### groupoids

Continuing from the previous post, we’ll take a detour in a different direction. The physics-oriented talks were by Martin Wolf, Sam Palmer, Thomas Strobl, and Patricia Ritter. Since my background in this subject isn’t particularly physics-y, I’ll do my best to summarize the ones that had obvious connections to other topics, but may be getting things wrong or unbalanced here…

### Dirac Sigma Models

Thomas Strobl’s talk, “New Methods in Gauge Theory” (based on a whole series of papers linked to from the conference webpage), started with a discussion of of generalizing Sigma Models. Strobl’s talk was a bit high-level physics for me to do it justice, but I came away with the impression of a fairly large program that has several points of contact with more mathematical notions I’ll discuss later.

In particular, Sigma models are physical theories in which a field configuration on spacetime $\Sigma$ is a map $X : \Sigma \rightarrow M$ into some target manifold, or rather $(M,g)$, since we need a metric to integrate and find differentials. Given this, we can define the crucial physics ingredient, an action functional
$S[X] = \int_{\Sigma} g_{ij} dX^i \wedge (\star d X^j)$
where the $dX^i$ are the differentials of the map into $M$.

In string theory, $\Sigma$ is the world-sheet of a string and $M$ is ordinary spacetime. This generalizes the simpler example of a moving particle, where $\Sigma = \mathbb{R}$ is just its worldline. In that case, minimizing the action functional above says that the particle moves along geodesics.

The big generalization introduced is termed a “Dirac Sigma Model” or DSM (the paper that introduces them is this one).

In building up to these DSM, a different generalization notes that if there is a group action $G \rhd M$ that describes “rigid” symmetries of the theory (for Minkowski space we might pick the Poincare group, or perhaps the Lorentz group if we want to fix an origin point), then the action functional on the space $Maps(\Sigma,M)$ is invariant in the direction of any of the symmetries. One can use this to reduce $(M,g)$, by “gauging out” the symmetries to get a quotient $(N,h)$, and get a corresponding $S_{gauged}$ to integrate over $N$.

To generalize this, note that there’s an action groupoid associated with $G \rhd M$, and replace this with some other (Poisson) groupoid instead. That is, one thinks of the real target for a gauge theory not as $M$, but the action groupoid $M \/\!\!\/ G$, and then just considers replacing this with some generic groupoid that doesn’t necessarily arise from a group of rigid symmetries on some underlying $M$. (In this regard, see the second post in this series, about Urs Schreiber’s talk, and stacks as classifying spaces for gauge theories).

The point here seems to be that one wants to get a nice generalization of this situation – in particular, to be able to go backward from $N$ to $M$, to deal with the possibility that the quotient $N$ may be geometrically badly-behaved. Or rather, given $(N,h)$, to find some $(M,g)$ of which it is a reduction, but which is better behaved. That means needing to be able to treat a Sigma model with symmetry information attached.

There’s also an infinitesimal version of this: locally, invariance means the Lie derivative of the action in the direction of any of the generators of the Lie algebra of $G$ – so called Killing vectors – is zero. So this equation can generalize to a case where there are vectors where the Lie derivative is zero – a so-called “generalized Killing equation”. They may not generate isometries, but can be treated similarly. What they do give, if you integrate these vectors, is a foliation of $M$. The space of leaves is the quotient $N$ mentioned above.

The most generic situation Thomas discussed is when one has a Dirac structure on $M$ – this is a certain kind of subbundle $D \subset TM \oplus T^*M$ of the tangent-plus-cotangent bundle over $M$.

### Supersymmetric Field Theories

Another couple of physics-y talks related higher gauge theory to some particular physics models, namely $N=(2,0)$ and $N=(1,0)$ supersymmetric field theories.

The first, by Martin Wolf, was called “Self-Dual Higher Gauge Theory”, and was rooted in generalizing some ideas about twistor geometry – here are some lecture notes by the same author, about how twistor geometry relates to ordinary gauge theory.

The idea of twistor geometry is somewhat analogous to the idea of a Fourier transform, which is ultimately that the same space of fields can be described in two different ways. The Fourier transform goes from looking at functions on a position space, to functions on a frequency space, by way of an integral transform. The Penrose-Ward transform, analogously, transforms a space of fields on Minkowski spacetime, satisfying one set of equations, to a set of fields on “twistor space”, satisfying a different set of equations. The theories represented by those fields are then equivalent (as long as the PW transform is an isomorphism).

The PW transform is described by a “correspondence”, or “double fibration” of spaces – what I would term a “span”, such that both maps are fibrations:

$P \stackrel{\pi_1}{\leftarrow} K \stackrel{\pi_2}{\rightarrow} M$

The general story of such correspondences is that one has some geometric data on $P$, which we call $Ob_P$ – a set of functions, differential forms, vector bundles, cohomology classes, etc. They are pulled back to $K$, and then “pushed forward” to $M$ by a direct image functor. In many cases, this is given by an integral along each fibre of the fibration $\pi_2$, so we have an integral transform. The image of $Ob_P$ we call $Ob_M$, and it consists of data satisfying, typically, some PDE’s.In the case of the PW transform, $P$ is complex projective 3-space $\mathbb{P}^3/\mathbb{P}^1$ and $Ob_P$ is the set of holomorphic principal $G$ bundles for some group $G$; $M$ is (complexified) Minkowski space $\mathbb{C}^4$ and the fields are principal $G$-bundles with connection. The PDE they satisfy is $F = \star F$, where $F$ is the curvature of the bundle and $\star$ is the Hodge dual). This means cohomology on twistor space (which classifies the bundles) is related self-dual fields on spacetime. One can also find that a point in $M$ corresponds to a projective line in $P$, while a point in $P$ corresponds to a null plane in $M$. (The space $K = \mathbb{C}^4 \times \mathbb{P}^1$).

Then the issue to to generalize this to higher gauge theory: rather than principal $G$-bundles for a group, one is talking about a 2-group $\mathcal{G}$ with connection. Wolf’s talk explained how there is a Penrose-Ward transform between a certain class of higher gauge theories (on the one hand) and an $N=(2,0)$ supersymmetric field theory (on the other hand). Specifically, taking $M = \mathbb{C}^6$, and $P$ to be (a subspace of) 6D projective space $\mathbb{P}^7 / \mathbb{P}^1$, there is a similar correspondence between certain holomorphic 2-bundles on $P$ and solutions to some self-dual field equations on $M$ (which can be seen as constraints on the curvature 3-form $F$ for a principal 2-bundle: the self-duality condition is why this only makes sense in 6 dimensions).

This picture generalizes to supermanifolds, where there are fermionic as well as bosonic fields. These turn out to correspond to a certain 6-dimensional $N = (2,0)$ supersymmetric field theory.

Then Sam Palmer gave a talk in which he described a somewhat similar picture for an $N = (1,0)$ supersymmetric theory. However, unlike the $N=(2,0)$ theory, this one gives, not a higher gauge theory, but something that superficially looks similar, but in fact is quite different. It ends up being a theory of a number of fields – form valued in three linked vector spaces

$\mathfrak{g}^* \stackrel{g}{\rightarrow} \mathfrak{h} \stackrel{h}{\rightarrow} \mathfrak{g}$

equipped with a bunch of maps that give the whole setup some structure. There is a collection of seven fields in groups (“multiplets”, in physics jargon) valued in each of these spaces. They satisfy a large number of identities. It somewhat resembles the higher gauge theory that corresponds to the $N=(1,0)$ case, so this situation gets called a “$(1,0)$-gauge model”.

There are some special cases of such a setup, including Courant-Dorfman algebras and Lie 2-algebras. The talk gave quite a few examples of solutions to the equations that fall out. The overall conclusion is that, while there are some similarities between $(1,0)$-gauge models and the way Higher Gauge Theory appears at the level of algebra-valued forms and the equations they must satisfy, there are some significant differences. I won’t try to summarize this in more depth, because (a) I didn’t follow the nitty-gritty technical details very well, and (b) it turns out to be not HGT, but some new theory which is less well understood at summary-level.

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)

To continue from the previous post

Twisted Differential Cohomology

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

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

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

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

Fusion Categories

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

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

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

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

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

Quantizing with Higher Categories

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

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

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

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

$A \leftarrow X \rightarrow B$

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

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

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

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

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

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

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

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.

Well, it’s been a while, but it’s now a new semester here in Hamburg, and I wanted to go back and look at some of what we talked about in last semester’s research seminar. This semester, Susama Agarwala and I are sharing the teaching in a topics class on “Category Theory for Geometry“, in which I’ll be talking about categories of sheaves, and building up the technology for Susama to talk about Voevodsky’s theory of motives (enough to give a starting point to read something like this).

As for last semester’s seminar, one of the two main threads, the one which Alessandro Valentino and I helped to organize, was a look at some of the material needed to approach Jacob Lurie’s paper on the classification of topological quantum field theories. The idea was for the research seminar to present the basic tools that are used in that paper to a larger audience, mostly of graduate students – enough to give a fairly precise statement, and develop the tools needed to follow the proof. (By the way, for a nice and lengthier discussion by Chris Schommer-Pries about this subject, which includes more details on much of what’s in this post, check out this video.)

So: the key result is a slightly generalized form of the Cobordism Hypothesis.

### Cobordism Hypothesis

The sort of theory which the paper classifies are those which “extend down to a point”. So what does this mean? A topological field theory can be seen as a sort of “quantum field theory up to homotopy”, which abstract away any geometric information about the underlying space where the fields live – their local degrees of freedom.  We do this by looking only at the classes of fields up to the diffeomorphism symmetries of the space.  The local, geometric, information gets thrown away by taking this quotient of the space of solutions.

In spite of reducing the space of fields this way, we want to capture the intuition that the theory is still somehow “local”, in that we can cut up spaces into parts and make sense of the theory on those parts separately, and determine what it does on a larger space by gluing pieces together, rather than somehow having to take account of the entire space at once, indissolubly. This reasoning should apply to the highest-dimensional space, but also to boundaries, and to any figures we draw on boundaries when cutting them up in turn.

Carrying this on to the logical end point, this means that a topological quantum field theory in the fully extended sense should assign some sort of data to every geometric entity from a zero-dimensional point up to an $n$-dimensional cobordism.  This is all expressed by saying it’s an $n$-functor:

$Z : Bord^{fr}_n(n) \rightarrow nAlg$.

Well, once we know what this means, we’ll know (in principle) what a TQFT is.  It’s less important, for the purposes of Lurie’s paper, what $nAlg$ is than what $Bord^){fr}_n(n)$ is.  The reason is that we want to classify these field theories (i.e. functors).  It will turn out that $Bord_n(n)$ has the sort of structure that makes it easy to classify the functors out of it into any target $n$-category $\mathcal{C}$.  A guess about what kind of structure is actually there was expressed by Baez and Dolan as the Cobordism Hypothesis.  It’s been slightly rephrased from the original form to get a form which has a proof.  The version Lurie proves says:

The $(\infty,n)$-category $Bord^{fr}_n(n)$ is equivalent to the free symmetric monoidal $(\infty,n)$-category generated by one fully-dualizable object.

The basic point is that, since $Bord^{fr}_n(n)$ is a free structure, the classification means that the extended TQFT’s amount precisely to the choice of a fully-dualizable object of $\mathcal{C}$ (which includes a choice of a bunch of morphisms exhibiting the “dualizability”). However, to make sense of this, we need to have a suitable idea of an $(\infty,n)$-category, and know what a fully dualizable object is. Let’s begin with the first.

### $(\infty,n)$-Categories

In one sense, the Cobordism Hypothesis, which was originally made about $n$-categories at a time when these were only beginning to be defined, could be taken as a criterion for an acceptable definition. That is, it expressed an intuition which was important enough that any definition which wouldn’t allow one to prove the Cobordism Hypothesis in some form ought to be rejected. To really make it work, one had to bring in the “infinity” part of $(\infty,n)$-categories. The point here is that we are talking about category-like structures which have morphisms between objects, 2-morphisms between morphisms, and so on, with $j$-morphisms between $j-1$-morphisms for every possible degree. The inspiration for this comes from homotopy theory, where one has maps, homotopies of maps, homotopies of homotopies, etc.

Nowadays, there are several possible concrete models for $(\infty,n)$-categories (see this survey article by Julie Bergner for a summary of four of them). They are all equivalent definitions, in a suitable up-to-homotopy way, but for purposes of the proof, Lurie is taking the definition that an $(\infty,n)$-category is an n-fold complete Segal space. One theme that shows up in all the definitions is that of simplicial methods. (In our seminar, we started with a series of two talks introducing the notions of simplicial sets, simplicial objects in a category, and Kan complexes. If you don’t already know this, essentially everything we need is nicely explained in here.)

One of the underlying ideas is that a category $C$ can be associated with a simplicial set, its nerve $N(C)_{\bullet}$, where the set $N(C)_k$ of $k$-dimensional simplexes is just the set of composable $k$-tuples of morphisms in $C$. If $C$ is a groupoid (everything is invertible), then the simplicial set is a Kan complex – it satisfies some filling conditions, which ensure that any morphism has an inverse. Not every Kan complex is the nerve of a groupoid, but one can think of them as weak versions of groupoids – $\infty$-groupoids, or $(\infty,0)$-categories – where the higher morphisms may not be completely trivial (as with a groupoid), but where at least they’re all invertible. This leads to another desirable feature in any definition of $(\infty,n)$-category, which is the Homotopy Hypothesis: that the $(\infty,1)$-category of $(\infty,0)$-categories, also called $\infty$-groupoids, should be equivalent (in the same weak sense) to a category of Hausdorff spaces with some other nice properties, which we call $\mathbf{Top}$ for short. This is true of Kan complexes.

Thus, up to homotopy, specifying an $\infty$-groupoid is the same as specifying a space.

The data which defines a Segal space (which was however first explicitly defined by Charlez Rezk) is a simplicial space $X_{\bullet}$: for each $n$, there are spaces $X_n$, thought of as the space of composable $n$-tuples of morphisms. To keep things tame, we suppose that $X_0$, the space of objects, is discrete – that is, we have only a set of objects. Being a simplicial space means that the $X_n$ come equipped with a collection of face maps $d_i : X_n \rightarrow X_{n-1}$, which we should think of as compositions: to get from an $n$-tuple to an $(n-1)$-tuple of morphisms, one can compose two morphisms together at any of $(n-1)$ positions in the tuple.

One condition which a simplicial space has to satisfy to be a Segal space has to do with the “weakening” which makes a Segal space a weaker notion than just a category lies in the fact that the $X_n$ cannot be arbitrary, but must be homotopy equivalent to the “actual” space of $n$-tuples, which is a strict pullback $X_1 \times_{X_0} \dots \times_{X_0} X_1$. That is, in a Segal space, the pullback which defines these tuples for a category is weakened to be a homotopy pullback. Combining this with the various face maps, we therefore get a weakened notion of composition: $X_1 \times_{X_0} \dots \times_{X_0} X_1 \cong X_n \rightarrow X_1$. Because we start by replacing the space of $n$-tuples with the homotopy-equivalent $X_n$, the composition rule will only satisfy all the relations which define composition (associativity, for instance) up to homotopy.

To be complete, the Segal space must have a notion of equivalence for $X_{\bullet}$ which agrees with that for Kan complexes seen as $\infty$-groupoids. In particular, there is a sub-simplicial object $Core(X_{\bullet})$, which we understand to consist of the spaces of invertible $k$-morphisms. Since there should be nothing interesting happening above the top dimension, we ask that, for these spaces, the face and degeneracy maps are all homotopy equivalences: up to homotopy, the space of invertible higher morphisms has no new information.

Then, an $n$-fold complete Segal space is defined recursively, just as one might define $n$-categories (without the infinitely many layers of invertible morphisms “at the top”). In that case, we might say that a double category is just a category internal to $\mathbf{Cat}$: it has a category of objects, and a category of morphims, and the various maps and operations, such as composition, which make up the definition of a category are all defined as functors. That turns out to be the same as a structure with objects, horizontal and vertical morphisms, and square-shaped 2-cells. If we insist that the category of objects is discrete (i.e. really just a set, with no interesting morphisms), then the result amounts to a 2-category. Then we can define a 3-category to be a category internal to $\mathbf{2Cat}$ (whose 2-category of objects is discrete), and so on. This approach really defines an $n$-fold category (see e.g. Chapter 5 of Cheng and Lauda to see a variation of this approach, due to Tamsamani and Simpson), but imposing the condition that the objects really amount to a set at each step gives exactly the usual intuition of a (strict!) $n$-category.

This is exactly the approach we take with $n$-fold complete Segal spaces, except that some degree of weakness is automatic. Since a C.S.S. is a simplicial object with some properties (we separately define objects of $k$-tuples of morphisms for every $k$, and all the various composition operations), the same recursive approach leads to a definition of an “$n$-fold complete Segal space” as simply a simplicial object in $(n-1)$-fold C.S.S.’s (with the same properties), such that the objects form a set. In principle, this gives a big class of “spaces of morphisms” one needs to define – one for every $n$-fold product of simplexes of any dimension – but all those requirements that any space of objects “is just a set” (i.e. is homotopy-equivalent to a discrete set of points) simplifies things a bit.

### Cobordism Category as $(\infty,n)$-Category

So how should we think of cobordisms as forming an $(\infty,n)$-category? There are a few stages in making a precise definition, but the basic idea is simple enough. One starts with manifolds and cobordisms embedded in some fixed finite-dimensional vector space $V \times \mathbb{R}^n$, and then takes a limit over all $V$. In each $V \times \mathbb{R}^n$, the coordinates of the $\mathbb{R}^n$ factor give $n$ ways of cutting the cobordism into pieces, and gluing them back together defines composition in a different direction. Now, this won’t actually produce a complete Segal space: one has to take a certain kind of completion. But the idea is intuitive enough.

We want to define an $n$-fold C.S.S. of cobordisms (and cobordisms between cobordisms, and so on, up to $n$-morphisms). To start with, think of the case $n=1$: then the space of objects of $Bord^{fr}_1(1)$ consists of all embeddings of a $(d-1)$-dimensional manifold into $V$. The space of $k$-simplexes (of $k$-tuples of morphisms) consists of all ways of cutting up a $d$-dimensional cobordism embedded in $V \times \mathbb{R}$ by choosing $t_0, \dots , t_{k-2}$, where we think of the cobordism having been glued from two pieces, where at the slice $V \times {t_i}$, we have the object where the two pieces were composed. (One has to be careful to specify that the Morse function on the cobordisms, got by projection only $\mathbb{R}$, has its critical points away from the $t_i$ – the generic case – to make sure that the objects where gluing happens are actual manifolds.)

Now, what about the higher morphisms of the $(\infty,1)$-category? The point is that one needs to have an $\infty$-groupoid – that is, a space! – of morphisms between two cobordisms $M$ and $N$. To make sense of this, we just take the space $Diff(M,N)$ of diffeomorphisms – not just as a set of morphisms, but including its topology as well. The higher morphisms, therefore, can be thought of precisely as paths, homotopies, homotopies between homotopies, and so on, in these spaces. So the essential difference between the 1-category of cobordisms and the $(\infty,1)$-category is that in the first case, morphisms are diffeomorphism classes of cobordisms, whereas in the latter, the higher morphisms are made precisely of the space of diffeomorphisms which we quotient out by in the first case.

Now, $(\infty,n)$-categories, can have non-invertible morphisms between morphisms all the way up to dimension $n$, after which everything is invertible. An $n$-fold C.S.S. does this by taking the definition of a complete Segal space and copying it inside $(n-1)$-fold C.S.S’s: that is, one has an $(n-1)$-fold Complete Segal Space of $k$-tuples of morphisms, for each $k$, they form a simplicial object, and so forth.

Now, if we want to build an $(\infty,n)$-category $Bord^{fr}_n(n)$ of cobordisms, the idea is the same, except that we have a simplicial object, in a category of simplicial objects, and so on. However, the way to define this is essentially similar. To specify an $n$-fold C.S.S., we have to specify a whole collection of spaces associated to cobordisms equipped with embeddings into $V \times \mathbb{R}^n$. In particular, for each tuple $(k_1,\dots,k_n)$, we have the space of such embeddings, such that for each $i = 1 \dots n$ one has $k_i$ special points $t_{i,j}$ along the $i^{th}$ coordinate axis. These are the ways of breaking down a given cobordism into a composite of $k_i +1$ pieces. Again, one has to make sure that these critical points of the Morse functions defined by the projections onto these coordinate axes avoid these special $t_{i,j}$ which define the manifolds where gluing takes place. The composition maps which make these into a simplical object are quite natural – they just come by deleting special points.

Finally, we take a limit over all $V$ (to get around limits to embeddings due to the dimension of $V$). So we know (at least abstractly) what the $(\infty,n)$-category of cobordisms should be. The cobordism hypothesis claims it is equivalent to one defined in a free, algebraically-flavoured way, namely as the free symmetric monoidal $(\infty,n)$-category on a fully-dualizable object. (That object is “the point” – which, up to the kind of homotopically-flavoured equivalence that matters here, is the only object when our highest-dimensional cobordisms have dimension $n$).

### Dualizability

So what does that mean, a “fully dualizable object”?

First, to get the idea, let’s think of the 1-dimensional example.  Instead of “$(\infty,n)$-category”, we would like to just think of this as a statement about a category.  Then $Bord^{fr}_1(1)$ is the 1-category of framed bordisms. For a manifold (or cobordism, which is a manifold with boundary), a framing is a trivialization of the tangent bundle.  That is, it amounts to a choice of isomorphism at each point between the tangent space there and the corresponding $\mathbb{R}^n$.  So the objects of $Bord^{fr}_1(1)$ are collections of (signed) points, and the morphisms are equivalence classes of framed 1-dimensional cobordisms.  These amount to oriented 1-manifolds with boundary, where the points (objects) on the boundary are the source and target of the cobordism.

Now we want to classify what TQFT’s live on this category.  These are functors $Z : Bord^{fr}_1(1)$.  We have two generating objects, $+$ and $-$, the two signed points.  A TQFT must assign these objects vector spaces, which we’ll call $V$ and $W$.  Collections of points get assigned tensor products of all the corresponding vector spaces, since the functor is monoidal, so knowing these two vector spaces determines what $Z$ does to all objects.

What does $Z$ do to morphisms?  Well, some generating morphsims of interest are cups and caps: these are lines which connect a positive to a negative point, but thought of as cobordisms taking two points to the empty set, and vice versa.  That is, we have an evaluation:This statement is what is generalized to say that $n$-dimensional TQFT’s are classified by “fully” dualizable objects.

$ev: W \otimes V \rightarrow \mathbb{C}$

and a coevaluation:

$coev: \mathbb{C} \rightarrow V \otimes W$

Now, since cobordisms are taken up to equivalence, which in particular includes topological deformations, we get a bunch of relations which these have to satisfy.  The essential one is the “zig-zag” identity, reflecting the fact that a bent line can be straightened out, and we have the same 1-morphism in $Born^{fr}_1(1)$.  This implies that:

$(ev \otimes id) \circ (id \otimes coev) : W \rightarrow W \otimes V \otimes W \rightarrow W$

is the same as the identity.  This in turn means that the evaluation and coevaluation maps define a nondegenerate pairing between $V$ and $W$.  The fact that this exists means two things.  First, $W$ is the dual of $V$: $W \cong V*$.  Second, this only makes sense if both $V$ and its dual are finite dimensional (since the evaluation will just be the trace map, which is not even defined on the identity if $V$ is infinite dimensional).

On the other hand, once we know, $V$, this determines $W \cong V*$ up to isomorphism, as well as the evaluation and coevaluation maps.  In fact, this turns out to be enough to specify $Z$ entirely.  The classification then is: 1-D TQFT’s are classified by finite-dimensional vector spaces $V$.  Crucially, what made finiteness important is the existence of the dual $V*$ and the (co)evaluation maps which express the duality.

In an $(\infty,n)$-category, to say that an object is “fully dualizable” means more that the object has a dual (which, itself, implies the existence of the morphisms $ev$ and $coev$). It also means that $ev$ and $coev$ have duals themselves – or rather, since we’re talking about morphisms, “adjoints”. This in turn implies the existence of 2-morphisms which are the unit and counit of the adjunctions (the defining properties are essentially the same as those for morphisms which define a dual). In fact, every time we get a morphism of degree less than $n$ in this process, “fully dualizable” means that it too must have a dual (i.e. an adjoint).

This does run out eventually, though, since we only require this goes up to dimension $(n-1)$: the $n$-morphisms which this forces to exist (quite a few) aren’t required to have duals. This is good, because if they were, since all the higher morphisms available are invertible, this would mean that the dual $n$-morphisms would actually be weak inverses (that is, their composite is isomorphic to the identity)… But that would mean that the dual $(n-1)$-morphisms which forced them to exist would also be weak inverses (their composite would be weakly isomorphic to the identity)… and so on! In fact, if the property of “having duals” didn’t stop, then everything would be weakly invertible: we’d actually have a (weak) $\infty$-groupoid!

### Classifying TQFT

So finally, the point of the Cobordism Hypothesis is that a (fully extended) TQFT is a functor $Z$ out of this $nBord^{fr}_n(n)$ into some target $(\infty,1)$-category $\mathcal{C}$. There are various options, but whatever we pick, the functor must assign something in $\mathcal{C}$ to the point, say $Z(pt)$, and something to each of $ev$ and $coev$, as well as all the higher morphisms which must exist. Then functoriality means that all these images have to again satisfy the properties which make $Z(pt)$ a fully dualizable object. Furthermore, since $nBord^{fr}_n(n)$ is the free gadget with all these properties on the single object $pt$, this is exactly what it means that $Z$ is a functor. Saying that $Z(pt)$ is fully dualizable, by implication, includes all the choices of morphisms like $Z(ev)$ etc. which show it as fully dualizable. (Conceivably one could make the same object fully dualizable in more than one way – these would be different functors).

So an extended $n$-dimensional TQFT is exactly the choice of a fully dualizable object $Z(pt) \in \mathcal{C}$, for some $(\infty,n)$-category $\mathcal{C}$. This object is “what the TQFT assigns to a point”, but if we understand the structure of the object as a fully dualizable object, then we know what the TQFT assigns to any other manifold of any dimension up to $n$, the highest dimension in the theory. This is how this algebraic characterization of cobordisms helps to classify such theories.

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!

I’ve written here before about building topological quantum field theories using groupoidification, but I haven’t yet gotten around to discussing a refinement of this idea, which is in the most recent version of my paper on the subject.  I also gave a talk about this last year in Erlangen. The main point of the paper is to pull apart some constructions which are already fairly well known into two parts, as part of setting up a category which is nice for supporting models of fairly general physical systems, using an extension of the  concept of groupoidification. So here’s a somewhat lengthy post which tries to unpack this stuff a bit.

Factoring TQFT

The older version of this paper talked about the untwisted version of the Dijkgraaf-Witten (DW for short) model, which is a certain kind of TQFT based on a gauge theory with a finite gauge group.  (Freed and Quinn put it as: “Chern-Simons theory with finite gauge group”).  The new version gets the general – that is, the twisted – form in the same way: factoring the theory into two parts. So, the DW model, which was originally described by Dijkgraaf and Witten in terms of a state-sum, is a functor

$Z : 3Cob \rightarrow Vect$

The “twisting” is the point of their paper, “Topological Gauge Theories and Group Cohomology”.  The twisting has to do with the action for some physical theory. Now, for a gauge theory involving flat connections, the kind of gauge-theory actions which involve the curvature of a connection make no sense: the curvature is zero.  So one wants an action which reflects purely global features of connections.  The cohomology of the gauge group is where this comes from.

Now, the machinery I describe is based on a point of view which has been described in a famous paper by Freed, Hopkins, Lurie and Teleman (FHLT for short – see further discussion here) in terms in which the two stages are called the “classical field theory” (which has values in groupoids), and the “quantization functor”, which takes one into Hilbert spaces.

Actually, we really want to have an “extended” TQFT: a TQFT gives a Hilbert space for each 2D manifold (“space”), and a linear map for a 3D cobordism (“spacetime”) between them. An extended TQFT will assign (higher) algebraic data to lower-dimension boundaries still.  My paper talks only about the case where we’ve extended down to codimension 2, whereas FHLT talk about extending “down to a point”. The point of this first stopping point is to unpack explicitly and computationally what the factorization into two parts looks like at the first level beyond the usual TQFT.

In the terminology I use, the classical field theory is:

$A^{\omega} : nCob_2 \rightarrow Span_2(Gpd)^{U(1)}$

This depends on a cohomology class $[\omega] \in H^3(G,U(1))$. The “quantization functor” (which in this case I call “2-linearization”):

$\Lambda^{U(1)} : Span_2(Gpd)^{U(1)} \rightarrow 2Vect$

The middle stage involves the monoidal 2-category I call $Span_2(Gpd)^{U(1)}$.  (In FHLT, they use different terminology, for instance “families” rather than “spans”, but the principle is the same.)

Freed and Quinn looked at the quantization of the “extended” DW model, and got a nice geometric picture. In it, the action is understood as a section of some particular line-bundle over a moduli space. This geometric picture is very elegant once you see how it works, which I found was a little easier in light of a factorization through $Span_2(Gpd)$.

This factorization isolates the geometry of this particular situation in the “classical field theory” – and reveals which of the features of their setup (the line bundle over a moduli space) are really part of some more universal construction.

In particular, this means laying out an explicit definition of both $Span_2(Gpd)^{U(1)}$ and $\Lambda^{U(1)}$.

2-Linearization Recalled

While I’ve talked about it before, it’s worth a brief recap of how 2-linearization works with a view to what happens when you twist it via groupoid cohomology. Here we have a 2-category $Span(Gpd)$, whose objects are groupoids ($A$, $B$, etc.), whose morphisms are spans of groupoids:

$A \stackrel{s}{\leftarrow} X \stackrel{t}{\rightarrow} B$

and whose 2-morphisms are spans of span-maps (taken up to isomorphism), which look like so:

(And, by the by: how annoying that WordPress doesn’t appear to support xypic figures…)

These form a (symmetric monoidal) 2-category, where composition of spans works by taking weak pullbacks.  Physically, the idea is that a groupoid has objects which are configurations (in the cause of gauge theory, connections on a manifold), and morphisms which are symmetries (gauge transformations, in this case).  Then a span is a groupoid of histories (connections on a cobordism, thought of as spacetime), and the maps $s,t$ pick out its starting and ending configuration.  That is, $A = A_G(S)$ is the groupoid of flat $G$-connections on a manifold $S$, and $X = A_G(\Sigma)$ is the groupoid of flat $G$-connections on some cobordism $\Sigma$, of which $S$ is part of the boundary.  So any such connection can be restricted to the boundary, and this restriction is $s$.

Now 2-linearization is a 2-functor:

$\Lambda : Span_2(Gpd)^{U(1)} \rightarrow 2Vect$

It gives a 2-vector space (a nice kind of category) for each groupoid $G$.  Specifically, the category of its representations, $Rep(G)$.  Then a span turns into a functor which comes from “pulling” back along $s$ (the restricted representation where $X$ acts by first applying $s$ then the representation), then “pushing” forward along $t$ (to the induced representation).

What happens to the 2-morphisms is conceptually more complicated, but it depends on the fact that “pulling” and “pushing” are two-sided adjoints. Concretely, it ends up being described as a kind of “sum over histories” (where “histories” are the objects of $Y$), which turns out to be exactly the path integral that occurs in the TQFT.

Or at least, it’s the path integral when the action is trivial! That is, if $S=0$, so that what’s integrated over paths (“histories”) is just $e^{iS}=1$. So one question is: is there a way to factor things in this way if there’s a nontrivial action?

Cohomological Twisting

The answer is by twisting via cohomology. First, let’s remember what that means…

We’re talking about groupoid cohomology for some groupoid $G$ (which you can take to be a group, if you like).  “Cochains” will measure how much some nice algebraic fact, such as being a homomorphism, or being associative, “fails to occur”.  “Twisting by a cocycle” is a controlled way to force some such failure to happen.

So, an $n$-cocycle is some function of $n$ composable morphisms of $G$ (or, if there’s only one object, “group elements”, which amounts to the same thing).  It takes values in some group of coefficients, which for us is always $U(1)$

The trivial case where $n=0$ is actually slightly subtle: a 0-cocycle is an invariant function on the objects of a groupoid. (That is, it takes the same value on any two objects related by an (iso)morphism. (Think of the object as a sequence of zero composable morphisms: it tells you where to start, but nothing else.)

The case $n=1$ is maybe a little more obvious. A 1-cochain $f \in Z^1_{gpd}(G,U(1))$ can measure how a function $h$ on objects might fail to be a 0-cocycle. It is a $U(1)$-valued function of morphisms (or, if you like, group elements).  The natural condition to ask for is that it be a homomorphism:

$f(g_1 \circ g_2) = f(g_1) f(g_2)$

This condition means that a cochain $f$ is a cocycle. They form an abelian group, because functions satisfying the cocycle condition are closed under pointwise multiplication in $U(1)$. It will automatically by satisfied for a coboundary (i.e. if $f$ comes from a function $h$ on objects as $f(g) = \delta h (g) = h(t(g)) - h(s(g))$). But not every cocycle is a coboundary: the first cohomology $H^1(G,U(1))$ is the quotient of cocycles by coboundaries. This pattern repeats.

It’s handy to think of this condition in terms of a triangle with edges $g_1$, $g_2$, and $g_1 \circ g_2$.  It says that if we go from the source to the target of the sequence $(g_1, g_2)$ with or without composing, and accumulate $f$-values, our $f$ gives the same result.  Generally, a cocycle is a cochain satisfying a “coboundary” condition, which can be described in terms of an $n$-simplex, like this triangle. What about a 2-cocycle? This describes how composition might fail to be respected.

So, for instance, a twisted representation $R$ of a group is not a representation in the strict sense. That would be a map into $End(V)$, such that $R(g_1) \circ R(g_2) = R(g_1 \circ g_2)$.  That is, the group composition rule gets taken directly to the corresponding rule for composition of endomorphisms of the vector space $V$.  A twisted representation $\rho$ only satisfies this up to a phase:

$\rho(g_1) \circ \rho(g_2) = \theta(g_1,g_2) \rho(g_1 \circ g_2)$

where $\theta : G^2 \rightarrow U(1)$ is a function that captures the way this “representation” fails to respect composition.  Still, we want some nice properties: $\theta$ is a “cocycle” exactly when this twisting still makes $\rho$ respect the associative law:

$\rho(g_1) \rho( g_2 \circ g_3) = \rho( g_1 \circ g_2) \circ \rho( g_3)$

Working out what this says in terms of $\theta$, the cocycle condition says that for any composable triple $(g_1, g_2, g_3)$ we have:

$\theta( g_1, g_2 \circ g_3) \theta (g_2,g_3) = \theta(g_1,g_2) \theta(g_1 \circ g_2, g_3)$

So $H^2_{grp}(G,U(1))$ – the second group-cohomology group of $G$ – consists of exactly these $\theta$ which satisfy this condition, which ensures we have associativity.

Given one of these $\theta$ maps, we get a category $Rep^{\theta}(G)$ of all the $\theta$-twisted representations of $G$. It behaves just like an ordinary representation category… because in fact it is one! It’s the category of representations of a twisted version of the group algebra of $G$, called $C^{\theta}(G)$. The point is, we can use $\theta$ to twist the convolution product for functions on $G$, and this is still an associative algebra just because $\theta$ satisfies the cocycle condition.

The pattern continues: a 3-cocycle captures how some function of 2 variable may fail to be associative: it specifies an associator map (a function of three variables), which has to satisfy some conditions for any four composable morphisms. A 4-cocycle captures how a map might fail to satisfy this condition, and so on. At each stage, the cocycle condition is automatically satisfied by coboundaries. Cohomology classes are elements of the quotient of cocycles by coboundaries.

So the idea of “twisted 2-linearization” is that we use this sort of data to change 2-linearization.

Twisted 2-Linearization

The idea behind the 2-category $Span(Gpd)^{U(1)}$ is that it contains $Span(Gpd)$, but that objects and morphisms also carry information about how to “twist” when applying the 2-linearization $\Lambda$.  So in particular, what we have is a (symmetric monoidal) 2-category where:

• Objects consist of $(A, \theta)$, where $A$ is a groupoid and $\theta \in Z^2(A,U(1))$
• Morphisms from $A$ to $B$ consist of a span $(X,s,t)$ from $A$ to $B$, together with $\alpha \in Z^1(X,U(1))$
• 2-Morphisms from $X_1$ to $X_2$ consist of a span $(Y,\sigma,\tau)$ from $X$, together with $\beta \in Z^0(Y,U(1))$

The cocycles have to satisfy some compatibility conditions (essentially, pullbacks of the cocycles from the source and target of a span should land in the same cohomology class).  One way to see the point of this requirement is to make twisted 2-linearization well-defined.

One can extend the monoidal structure and composition rules to objects with cocycles without too much trouble so that $Span(Gpd)$ is a subcategory of $Span(Gpd)^{U(1)}$. The 2-linearization functor extends to $\Lambda^{U(1)} : Span(Gpd)^{U(1)} \rightarrow 2Vect$:

• On Objects: $\Lambda^{U(1)} (A, \theta) = Rep^{\theta}(A)$, the category of $\theta$-twisted representation of $A$
• On Morphisms: $\Lambda^{U(1)} ( (X,s,t) , \alpha )$ comes by pulling back a twisted representation in $Rep^{\theta_A}(A)$ to one in $Rep^{s^{\ast}\theta_A}(X)$, pulling it through the algebra map “multiplication by $\alpha$“, and pushing forward to $Rep^{\theta_B}(B)$
• On 2-Morphisms: For a span of span maps, one uses the usual formula (see the paper for details), but a sum over the objects $y \in Y$ picks up a weight of $\beta(y)$ at each object

When the cocycles are trivial (evaluate to 1 always), we get back the 2-linearization we had before. Now the main point here is that the “sum over histories” that appears in the 2-morphisms now carries a weight.

So the twisted form of 2-linearization uses the same “pull-push” ideas as 2-linearization, but applied now to twisted representations. This twisting (at the object level) uses a 2-cocycle. At the morphism level, we have a “twist” between “pull” and “push” in constructing . What the “twist” actually means depends on which cohomology degree we’re in – in other words, whether it’s applied to objects, morphisms, or 2-morphisms.

The “twisting” by a 0-cocycle just means having a weight for each object – in other words, for each “history”, or connection on spacetime, in a big sum over histories. Physically, the 0-cocycle is playing the role of the Lagrangian functional for the DW model. Part of the point in the FHLT program can be expressed by saying that what Freed and Quinn are doing is showing how the other cocycles are also the Lagrangian – as it’s seen at higher codimension in the more “local” theory.

For a TQFT, the 1-cocycles associated to morphisms describe how to glue together values for the Lagrangian that are associated to histories that live on different parts of spacetime: the action isn’t just a number. It is a number only “locally”, and when we compose 2-morphisms, the 0-cocycle on the composite picks up a factor from the 1-morphism (or 0-morphism, for a horizontal composite) where they’re composed.

This has to do with the fact that connections on bits of spacetime can be glued by particular gauge transformations – that is, morphisms of the groupoid of connections. Just as the gauge transformations tell how to glue connections, the cocycles associated to them tell how to glue the actions. This is how the cohomological twisting captures the geometric insight that the action is a section of a line bundle – not just a function, which is a section of a trivial bundle – over the moduli space of histories.

So this explains how these cocycles can all be seen as parts of the Lagrangian when we quantize: they explain how to glue actions together before using them in a sum-over histories. Gluing them this way is essential to make sure that $\Lambda^{U(1)}$ is actually a functor. But if we’re really going to see all the cocycles as aspects of “the action”, then what is the action really? Where do they come from, that they’re all slices of this bigger thing?

Twisting as Lagrangian

Now the DW model is a 3D theory, whose action is specified by a group-cohomology class $[\omega] \in H^3_{grp}(G,U(1))$. But this is the same thing as a class in the cohomology of the classifying space: $[\omega] \in H^3(BG,U(1))$. This takes a little unpacking, but certainly it’s helpful to understand that what cohomology classes actually classify are… gerbes. So another way to put a key idea of the FHLT paper, as Urs Schreiber put it to me a while ago, is that “the action is a gerbe on the classifying space for fields“.

What does this mean?

This map is given as a path integral over all connections on the space(-time) $S$, which is actually just a sum, since the gauge group is finite and so all the connections are flat.  The point is that they’re described by assigning group elements to loops in $S$:

$A : \pi_1(M) \rightarrow G$

But this amounts to the same thing as a map into the classifying space of $G$:

$f_A : M \rightarrow BG$

This is essentially the definition of $BG$, and it implies various things, such as the fact that $BG$ is a space whose fundamental group is $G$, and has all other homotopy groups trivial. That is, $BG$ is the Eilenberg-MacLane space $K(G,1)$. But the point is that the groupoid of connections and gauge transformations on $S$ just corresponds to the mapping space $Maps(S,BG)$. So the groupoid cohomology classes we get amount to the same thing as cohomology classes on this space. If we’re given $[\omega] \in H^3(BG,U(1))$, then we can get at these by “transgression” – which is very nicely explained in a paper by Simon Willerton.

The essential idea is that a 3-cocycle $\omega$ (representing the class $[\omega]$) amounts to a nice 3-form on $BG$ which we can integrate over a 3-dimentional submanifold to get a number. For a $d$-dimensional $S$, we get such a 3-manifold from a $(3-d)$-dimensional submanifold of $Maps(S,BG)$: each point gives a copy of $S$ in $BG$. Then we get a $(3-d)$-cocycle on $Maps(S,BG)$ whose values come from integrating $\omega$ over this image. Here’s a picture I used to illustrate this in my talk:

Now, it turns out that this gives 2-cocycles for 1-manifolds (the objects of $3Cob_2$, 1-cocycles on 2D cobordisms between them, and 0-cocycles on 3D cobordisms between these cobordisms. The cocycles are for the groupoid of connections and gauge transformations in each case. In fact, because of Stokes’ theorem in $BG$, these have to satisfy all the conditions that make them into objects, morphisms, and 2-morphisms of $Span^{U(1)}(Gpd)$. This is the geometric content of the Lagrangian: all the cocycles are really “reflections” of $\omega$ as seen by transgression: pulling back along the evaluation map $ev$ from the picture. Then the way you use it in the quantization is described exactly by $\Lambda^{U(1)}$.

What I like about this is that $\Lambda^{U(1)}$ is a fairly universal sort of thing – so while this example gets its cocycles from the nice geometry of $BG$ which Freed and Quinn talk about, the insight that an action is a section of a (twisted) line bundle, that actions can be glued together in particular ways, and so on… These presumably can be moved to other contexts.

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