,x_a]$. Aaron described how to categorify this presentation of the NilHecke algebra with a string-diagram calculus.

So anyway, there were a number of talks about the explosion of work within this general program – for instance, Marco Mackaay’s which I mentioned, as well as that of Pedro Vaz about the same project. One aspect of this program is that the relatively free “string diagram categories” are sometimes replaced with categories where the objects are bimodules and morphisms are bimodule homomorphisms. Making the relationship precise is then a matter of proving these satisfy exactly the relations on a “free” category which one wants, but sometimes they’re a good setting to prove one has a nice categorification. Thus, Ben Elias and Geordie Williamson gave two parts of one talk about “Soergel Bimodules and Kazhdan-Lusztig Theory” (see a blog post by Ben Webster which gives a brief intro to this notion, including pointing out that Soergel bimodules give a categorification of the Hecke algebra).

One of the reasons for doing this sort of thing is that one gets invariants for manifolds from algebras – in particular, things like the Jones polynomial, which is related to the Temperley-Lieb algebra. A categorification of it is Khovanov homology (which gives, for a manifold, a complex, with the property that the graded Euler characteristic of the complex is the Jones polynomial). The point here is that categorifying the algebra lets you raise the dimension of the kind of manifold your invariants are defined on.

So, for instance, Scott Morrison described “Invariants of 4-Manifolds from Khonanov Homology“.  This was based on a generalization of the relationship between TQFT’s and planar algebras.  The point is, planar algebras are described by the composition of diagrams of the following form: a big circle, containing some number of small circles.  The boundaries of each circle are labelled by some number of marked points, and the space between carries curves which connect these marked points in some way.  One composes these diagrams by gluing big circles into smaller circles (there’s some further discussion here including a picture, and much more in this book here).  Scott Morrison described these diagrams as “spaghetti and meatball” diagrams.  Planar algebras show up by associating a vector spaces to “the” circle with $n$ marked points, and linear maps to each way (up to isotopy) of filling in edges between such circles.  One can think of the circles and marked-disks as a marked-cobordism category, and so a functorial way of making these assignments is something like a TQFT.  It also gives lots of vector spaces and lots of linear maps that fit together in a particular way described by this category of marked cobordisms, which is what a “planar algebra” actually consists of.  Clearly, these planar algebras can be used to get some manifold invariants – namely the “TQFT” that corresponds to them.

Scott Morrison’s talk described how to get invariants of 4-dimensional manifolds in a similar way by boosting (almost) everything in this story by 2 dimensions.  You start with a 4-ball, whose boundary is a 3-sphere, and excise some number of 4-balls (with 3-sphere boundaries) from the interior.  Then let these 3D boundaries be “marked” with 1-D embedded links (think “knots” if you like).  These 3-spheres with embedded links are the objects in a category.  The morphisms are 4-balls which connect them, containing 2D knotted surfaces which happen to intersect the boundaries exactly at their embedded links.  By analogy with the image of “spaghetti and meatballs”, where the spaghetti is a collection of 1D marked curves, Morrison calls these 4-manifolds with embedded 2D surfaces “lasagna diagrams” (which generalizes to the less evocative case of “$(n,k)$ pasta diagrams”, where we’ve just mentioned the $(2,1)$ and $(4,2)$ cases, with $k$-dimensional “pasta” embedded in $n$-dimensional balls).  Then the point is that one can compose these pasta diagrams by gluing the 4-balls along these marked boundaries.  One then gets manifold invariants from these sorts of diagrams by using Khovanov homology, which assigns to

Ben Webster talked about categorification of Lie algebra representations, in a talk called “Categorification, Lie Algebras and Topology“. This is also part of categorifying manifold invariants, since the Reshitikhin-Turaev Invariants are based on some monoidal category, which in this case is the category of representations of some algebra.  Categorifying this to a 2-category gives higher-dimensional equivalents of the RT invariants.  The idea (which you can check out in those slides) is that this comes down to describing the analog of the “highest-weight” representations for some Lie algebra you’ve already categorified.

The Lie theory point here, you might remember, is that representations of Lie algebras $\mathfrak{g}$ can be analyzed by decomposing them into “weight spaces” $V_{\lambda}$, associated to weights $\lambda : \mathfrak{g} \rightarrow \mathbf{k}$ (where $\mathbf{k}$ is the base field, which we can generally assume is $\mathbb{C}$).  Weights turn Lie algebra elements into scalars, then.  So weight spaces generalize eigenspaces, in that acting by any element $g \in \mathfrak{g}$ on a “weight vector” $v \in V_{\lambda}$ amounts to multiplying by $\lambda{g}$.  (So that $v$ is an eigenvector for each $g$, but the eigenvalue depends on $g$, and is given by the weight.)  A weight can be the “highest” with respect to a natural order that can be put on weights ($\lambda \geq \mu$ if the difference is a nonnegative combination of simple weights).  Then a “highest weight representation” is one which is generated under the action of $\mathfrak{g}$ by a single weight vector $v$, the “highest weight vector”.

The point of the categorification is to describe the representation in the same terms.  First, we introduce a special strand (which Ben Webster draws as a red strand) which represents the highest weight vector.  Then we say that the category that stands in for the highest weight representation is just what we get by starting with this red strand, and putting all the various string diagrams of the categorification of $\mathfrak{g}$ next to it.  One can then go on to talk about tensor products of these representations, where objects are found by amalgamating several such diagrams (with several red strands) together.  And so on.  These categorified representations are then supposed to be usable to give higher-dimensional manifold invariants.

Now, the flip side of higher categories that reproduce ordinary representation theory would be the representation theory of higher categories in their natural habitat, so to speak. Presumably there should be a fairly uniform picture where categorifications of normal representation theory will be special cases of this. Vlodymyr Mazorchuk gave an interesting talk called 2-representations of finitary 2-categories.  He gave an example of one of the 2-categories that shows up a lot in these Khovanov-Lauda categorifications, the 2-category of Soergel Bimodules mentioned above.  This has one object, which we can think of as a category of modules over the algebra $\mathbb{C}[x_1, \dots, x_n]/I$ (where I  is some ideal of homogeneous symmetric polynomials).  The morphisms are endofunctors of this category, which all amount to tensoring with certain bimodules – the irreducible ones being the Soergel bimodules.  The point of the talk was to explain the representations of 2-categories $\mathcal{C}$ – that is, 2-functors from $\mathcal{C}$ into some “classical” 2-category.  Examples would be 2-categories like “2-vector spaces”, or variants on it.  The examples he gave: (1) [small fully additive $\mathbf{k}$-linear categories], (2) the full subcategory of it with finitely many indecomposible elements, (3) [categories equivalent to module categories of finite dimensional associative $\mathbf{k}$-algebras].  All of these have some claim to be a 2-categorical analog of [vector spaces].  In general, Mazorchuk allowed representations of “FIAT” categories: Finitary (Two-)categories with Involutions and Adjunctions.

Part of the process involved getting a “multisemigroup” from such categories: a set $S$ with an operation which takes pairs of elements, and returns a subset of $S$, satisfying some natural associativity condition.  (Semigroups are the case where the subset contains just one element – groups are the case where furthermore the operation is invertible).  The idea is that FIAT categories have some set of generators – indecomposable 1-morphisms – and that the multisemigroup describes which indecomposables show up in a composite.  (If we think of the 2-category as a monoidal category, this is like talking about a decomposition of a tensor product of objects).  So, for instance, for the 2-category that comes from the monoidal category of $\mathfrak{sl}(2)$ modules, we get the semigroup of nonnegative integers.  For the Soergel bimodule 2-category, we get the symmetric group.  This sort of thing helps characterize when two objects are equivalent, and in turn helps describe 2-representations up to some equivalence.  (You can find much more detail behind the link above.)

On the more classical representation-theoretic side of things, Joel Kamnitzer gave a talk called “Spiders and Buildings”, which was concerned with some geometric and combinatorial constructions in representation theory.  These involved certain trivalent planar graphs, called “webs”, whose edges carry labels between 1 and $(n-1)$.  They’re embedded in a disk, and the outgoing edges, with labels $(k_1, \dots, k_m)$ determine a representation space for a group $G$, say $G = SL_n$, namely the tensor product of a bunch of wedge products, $\otimes_j \wedge^{k_j} \mathbb{C}^n$, where $SL_n$ acts on $\mathbb{C}^n$ as usual.  Then a web determines an invariant vector in this space.  This comes about by having invariant vectors for each vertex (the basic case where $m =3$), and tensoring them together.  But the point is to interpret this construction geometrically.  This was a bit outside my grasp, since it involves the Langlands program and the geometric Satake correspondence, neither of which I know much of anything about, but which give geometric/topological ways of constructing representation categories.  One thing I did pick up is that it uses the “Langlands dual group” $\check{G}$ of $G$ to get a certain metric space called $Gn_{\check{G}}$.  Then there’s a correspondence between the category of representations of $G$ and the category of (perverse, constructible) sheaves on this space.  This correspondence can be used to describe the vectors that come out of these webs.

Jim Dolan gave a couple of talks while I was there, which actually fit together as two parts of a bigger picture – one was during the workshop itself, and one at the logic seminar on the following Monday. It helped a lot to see both in order to appreciate the overall point, so I’ll mix them a bit indiscriminately. The first was called “Dimensional Analysis is Algebraic Geometry”, and the second “Toposes of Quasicoherent Sheaves on Toric Varieties”. For the purposes of the logic seminar, he gave the slogan of the second talk as “Algebraic Geometry is a branch of Categorical Logic”. Jim’s basic idea was inspired by Bill Lawvere’s concept of a “theory”, which is supposed to extend both “algebraic theories” (such as the “theory of groups”) and theories in the sense of physics.  Any given theory is some structured category, and “models” of the theory are functors into some other category to represent it – it thus has a functor category called its “moduli stack of models”.  A physical theory (essentially, models which depict some contents of the universe) has some parameters.  The “theory of elastic scattering”, for instance, has the masses, and initial and final momenta, of two objects which collide and “scatter” off each other.  The moduli space for this theory amounts to assignments of values to these parameters, which must satisfy some algebraic equations – conservation of energy and momentum (for example, $\sum_i m_i v_i^{in} = \sum_i m_i v_i^{out}$, where $i \in 1, 2$).  So the moduli space is some projective algebraic variety.  Jim explained how “dimensional analysis” in physics is the study of line bundles over such varieties (“dimensions” are just such line bundles, since a “dimension” is a 1-dimensional sort of thing, and “quantities” in those dimensions are sections of the line bundles).  Then there’s a category of such bundles, which are organized into a special sort of symmetric monoidal category – in fact, it’s contrained so much it’s just a graded commutative algebra.

In his second talk, he generalized this to talk about categories of sheaves on some varieties – and, since he was talking in the categorical logic seminar, he proposed a point of view for looking at algebraic geometry in the context of logic.  This view could be summarized as: Every (generalized) space studied by algebraic geometry “is” the moduli space of models for some theory in some doctrine.  The term “doctrine” is Bill Lawvere’s, and specifies what kind of structured category the theory and the target of its models are supposed to be (and of course what kind of functors are allowed as models).  Thus, for instance, toposes (as generalized spaces) are supposed to be thought of as “geometric theories”.  He explained that his “dimensional analysis doctrine” is a special case of this.  As usual when talking to Jim, I came away with the sense that there’s a very large program of ideas lurking behind everything he said, of which only the tip of the iceberg actually made it into the talks.

Next post, when I have time, will talk about the meeting at Erlangen…

Now for a more sketchy bunch of summaries of some talks presented at the HGTQGR workshop.  I’ll organize this into a few themes which appeared repeatedly and which roughly line up with the topics in the title: in this post, variations on TQFT, plus 2-group and higher forms of gauge theory; in the next post, gerbes and cohomology, plus talks on discrete models of quantum gravity and suchlike physics.

## TQFT and Variations

I start here for no better reason than the personal one that it lets me put my talk first, so I’m on familiar ground to start with, for which reason also I’ll probably give more details here than later on.  So: a TQFT is a linear representation of the category of cobordisms – that is, a (symmetric monoidal) functor $nCob \rightarrow Vect$, in the notation I mentioned in the first school post.  An Extended TQFT is a higher functor $nCob_k \rightarrow k-Vect$, representing a category of cobordisms with corners into a higher category of k-Vector spaces (for some definition of same).  The essential point of my talk is that there’s a universal construction that can be used to build one of these at $k=2$, which relies on some way of representing $nCob_2$ into $Span(Gpd)$, whose objects are groupoids, and whose morphisms in $Hom(A,B)$ are pairs of groupoid homomorphisms $A \leftarrow X \rightarrow B$.  The 2-morphisms have an analogous structure.  The point is that there’s a 2-functor $\Lambda : Span(Gpd) \rightarrow 2Vect$ which is takes representations of groupoids, at the level of objects; for morphisms, there is a “pull-push” operation that just uses the restricted and induced representation functors to move a representation across a span; the non-trivial (but still universal) bit is the 2-morphism map, which uses the fact that the restriction and induction functors are bi-ajdoint, so there are units and counits to use.  A construction using gauge theory gives groupoids of connections and gauge transformations for each manifold or cobordism.  This recovers a form of the Dijkgraaf-Witten model.  In principle, though, any way of getting a groupoid (really, a stack) associated to a space functorially will give an ETQFT this way.  I finished up by suggesting what would need to be done to extend this up to higher codimension.  To go to codimension 3, one would assign an object (codimension-3 manifold) a 3-vector space which is a representation 2-category of 2-groupoids of connections valued in 2-groups, and so on.  There are some theorems about representations of n-groupoids which would need to be proved to make this work.

The fact that different constructions can give groupoids for spaces was used by the next speaker, Thomas Nicklaus, whose talk described another construction that uses the $\Lambda$ I mentioned above.  This one produces “Equivariant Dijkgraaf-Witten Theory”.  The point is that one gets groupoids for spaces in a new way.  Before, we had, for a space $M$ a groupoid $\mathcal{A}_G(M)$ whose objects are $G$-connections (or, put another way, bundles-with-connection) and whose morphisms are gauge transformations.  Now we suppose that there’s some group $J$ which acts weakly (i.e. an action defined up to isomorphism) on $\mathcal{A}_G(M)$.  We think of this as describing “twisted bundles” over $M$.  This is described by a quotient stack $\mathcal{A}_G // J$ (which, as a groupoid, gets some extra isomorphisms showing where two objects are related by the $J$-action).  So this gives a new map $nCob \rightarrow Span(Gpd)$, and applying $\Lambda$ gives a TQFT.  The generating objects for the resulting 2-vector space are “twisted sectors” of the equivariant DW model.  There was some more to the talk, including a description of how the DW model can be further mutated using a cocycle in the group cohomology of $G$, but I’ll let you look at the slides for that.

Next up was Jamie Vicary, who was talking about “(1,2,3)-TQFT”, which is another term for what I called “Extended” TQFT above, but specifying that the objects are 1-manifolds, the morphisms 2-manifolds, and the 2-morphisms are 3-manifolds.  He was talking about a theorem that identifies oriented TQFT’s of this sort with “anomaly-free modular tensor categories” – which is widely believed, but in fact harder than commonly thought.  It’s easy enough that such a TQFT $Z$ corresponds to a MTC – it’s the category $Z(S^1)$ assigned to the circle.  What’s harder is showing that the TQFT’s are equivalent functors iff the categories are equivalent.  This boils down, historically, to the difficulty of showing the category is rigid.  Jamie was talking about a project with Bruce Bartlett and Chris Schommer-Pries, whose presentation of the cobordism category (described in the school post) was the basis of their proof.

Part of it amounts to giving a description of the TQFT in terms of certain string diagrams.  Jamie kindly credited me with describing this point of view to him: that the codimension-2 manifolds in a TQFT can be thought of as “boundaries in space” – codimension-1 manifolds are either time-evolving boundaries, or else slices of space in which the boundaries live; top-dimension cobordisms are then time-evolving slices of space-with-boundary.  (This should be only a heuristic way of thinking – certainly a generic TQFT has no literal notion of “time-evolution”, though in that (2+1) quantum gravity can be seen as a TQFT, there’s at least one case where this picture could be taken literally.)  Then part of their proof involves showing that the cobordisms can be characterized by taking vector spaces on the source and target manifolds spanned by the generating objects, and finding the functors assigned to cobordisms in terms of sums over all “string diagrams” (particle worldlines, if you like) bounded by the evolving boundaries.  Jamie described this as a “topological path integral”.  Then one has to describe the string diagram calculus – ridigidy follows from the “yanking” rule, for instance, and this follows from Morse theory as in Chris’ presentation of the cobordism category.

There was a little more discussion about what the various properties (proved in a similar way) imply.  One is “cloaking” – the fact that a 2-morphism which “creates a handle” is invisible to the string diagrams in the sense that it introduces a sum over all diagrams with a string “looped” around the new handle, but this sum gives a result that’s equal to the original map (in any “pivotal” tensor category, as here).

Chronologically before all these, one of the first talks on such a topic was by Rafael Diaz, on Homological Quantum Field Theory, or HLQFT for short, which is a rather different sort of construction.  Remember that Homotopy QFT, as described in my summary of Tim Porter’s school sessions, is about linear representations of what I’ll for now call $Cob(d,B)$, whose morphisms are $d$-dimensional cobordisms equipped with maps into a space $B$ up to homotopy.  HLQFT instead considers cobordisms equipped with maps taken up to homology.

Specifically, there’s some space $M$, say a manifold, with some distinguished submanifolds (possibly boundary components; possibly just embedded submanifolds; possibly even all of $M$ for a degenerate case).  Then we define $Cob_d^M$ to have objects which are $(d-1)$-manifolds equipped with maps into $M$ which land on the distinguished submanifolds (to make composition work nicely, we in fact assume they map to a single point).  Morphisms in $Cob_d^M$ are trickier, and look like $(N,\alpha, \xi)$: a cobordism $N$ in this category is likewise equipped with a map $\alpha$ from its boundary into $M$ which recovers the maps on its objects.  That $\xi$ is a homology class of maps from $N$ to $M$, which agrees with $\alpha$.  This forms a monoidal category as with standard cobordisms.  Then HLQFT is about representations of this category.  One simple case Rafael described is the dimension-1 case, where objects are (ordered sets of) points equipped with maps that pick out chosen submanifolds of $M$, and morphisms are just braids equipped with homology classes of “paths” joining up the source and target submanifolds.  Then a representation might, e.g., describe how to evolve a homology class on the starting manifold to one on the target by transporting along such a path-up-to-homology.  In higher dimensions, the evolution is naturally more complicated.

A slightly looser fit to this section is the talk by Thomas Krajewski, “Quasi-Quantum Groups from Strings” (see this) – he was talking about how certain algebraic structures arise from “string worldsheets”, which are another way to describe cobordisms.  This does somewhat resemble the way an algebraic structure (Frobenius algebra) is related to a 2D TQFT, but here the string worldsheets are interacting with 3-form field, $H$ (the curvature of that 2-form field $B$ of string theory) and things needn’t be topological, so the result is somewhat different.

Part of the point is that quantizing such a thing gives a higher version of what happens for quantizing a moving particle in a gauge field.  In the particle case, one comes up with a line bundle (of which sections form the Hilbert space) and in the string case one comes up with a gerbe; for the particle, this involves associated 2-cocycle, and for the string a 3-cocycle; for the particle, one ends up producing a twisted group algebra, and for the string, this is where one gets a “quasi-quantum group”.  The algebraic structures, as in the TQFT situation, come from, for instance, the “pants” cobordism which gives a multiplication and a comultiplication (by giving maps $H \otimes H \rightarrow H$ or the reverse, where $H$ is the object assigned to a circle).

There is some machinery along the way which I won’t describe in detail, except that it involves a tricomplex of forms – the gradings being form degree, the degree of a cocycle for group cohomology, and the number of overlaps.  As observed before, gerbes and their higher versions have transition functions on higher numbers of overlapping local neighborhoods than mere bundles.  (See the paper above for more)

## Higher Gauge Theory

The talks I’ll summarize here touch on various aspects of higher-categorical connections or 2-groups (though at least one I’ll put off until later).  The division between this and the section on gerbes is a little arbitrary, since of course they’re deeply connected, but I’m making some judgements about emphasis or P.O.V. here.

Apart from giving lectures in the school sessions, John Huerta also spoke on “Higher Supergroups for String Theory”, which brings “super” (i.e. $\mathbb{Z}_2$-graded) objects into higher gauge theory.  There are “super” versions of vector spaces and manifolds, which decompose into “even” and “odd” graded parts (a.k.a. “bosonic” and “fermionic” parts).  Thus there are “super” variants of Lie algebras and Lie groups, which are like the usual versions, except commutation properties have to take signs into account (e.g. a Lie superalgebra’s bracket is commutative if the product of the grades of two vectors is odd, anticommutative if it’s even).  Then there are Lie 2-algebras and 2-groups as well – categories internal to this setting.  The initial question has to do with whether one can integrate some Lie 2-algebra structures to Lie 2-group structures on a spacetime, which depends on the existence of some globally smooth cocycles.  The point is that when spacetime is of certain special dimensions, this can work, namely dimensions 3, 4, 6, and 10.  These are all 2 more than the real dimensions of the four real division algebras, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$.  It’s in these dimensions that Lie 2-superalgebras can be integrated to Lie 2-supergroups.  The essential reason is that a certain cocycle condition will hold because of the properties of a form on the Clifford algebras that are associated to the division algebras.  (John has some related material here and here, though not about the 2-group case.)

Since we’re talking about higher versions of Lie groups/algebras, an important bunch of concepts to categorify are those in representation theory.  Derek Wise spoke on “2-Group Representations and Geometry”, based on work with Baez, Baratin and Freidel, most fully developed here, but summarized here.  The point is to describe the representation theory of Lie 2-groups, in particular geometrically.  They’re to be represented on (in general, infinite-dimensional) 2-vector spaces of some sort, which is chosen to be a category of measurable fields of Hilbert spaces on some measure space, which is called $H^X$ (intended to resemble, but not exactly be the same as, $Hilb^X$, the space of “functors into $Hilb$ from the space $X$, the way Kapranov-Voevodsky 2-vector spaces can be described as $Vect^k$).  The first work on this was by Crane and Sheppeard, and also Yetter.  One point is that for 2-groups, we have not only representations and intertwiners between them, but 2-intertwiners between these.  One can describe these geometrically – part of which is a choice of that measure space $(X,\mu)$.

This done, we can say that a representation of a 2-group is a 2-functor $\mathcal{G} \rightarrow H^X$, where $\mathcal{G}$ is seen as a one-object 2-category.  Thinking about this geometrically, if we concretely describe $\mathcal{G}$ by the crossed module $(G,H,\rhd,\partial)$, defines an action of $G$ on $X$, and a map $X \rightarrow H^*$ into the character group, which thereby becomes a $G$-equivariant bundle.  One consequence of this description is that it becomes possible to distinguish not only irreducible representations (bundles over a single orbit) and indecomposible ones (where the fibres are particularly simple homogeneous spaces), but an intermediate notion called “irretractible” (though it’s not clear how much this provides).  An intertwining operator between reps over $X$ and $Y$ can be described in terms of a bundle of Hilbert spaces – which is itself defined over the pullback of $X$ and $Y$ seen as $G$-bundles over $H^*$.  A 2-intertwiner is a fibre-wise map between two such things.  This geometric picture specializes in various ways for particular examples of 2-groups.  A physically interesting one, which Crane and Sheppeard, and expanded on in that paper of [BBFW] up above, deals with the Poincaré 2-group, and where irreducible representations live over mass-shells in Minkowski space (or rather, the dual of $H \cong \mathbb{R}^{3,1}$).

Moving on from 2-group stuff, there were a few talks related to 3-groups and 3-groupoids.  There are some new complexities that enter here, because while (weak) 2-categories are all (bi)equivalent to strict 2-categories (where things like associativity and the interchange law for composing 2-cells hold exactly), this isn’t true for 3-categories.  The best strictification result is that any 3-category is (tri)equivalent to a Gray category – where all those properties hold exactly, except for the interchange law $(\alpha \circ \beta) \cdot (\alpha ' \circ \beta ') = (\alpha \cdot \alpha ') \circ (\beta \circ \beta ')$ for horizontal and vertical compositions of 2-cells, which is replaced by an “interchanger” isomorphism with some coherence properties.  John Barrett gave an introduction to this idea and spoke about “Diagrams for Gray Categories”, describing how to represent morphisms, 2-morphisms, and 3-morphisms in terms of higher versions of “string” diagrams involving (piecewise linear) surfaces satisfying some properties.  He also carefully explained how to reduce the dimensions in order to make them both clearer and easier to draw.  Bjorn Gohla spoke on “Mapping Spaces for Gray Categories”, but since it was essentially a shorter version of a talk I’ve already posted about, I’ll leave that for now, except to point out that it linked to the talk by Joao Faria Martins, “3D Holonomy” (though see also this paper with Roger Picken).

The point in Joao’s talk starts with the fact that we can describe holonomies for 3-connections on 3-bundles valued in Gray-groups (i.e. the maximally strict form of a general 3-group) in terms of Gray-functors $hol: \Pi_3(M) \rightarrow \mathcal{G}$.  Here, $\Pi_3(M)$ is the fundamental 3-groupoid of $M$, which turns points, paths, homotopies of paths, and homotopies of homotopies into a Gray groupoid (modulo some technicalities about “thin” or “laminated”  homotopies) and $\mathcal{G}$ is a gauge Gray-group.  Just as a 2-group can be represented by a crossed module, a Gray (3-)group can be represented by a “2-crossed module” (yes, the level shift in the terminology is occasionally confusing).  This is a chain of groups $L \stackrel{\delta}{\rightarrow} E \stackrel{\partial}{\rightarrow} G$, where $G$ acts on the other groups, together with some structure maps (for instance, the Peiffer commutator for a crossed module becomes a lifting $\{ ,\} : E \times E \rightarrow L$) which all fit together nicely.  Then a tri-connection can be given locally by forms valued in the Lie algebras of these groups: $(\omega , m ,\theta)$ in  $\Omega^1 (M,\mathfrak{g} ) \times \Omega^2 (M,\mathfrak{e}) \times \Omega^3(M,\mathfrak{l})$.  Relating the global description in terms of $hol$ and local description in terms of $(\omega, m, \theta)$ is a matter of integrating forms over paths, surfaces, or 3-volumes that give the various $j$-morphisms of $\Pi_3(M)$.  This sort of construction of parallel transport as functor has been developed in detail by Waldorf and Schreiber (viz. these slides, or the full paper), some time ago, which is why, thematically, they’re the next two speakers I’ll summarize.

Konrad Waldorf spoke about “Abelian Gauge Theories on Loop Spaces and their Regression”.  (For more, see two papers by Konrad on this)  The point here is that there is a relation between two kinds of theories – string theory (with $B$-field) on a manifold $M$, and ordinary $U(1)$ gauge theory on its loop space $LM$.  The relation between them goes by the name “regression” (passing from gauge theory on $LM$ to string theory on $M$), or “transgression”, going the other way.  This amounts to showing an equivalence of categories between [principal $U(1)$-bundles with connection on $LM$] and [$U(1)$-gerbes with connection on $M$].  This nicely gives a way of seeing how gerbes “categorify” bundles, since passing to the loop space – whose points are maps $S^1 \rightarrow M$ means a holonomy functor is now looking at objects (points in $LM$) which would be morphisms in the fundamental groupoid of $M$, and morphisms which are paths of loops (surfaces in $M$ which trace out homotopies).  So things are shifted by one level.  Anyway, Konrad explained how this works in more detail, and how it should be interpreted as relating connections on loop space to the $B$-field in string theory.

Urs Schreiber kicked the whole categorification program up a notch by talking about $\infty$-Connections and their Chern-Simons Functionals .  So now we’re getting up into $\infty$-categories, and particularly $\infty$-toposes (see Jacob Lurie’s paper, or even book if so inclined to find out what these are), and in particular a “cohesive topos”, where derived geometry can be developed (Urs suggested people look here, where a bunch of background is collected). The point is that $\infty$-topoi are good for talking about homotopy theory.  We want a setting which allows all that structure, but also allows us to do differential geometry and derived geometry.  So there’s a “cohesive” $\infty$-topos called $Smooth\infty Gpds$, of “sheaves” (in the $\infty$-topos sense) of $\infty$-groupoids on smooth manifolds.  This setting is the minimal common generalization of homotopy theory and differential geometry.

This is about a higher analog of this setup: since there’s a smooth classifying space (in fact, a Lie groupoid) for $G$-bundles, $BG$, there’s also an equivalence between categories $G-Bund$ of $G$-principal bundles, and $SmoothGpd(X,BG)$ (of functors into $BG$).  Moreover, there’s a similar setup with $BG_{conn}$ for bundles with connection.  This can be described topologically, or there’s also a “differential refinement” to talk about the smooth situation.  This equivalence lives within a category of (smooth) sheaves of groupoids.  For higher gauge theory, we want a higher version as in $Smooth \infty Gpds$ described above.  Then we should get an equivalence – in this cohesive topos – of $hom(X,B^n U(1))$ and a category of $U(1)$$(n-1)$-gerbes.

Then the part about the  “Chern-Simons functionals” refers to the fact that CS theory for a manifold (which is a kind of TQFT) is built using an action functional that is found as an integral of the forms that describe some $U(1)$-connection over the manifold.  (Then one does a path-integral of this functional over all connections to find partition functions etc.)  So the idea is that for these higher $U(1)$-gerbes, whose classifying spaces we’ve just described, there should be corresponding functionals.  This is why, as Urs remarked in wrapping up, this whole picture has an explicit presentation in terms of forms.  Actually, in terms of Cech-cocycles (due to the fact we’re talking about gerbes), whose coefficients are taken in sheaves of complexes (this is the derived geometry part) of differential forms whose coefficients are in $L_\infty$-algebroids (the $\infty$-groupoid version of Lie algebras, since in general we’re talking about a theory with gauge $\infty$-groupoids now).

Whew!  Okay, that’s enough for this post.  Next time, wrapping up blogging the workshop, finally.

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