QGQG Nottingham

Well, a week ago I got back from England, where I spent a week at the University of Nottingham at the conference “Quantum Gravity and Quantum Geometry 2008″, and a weekend visiting friends in London. London was enjoyable, though surprisingly expensive. It’s strange, when so many things are traded globally, that prices differ so much from place to place - the standard rule being to imagine that all prices in Pounds are actually in dollars, and they seem quite familiar. Clearly not everything is affected by trade, with restaurant meals among them. In any case, it was quite interesting to come come from London, Ontario to London, England, and walk around all the places whose names show up attached to completely dissimilar landmarks in the Canadian version.

As for the conference, it was a great experience. This was an outgrowth of the “LOOPS” series of conferences. The only one of those I’d been to previously was LOOPS ‘05 at the Albert Einstein Institute, in Germany. At that time the conference was a little more focused on some particular approaches to quantum gravity (though there was still a whole range of talks). This year, there seemed to have been some attempt to broaden the conference a little - one result being that there must have been about 200 people attending, with something on the order of 90 talks, most of them half-hour talks in the parallel sessions. As a result, I saw less than half of what was going on. However, there were some broad subject areas, such as loop quantum gravity, spin foam and combinatorial quantization, noncommutative geometry, quantum groups, as well as some less readily classifiable talks.

In one talk on the first day, Carlo Rovelli discussed the relation between the Loop Quantum Gravity and spin-foam approaches to a theory of 4D quantum gravity. In particular, he was talking about the fact that the two approaches agree with each other in 3D, but it’s not so clear they do in 4D - or at least, it’s not clear what the spin foam model is that does this in 4D. This is part of what’s behind the program to improve the Barrett-Crane spin foam model for 4D gravity. It has various technical problems as well, which various more technical talks got into in more detail later in the conference. Rovelli was describing work on the new models which agree with LQG. Various other people have done work on this, including (among others) Freidel (who talked about that in his own talk later) and Krasnov, and Engle, Pereira and Rovelli. Florian Conrady also talked about these new models later on. I know Igor Khavkine, just graduating here at Western, has also done some work on these.

Another talk based off the successes of these models was by Abhay Ashtekar, about Loop Quantum Cosmology - that is, applying loop QG methods to the universe as a whole - a quantum version of the Friedman-Robertson-Walker universe. What’s interesting about this is that they’re doing numerical and analytic simulations, and predicting something that otherwise has usually been added as a “what-if” afterthougoht. Namely, such a universe behaves a lot like classical FRW, except near the “big bang”, classically a singularity, where quantum geometric effects prevent that from happening. Continuing through the other side, one sees a collapsing universe - an overall “bounce” effect. An interesting prediction, if hard to check.

In any case, I was bombarded by a whole range of other talks on other points of view. Starting from the very first talk, by Vincent Rivasseau, there were several talks presenting noncommutative geometry, Alain Connes-style, as a setting for a quantum theory of gravity. There’s certainly an appeal to the idea of replacing measure-theoretic and topological information about spacetime with a quantum algebra of observables - just write the theory in quantum terms from the start, giving up the usual differential geometry for its noncommutative version. Rivasseau presented, among other things, the idea of QFT as weighted species, in the sense of Joyal’s combinatorial species. I thought this was great, since I looked at just that idea for the simplest QFT of all, the quantum harmonic oscillator.

(Speaking of which, I had some interesting conversations with Jamie Vicary in which I finally “got” part of what he did with his own paper about the oscillator - which is to show how “taking Fock space” for a quantum system is a monad, namely the monad associated with the “free commutative monoid” functor, and its adjoint.)

Shahn Majid, whom I knew as the author of some well-known books on quantum groups, also spoke about this C*-algebra approach to geometry, and quantum gravity. : begin with a space, like a manifold, or better yet a fibre bundle, which is where a lot of physics gets done, and look at the algebra of forms on it. It has nice properties (it’s a differential graded algebra, etc.), including being commutative. One can deform these to noncommutative algebras that are quite nice - “q-deformation” assumes the commutators between elements depend on some parameter q, so the old picture where q=0 is simply a special case.

So then one thing is to develop a deformed version of classical things from geometry and analysis - for example, the Fourier transform. Even in the big purple book on quantum groups, he outlined what this approach consists of: a criterion for a quantum theory of gravity, that it should be algebraically “self-dual”, under exchange of “position” and “momentum” variables. (That is, under a Fourier transform - being its own Fourier dual).

Well, speaking of quantum groups, I should mention Aaron Lauda’s talk on categorifying them - specifically, on categorifying “deformed classical Lie groups”, like (a q-deformed version of the universal enveloping algebra , which for is the algebra where the Lie bracket of is a genuine commutator). He described a graphical calculus - a particular kind of string diagram, with some relations on them - which is a categorification of the quantum group. In fact, as sometimes happens, it categorifies a specific presentation of the algebra in terms of some generators and relations.

An appealing thing about these string diagram methods and so forth is that it suggests why these algebraic gadgets - quantum groups, in this case - are good at encoding topological information about tangles, braids, knots, and so on. If diagrams that involve those shapes categorify (read “model the underlying structure of”) quantum groups, then it makes sense that quantum groups to give invariants for them.

Along similar lines, Joao Faria Martins talked about invariants for “welded virtual knots”, and for knotted surfaces from crossed modules (read “2-groups”, if you’re so inclined - they are equivalent). Martins also published a paper with Tim Porter about related work, which in turn builds on David Yetter’s, on a class of manifold invariants. Their paper talks about “extending the Dijkgraaf-Witten model to categorical groups” (Urs Schreiber, possibly among others, rephrased that to call it a “categorification of the Dijkgraaf-Witten model”. The DW model is the TQFT foundation for my own look at extending (read, “categorifying”) TQFT’s based on gauge theory using a group - (finite, for the DW model). These are categorifications in two different directions, though: one, from a gauge group to a gauge 2-group, the other from a TQFT - a functor - to a 2-functor given by a group. Probably for 4 dimensions and higher, the 2-group version or higher is the most interesting to study.

In fact, there was a fair bevy of talks relating to categorical methods in quantum geometry. For example, Jamie Vicary gave a talk introducing a “categorical framework for quantum algebra”, by means of non-threatening string diagrams. These can be used to show the axioms for a “-monoidal category”. Not incidentally to all this, he also shows that in finite dimensions, at least, a -algebra is “the same thing as” a -Frobenius algebra.

Benjamin Bahr gave another talk dealing with categorical issues - namely, how to get measures on certain groupoids, such as, indeed, the groupoid of connections on a manifold. In fact, he treated various cases under the same framework: flat and non-flat connections, on manifolds and on graphs - and others.

In all, I was pleasantly surprised by the mix of the physically and mathematically inclined points of view, and the trip itself was a lot of fun.

 

Two Kinds of Categorification And Other Stuff

I’d just like to post something about a conceptual clarification that came up recently. Last week I gave the first of a couple of talks in the Algebra seminar in our department, about the ideas of structure types and stuff types, more or less as outlined in this paper which I put out a couple of years ago. It summarizes and traipses a little way beyond the matter of the 2003/2004 quantum gravity seminar at UCR, whence on this paper by John Baez and Jim Dolan, and even further back on work by André Joyal, particularly in the paper “Foncteurs analytiques et espèces de structures“, which regrettably doesn’t seem to be available either online. (I gave a blackboard version of the talk, but it was an expanded form of this one hour version.)

(Semantic side note: these espèces de structures are often referred to as “combinatorial species” in English. This is the more common translation than “structure type”, but unfortunately, it doesn’t capture the modifier “de structures“, instead choosing the more generic “combinatorial”, which makes it hard to distinguish “structure types” from “stuff types” in the Baez-Dolan sense. Also, “species” is probably over-specific as a translation of “espèces” in a way that “type” isn’t. The generic sense of “species” as “a kind of” in English is a bit recherché.)

In any case, what I’m interested in this post is the sense in which stuff types give a “categorification” of a vector space. In a nutshell, a stuff type is a groupoid over (the groupoid whose objects are finite sets, and whose morphisms are bijections). That is, it’s really a functor , which we call the “underlying set” functor. For example, consider the groupoid of all binary trees, where the underlying set is the set of nodes (or, a different example, the set of leaves). Any isomorphism between two such trees gives a bijection between the underlying sets, so this actually is a functor. Or one could take the functor , where the “underlying set” of a pair of sets is just , and likewise for morphisms. (Notice that different bijections “up above” in the bundle may give the same bijection “below” - in cases where this doesn’t happen, we have one of Joyal’s “structure types”). In some ways, it’s better to think of it as a bundle of groupoids - one fibre over each object in

The thing is, that map gives an invariant for objects in the category of groupoids, but not a complete invariant. Unlike, say, finite sets and the natural numbers. Natural numbers correspond exactly to isomorphism classes of sets - not so with groupoid cardinalities. So there’s an equivalence relation, and reducing the object set modulo that equivalence relation gives a structure - but it’s not the minimal throwing-away of information about objects that taking isomorphism classes would be.

But in any case, it’s the whole category of groupoids (over ) which gets “decategorified” down to a vector space, in that world. There is a concept of groupoid cardinality, which is given by Baez and Dolan in the paper above, and which is also linked to Tom Leinster’s definition of the Euler characteristic of a category. This adds up, over all the isomorphism classes of objects, , the reciprocals of the sizes of automorphism groups. Reasons why this is the nicest concept of cardinality are described in some of those references, but all that really matters here is that groupoid cardinality gets along with disjoint unions of groupoids (corresponding to sums of cardinalitys), and products of groupoids (which get the product of the two cardinalities). That is, the categorical coproduct and product, respectively, define operations on the set of cardinalities!

In particular, taking stuff types - groupoids over , we can take the cardinalities of the fibres over sets of each size giving the coordinate in a vector. So then is, the slice category has this “cardinality” on objects into a set, and the structure of the category gives well-defined operations on this set, turning it into a vector space. In fact, there’s an operation (weak pullback) which makes it an inner product space. (To make this work in complex cardinalities takes some fudging with phases in , but it can be done.)

The details are interesting, and I’m coming back to looking at some of this again, but what I want to point out at the moment is a more fundamental point, which has to do with the offhanded use of the handy, but imprecise, term “categorify”. With the category of (-) stuff types, we have a category with a “decategorification” map that compresses it into a vector space. This sure sounds like a “categorified vector space”. In fact, this seems to be what people who hear the term “categorification” often want it to mean: I look for a categorification of mathematical object X by finding a category which, secretly, looks like X.

The problem is, there’s another concept attached to the phrase “categorified vector space”, namely that of 2-vector space in the sense of Kapranov and Voevodski, as discussed, say, here. There’s a different level of abstraction at work here. The specific category of stuff types provides a categorification (if that indeed is the right word to use) of a specific vector space. The concept of a KV 2-vector space categorifies the concept of a regular vector space in a particular way: putting “additive” structure on objects, and “C-linear” structure on morphisms. (The Baez-Crans version does the same job in a different way).

You don’t think of a specific KV 2-vector space “decategorifying to” a specific vector space. Indeed, just taking the “minimal” equivalence relation - isomorphism classes of objects - what we get from a KV 2-vector space is more like an -module (over a rig, not a ring). Basically, 2-vectors have components which are vector spaces, and therefore classified by their dimension. The relationship between THIS kind of 2-vector space and the non-categorified concept is that real vector spaces show up as the -sets in a KV 2-vector space.

Elucidating exactly what’s going on with these two forms of categorification would be nice - perhaps somebody’s done it, but if so, I don’t know who. I also don’t know any nice conditions that tell you when you have a “category that can be mistaken for a vector space”, like stuff types: a good characterization of these things would be nice. Or again: both versions of “categorification” of vector space have special relationships to groupoids - but of two very different natures (in one, the groupoids can be interpreted as 2-vectors - in the other, there are whole 2-vector spaces associated to groupoids). Just a coincidence?

Another possibility that comes to mind would be to form some kind of hybrid structure - where the “vector spaces” which show up in the -sets in a KV 2-v.s. are secretly this fake-vector space type of category. Since both types seem to have physics-y ambitions, such a setup that combines both approaches is appealing, rather than a muddled and confusing competition for the term “categorification”.

I don’t have a good ending to this story, which is why this is a blog, not a book.

 

Comments on Gfest 07 - Groupoids and C*-algebras

So I’ve posted some slides from my talk at Groupoidfest. I also gave this talk here at Western in the Algebra seminar on Wednesday. It seemed to go over fairly well, although it was a bit of an outlier for the conference. However, I’m getting used to that consequence of trying to talk to both physicists and mathematicians. Anyway, after I got back from GFest (as they call it), it took me a few days to get caught up on lecturing and grading and so forth, but here are some slightly belated comments on what went on there. A lot of the content of the talks went over my head, as happens. However, at lunch of the first day, Arlan Ramsay gave me and a couple other beginning researchers some good advice about learning at conferences where you only grasp about 10% of what’s going on: be like a baby learning to walk. Don’t be afraid of looking stupid - just grab the 10% you understand, and then do it again. (Since I spent part the weekend watching a baby learn to walk, this was quite apropos).

So this I’ll comment a bit on some of the general themes I did manage to pick up, and in a subsequent post I may say more about some of the talks that seemed particularly relevant and/or comprehensible to me.GFest was held at the University of Iowa, in Iowa City - by happenstance, a friend from UCR, Erin Pearse, recently started there as a VIGRE postdoc, so I managed to stay with him and his family while I was in town, which was good. I was a little surprised at first that he was interested in sitting in on the talks at the conference, since his research is mostly in fractal geometry, and I didn’t initally see the relevance. However, I guess it shouldn’t have been too surprising, since part of the great thing about groupoids is their ability to represent symmetry. The kinds of fractals in question are the self-similar kind, which have various interesting types of symmetry.

In particular, Erin explained to me that the connection has something to do with shift operators. These operators, which shift a sequence of numbers and insert a new value in it, can be used iteratively to build up, for instance, the Cantor set. (Which is a set of sequences of 0’s and 2’s in ternary notation - the shift operators take you from a point in the whole set, to a point in one of its pieces, which resemble the whole.)

This was one reflection of a more general theme: since there’s a Hilbert space of sequences, namely , the shift operators can be taken as operators on a Hilbert space. So in particular, they generate an algebra of operators - a -algebra (see also some notes). The general theme is that most of the people at the GFest were interested in groupoids as a way of saying something about -algebras. I probably heard this term bandied about more than the actual term “groupoid” while I was there.

One reason my point of view was an outlier is that I was talking about finite, topologically discrete groupoids. However, this is kind of beside the point, since I’m really more interested in ones that come from Lie groups, and have some interesting topology. But I avoid getting into that so far because I’ve been postponing extending this stuff to smooth groupoids, since that leads to infinite-dimensional 2-Hilbert spaces, and gets more complicated than what I’ve been talking about so far. The theory of these does exist - Crane and Yetter develop a lot of the theory needed under the aegis of “measurable categories” - but it involves a lot more analysis.

In fact, while I’m used to thinking of groupoids as a special kind of category, a lot of the talk about them at GFest emphasized exactly this analysis a lot more. It seems to be bread-and-butter for people who work with groupoids arising in -algebras. Paul Muhly, who organized the conference, kindly gave me the current working draft of a book he’s writing on this stuff, where a lot of the important ideas people were using are collected together and explained. (Note that I’ve only started reading it, so I may be mistaking things here).

One point seems to be that these algebras coming from groupoids are related to the -algebras coming from transformation groups: situations where a (locally compact topological) group acts on a (locally compact Hausdorff) space . These can automatically be thought of as groupoids, taking objects to be points in the space, and morphisms from to to be group elements whose action takes to . Now as for -algebras, you can build them by taking algebras of compactly supported complex functions on . This becomes an algebra with the convolution product, given by integrating over the group (so we’re assuming has a nice invariant measure like Haar measure on a Lie group):

and the “star” operation is just complex conjugation.

You can do something similar for groupoids generally, since groupoids decompose into isomorphism classes, each of which looks just like a set with the action of some particular group on it. For this to really make sense, you must be talking about topological groupoids. Here, they think of groupoids as a set of all morphisms, with being the set of composable pairs. Given a topology on , this gets the subspace topology on the product. This is making use of the fact that objects of the groupoid needn’t be defined separately - they correspond to the “identity” morphisms (with ), which again gets the subspace topology automatically (which makes source and target maps continuous).

Then we’d like to again define a -algebra on using something like the above definition. But then we need to define a convolution product, and for that, we needed a Haar measure on the group. Fortunately, for topologically reasonable groups, you’re guaranteed to have one, and it’s unique (maybe up to a scalar multiple); unfortunately, you don’t have either existence or uniqueness guaranteed for groupoids. So instead you need to have a Haar system.

This is a family of measures on (the set of all morphisms), one for each object: , which we’ll use to do convolution at the which correspond to the object . The measure is supported on the component of the object . The whole system needs to have some nice properties. One is that for any function , the function taking to the integral of with respect to should be in 9. The other is that is equivariant, in the sense that if ,

(shifting which measure we use by is the same as shifting the function by ).

This is a bit obscure to me at the moment, but it’s clear enough that you need some family of measures to define a convolution. The first property just ensures that the algebra is closed under this product. The second is just the kind of property you should expect from groupoids: if you’ve defined something that’s not equivariant, you’re just asking for aggravation. So then finally, making a bunch of assumptions, such as that is locally compact, Hausdorff, and so on, we get , the set of smooth, compactly supported complex functions with a convolution product:

(where is the source object of the morphism ). The star operation is still complex conjugation.

So, while I’m running a bit long here, this is the basic setup behind most of what people were talking about at Groupoidfest. Either studying these -algebras in their own right, or using groupoids to think of already existing algebras as coming from this setup for some groupoid . The point, I suppose, is that representations of these algebras, and of the groupoids they come from, are closely related, just as representations of groups and their group algebras are.

This subject - representations of groupoids, is exactly what my talk was about, except that I ignored all the topology to simplify certain things. Right after my talk, Marius Ionescu gave one about irreducible representations of groupoid -algebras, which I’m trying to get up to speed on to see how these things are done in the case with more interesting topology. (For my purposes, it’ll also be necessary to understand infinite-dimensional 2-Hilbert spaces better, but that’s another story…) Maybe when I see what that’s about, I’ll say something further on that subject.

There were a number of other good talks - perhaps soon I’ll see if I can summarize what I gathered from some of them.

 

Spans and Vector Spaces - pt 2

In the last post, I was describing how you can represent spans of sets using vector spaces and linear maps, which turn out to be fairly special, in that they’re given by integer matrices in the obvious basis. Next I’d like to say a little about what happens if you step up one categorical level. This is something I gave a little talk on to our group at UWO on Wednesday, and will continue with next Wednesday. Here I’ll give a record of part of it.

Once again, part of the point here is that categories of spans are symmetric monoidal categories with duals - like categories of cobordisms (which can be interpreted as “pieces of spacetime” in a sufficienly loose sense), and also like categories of quantum processes (that is, whose objects are Hilbert spaces of states, and whose morphisms are linear maps - processes taking states to states).

So first, what do I mean by “move up a categorical level”?

We were talking about spans of, say, sets, like this: . To go up a categorical level, we can talk about spans of categories. The objects and now carry some extra information - they’re not just collections of elements, but they also tell us about how elements are related to each other. So then remember that spans of sets really want to form a bicategory, which we can cut down to a category by only thinking of them up to isomorphism. Well, likewise, spans of categories probably want to form a tricategory, which we can cut down to a bicategory in the same way. (Several people have studied them, but the only person I know who really seems to grok tricategories is Nick Gurski, though in this talk he tried to convince us that we all could have invented them ourselves. ) Before rushing off into realms involving the word “terrifying”, we should start by looking at what happens at the level of objects.

But first, why should we bother? Well, there’s a physical motivation: building vector spaces from sets plays a role in quantizing physical theories, where the sets are sets of classical states for some system. That is, when you quantize the system, you allow it to have states which are linear combinations - superpositions - of classical states. But saying you have just a set of states is limiting even in the classical situation. Sometimes - for instance, in gauge theory - there are actually lots of “configurations” of a system that are physically indistinguishable (because of some symmetry, which in that example is achieved by “gauge equivalence”), and so what’s usually done is to just look at the set of equivalence classes of configurations. But that throws away information we may want: it’s better to just take a category whose objects are states, and whose morphisms are the symmetries of the states.

For these to really be symmetries, they should be invertible, so we’re looking at a groupoid of states, . But then to quantize things, we can’t just take a vector space of all functions - as we did when was a set. Now we need to have something collecting together all the functors out of . These certainly form a category, so we want some kind of category which is “like” a vector space. By default it’s called a 2-vector space, since it now has an extra level of structure.

As I said before, this stuff isn’t so hard if you’re willing to ignore details until needed - so for now, I’ll just say that (Kapranov-Voevodsky) 2-vector spaces are categories which resemble , just as (finite-dimensional) vector spaces are sets resembling , for some . And just as the set of functions becomes a vector space, so does the category of functors become a 2-vector space when is a groupoid. (Josep Elgueta discusses in some depth what happens for a general category in this paper.)

What makes a groupoid special is that the two layers - objects and morphisms - both get along nicely with the operation of taking functors into . That is, it’s easy to describe such functors. It’s a little easier to talk about it for a skeletal groupoid: one with just one object in each isomorphism class. Fortunately, every groupoid is equivalent to one like this. So since I’ve figured out how to do pictures here, let’s see one of a functor :

This is one particular 2-vector in the 2-vector space I’m building. The picture is showing the following: the objects have groups of automorphisms, , indicated by the curved arrows. A functor assigns, to each object , a vector space (sketched roughly as squares), and for each automorphism of that object , a linear map . Since is a functor, these linear maps are chosen so that - so this is a -action on . In other words, for each , we have a representation of its automorphism group on the vector space .

A morphism between two such 2-vectors is a natural transformation of functors - for each , a linear map satisfying the usual naturality condition. As you might expect, this condition means that gives, for each , an intertwining operator between the two representations and . So it turns out that the 2-vector space is a product, taken over the objects , of the categories .

In particular, that if is just a set, thought of as a groupoid with only identity morphisms, then this is just , since any vector space is automatically a representation of the trivial group, and any linear map is an intertwining operator between such trivial representations.

Now, proving that this is a 2-vector space would involve giving a lot more details about what that actually means - and would involve some facts about representation theory, such as Schur’s Lemma - but at least we have some idea what the 2-vector space on a groupoid looks like.

Next up (pt 3): what about spans? What happened to spans, anyway? There was supposed to be an earth-shattering fact about spans! Then, that done, hopefully I’ll get back to looking at the physical interpretation of an extended TQFT.

 

TQFT and Gravity - pt 2

So last time I was describing this “matter without matter” idea and claiming that it has something to do with TQFT and the Ponzano-Regge model of quantum gravity. I’d like to get a little more detailed here.

To describe this in physics terms, it’s easiest to understand the point if, instead of using the (more technically accurate) terms “manifold”, “cobordism between manfolds”, and “cobordism with corners between cobordisms, I name-drop the terms “boundary”, “space”, and “spacetime”. But the caveat here is that these terms really imply a certain geometric structure which I’m not actually assuming is there: a specific geometric structure on these manifolds is a state of the theory. Furthermore, with Ponzano-Regge, we’re talking about Riemannian gravity - there’s no such thing as a “timelike” direction. So using the term “spacetime” is being rather optimistic that everything will work out in more physical settings - but it’s a helpful motivation.

At any rate, the way I describe it in the thesis, in dimensions the typical setup for an extended TQFT in the sense of a weak 2-functor into 2-Vect, one has “boundaries”, which are manifolds of dimension (in 3D, each boundary is some union of a bunch of circles, and in 4D it would be a union of surfaces, each with some genus). These are joined by “spaces” (cobordisms), of dimensions, which are in turn connected by “spacetimes” (with the above caveat). These cobordisms are, in particular, cospans in some category of spaces, and they give rise to spans of groupoids of configurations for a gauge theory.

In any case, how does this relate to gravity? The answer is by way of topological gauge theory: the extended TQFT in question has a lot to do with flat connections on manifolds (or indeed manifolds with boundary or corners), which is what topological gauge theory is about. One way to say what a flat connection is, is to say that it takes a path in the space , and gives an element of the gauge group (this is not the most well-known way to describe a flat connection - more on that in another post, but I’ll cite weeks 8 and 9 of the spring 2005 UCR Quantum Gravity Seminar for now).

If the gauge group represents the symmetries of something we’re transporting around the surface, this tells us how that thing is being transformed as we move it. For gravity, we take the gauge group to be the symmetries of a model spacetime - what spacetime “looks like locally”. For standard special relativity, this is the Lorentz group - the symmetries of Minkowski space. For 3D gravity, it’s (symmetries of Minkowski space with two space and one time dimension). For 3D Riemannian gravity, it’s the group of rotations in 3D. Actually, I lied: each of these has a double cover, and this is the gauge group (which allows for a spin structure. To simplify a lot of things in my thesis, I talk about the case where is some finite group, but eventually I’d like it to be , the double cover of the rotation group .

So we imagine the connection tells us how an observer would be rotated by the act of moving along a path. (There is a kind of trivialization of a bundle lurking behind this glib statement, but I’m putting that off). Now, some connections are physically the same, even though we describe them differently. They are related by gauge transformations, which are symmetries of the connections themselves. These amount to a way of changing the coordinate system in which we describe (say) our rotation: two rotations of 60 degrees around different axes are not “really” different, since the observer can turn one into the other by tilting her head. What’s traditionally done is to “mod out” by gauge transformations: take any two connections related in this way to be just the same, and throw away any information that distinguishes them. Instead, we can organize flat connections into a category - in fact, a groupoid - where the objects are the connections, and the morphisms are the gauge transformations. We can organize this into the category of functors from the fundamental groupoid of a manifold into the gauge group (thought of as a one-object category).

What’s the point - from a physical point of view - of keeping all the extra structure of these morphisms? To make a long story short, they’re what ends up allowing the theory to classify particles as having spins, not just masses. (Incidentally, I notice that Marni Sheppeard made a guest post on another blog arguing that category theory is useful to physics. Here is another example of how this can be so. Morphisms encode information that would be absent without them, and which has a straightforward physical meaning.)

How does this extra information appear? Well, first of all, what is a point particle, in this model? It’s represented as a boundary around a puncture in “space” - a circular boundary in a 2D surface of some shape or other. The fundamental groupoid of the circle has objects which are points of the circle, and morphisms which are (homotopy classes of) paths. There is an equivalence of categories between this and the fundmental group of the circle, which we can think of as a category with just one object (this is because the circle is a connected space).

Then we’re looking at a category of functors between a couple of one-object categories. Since , these are determined by the image of the generating path, “1″. So the groupoid of flat connections on this boundary has objects which correpond just to elements of . But wait! There’s more! You also get natural transformations between these functors! These amount to just conjugations relating elements of (those “coordinate transformations” I mentioned before). So the whole groupoid has objects corresponding to elements of , and morphisms for each such that . We call this whole groupoid by the name - or “ weakly modulo the adjoint action of .

This is also equivalent (as a category) to a smaller category I’ll call - the “skeleton” of , namely, a category with one object for each isomorphism class of objects in (i.e. each conjugacy class in ). Each of these has a group (the original category was a groupoid, so the new one is also) of automorphisms. This will be the same as the group of automorphisms of the corresponding object in - namely, the stabilizer subgroup of that element of , which, if is generically , except for a couple of exceptional points corresponding to 0-degree and 360-degree rotations.

Finally, a 2-vector in the 2-vector space assigned to the circle (which I like to think of as a “2-state”) is a functor from this into . Each such functor is a direct sum of a bunch of irreducible ones, and the irreducible ones assign a nontrivial vector space to just one object - and the group of automorphisms of that object are taken to a group of automorphisms of . That is, is specified by a conjugacy class of , and a representation of its stablizer subgroup. If , this is an angle and a spin. And in 3D gravity, the mass of a particle corresponds to an angle, because Einstein’s equation here says that space is locally flat, except where there is matter - where there is an amount of curvature proportional to the mass. This shows up as an “angle deficit” - an amount by which you end up rotated if you travel around the particle.

So that’s how you can see a “hole” in “space” as a point particle with mass and spin in this kind of extended TQFT. In higher dimensions, something similar happens, but the classification is more complicated, because in general the matter looks like “stringy” loops (this is something Derek Wise has looked at in his thesis). Also, above 3D, a theory of flat connections is no longer a theory of gravity, but rather something called BF theory - although in 4D it happens to be a limit of the theory of gravity as you allow Newton’s constant to approach zero. (That is, it describes the topological sector of the theory of gravity.)

What I haven’t yet explained is how this matter, which so far has the properties we might hope for, also gets to live in a spacetime governed by the Ponzano-Regge model. That means looking at what the extended TQFT does to the morphisms and 2-morphisms of the cobordism category - to “space” and to “spacetime”, and what the “2-linear maps” and “transformations” they give are like. Tune in next installment…

 

Coming Attractions

Due to the rapid-fire the nature of the blogosphere (or, in deference to John Armstrong, the “Blathysphere”, or maybe “blathyscape”), my blog (”blath”) has been discovered before I expected, and in particular before I’ve had the chance to put anything very interesting in it. So here I’ll just say something about “coming attractions” - a sort of mid-level executive summary of the next batch of things I expect to be working and commenting on. Also possibly later on I should have a math post or two about some talks I saw recently.

Since I graduated at UCR in June, I haven’t had much chance to do any actual work - partly because I broke my wrist in a bike accident, and lost the use of my writing hand for six weeks. Between that and the hassle of moving, I wasn’t able to do much but some reading. Now that the cast is off, I’ve been getting back to work. The first “real” research-related post I expect to make will be an announcement that a (slightly) polished version of my dissertation, “Extended TQFT’s and Quantum Gravity” has been released on the preprint archive - hopefully this week. That in turn should kick off some descriptions of what’s inside as I get more into the process of turning it into some smaller, more digestible papers.

These will fall, at first, into three parts:

1) A paper which has already been posted as math.CT/0611930, describing how to get a “double bicategory” of cobordisms with corners, and from that, a bicategory. Here I explain how cobordisms are cospans of manifolds with boundary, so the new structures are double cospans of manifolds with corners, and how that works.

This may end up being two parts. One is a decription of Dominic Verity’s notion of a “double bicategory”, an aside on how to interpret it as a special case of bicategories internal to , and how to get one from double spans (functors ). Marco Grandis has a pretty thorough description of these in this paper and its sequels, although our approaches are slightly different.

The second part has to do with how to apply this to cobordisms with corners (cobordisms between cobordisms) - also something Grandis discusses in the second paper of that series. I also need to show how to collapse the more complicated structure to a mere bicategory, in order to do what I will want to do in part (3) below.

There’s an issue here I’ll want to think about at some point, related to a question Aaron Lauda raised. The question was this. The category whose objects are 1-D manifolds and whose morphisms are 2D cobordisms between them has a nice abstract description. It is the free symmetric monoidal category with a Frobenius object.

In Aaron’s work with Hendryk Pfeiffer, they likewise described a category of “open closed strings”, which can have either 1-D manifolds or 1-D manifolds with boundary (collections of circles and line segments, basically) as objects, and cobordisms between them as morphisms. They showed this has a similar characterization, but with “Knowledgeable Frobenius” replacing “Frobenius” in the above. These have a nice description in terms of adjunctions, so Aaron was asking me if the same could be done for the double bicategory I talk about. That would need a concept of adjunction in double categories (or cubical n-categories, more generally). I don’t know what the state of understanding is on this.

More generally, it’s strange that “cobordisms of cobordisms” really wants to be a cubical 2-category in some sense, whereas, to do what I want to do with them (see below), I have to convert them into a globular one, to take functors into . I don’t know the best way to deal with this: is there a cubical version of , for example?

2) One part will deal with building 2-vector spaces from groupoids using functors into the category ; and 2-linear maps from spans of groupoids, using the pullback (composition) along an inclusion, and its (two-sided) adjoint. Along the way, it includes some proofs of well-known folklore theorems about 2-vector spaces which are hard to find anywhere. I plan to give a talk based on this at Groupoidfest ‘07 in Iowa City in November.

Soon enough - certainly before the Groupoidfest, I’ll have a bigger post about this stuff (and most likely post slides). The basic idea is that the category of functors from an essentially finite groupoid into is a Kapranov-Voevodsky 2-vector space - that is, a $\mathbbm{C}$-linear additive category which is generated by a finite number of simple objects. (The fact that this definition is equivalent to the one given by Kapranov and Voevodsky is one of those theorems which seems to be well known, but hard to track down). The finite number of simple objects correspond to the equivalence classes of . From a span of groupoids, it is possible to build a linear map between the corresponding 2-vector spaces.

The motivation for building 2-vector spaces on groupoids in the new work is to categorify the quantization of a classical system, but the two ways I’ve looked at are a bit different in how they accomplish it. Ignoring complications like symplectic geometry for the moment, the configuration space of a classical system is described as a set . Each element of the set is one possible state of the system. The corresponding quantum system will have states which live in - in particular, they are complex-valued functions on the set . And instead of being able to read off values like position, momentum, energy, and other features of the system by looking at the value these have at a single point, you need some algebra of operators on , whose eigenvalues are the values you can observe for the observable that corresponds to a given operator. In categorifying this, becomes a groupoid, in which the elements of the set can be related to each other - by “symmetries”. Instead of functions into the complex numbers, we take functors into , and obtain a 2-vector space of what I suppose should be called “2-states”. Given spans of groupoids, it becomes possible to get linear maps from one 2-vector space to another, using “pullback” and “pushforward” of these functors into .

I’ll say more about this later on, but one thing that I find perplexing about this is how (if at all), it relates to some earlier work I did in this paper on the categorified harmonic oscillator, which is heavily based on this paper by John Baez and Jim Dolan, which introduces “stuff types”. Both involve groupoids, and spans of groupoids giving rise to linear operators, as part of a categorification of some elementary quantum theory, but there are significant differences. At some point, I’d like to return to the question of whether they’re related, and if so, how.

3) One part uses the above to build an “extended TQFT”. A TQFT, or topological quantum field theory is a quantum field theory, in that it gives a Hilbert space of states for some field on a specifed “space” (i.e. manifold), and linear maps associated to “spacetimes” (cobordisms) joining them. It is topological, in that its states are topologically invariant - that is, they have no local degrees of freedom, only global ones. These started life in physics, but have fallen by the wayside there, and now mostly find life in the subject of quantum topology, where they give manifold invariants.

A TQFT can be described as a functor from a category of manifolds and cobordisms (see (1)) into . This way of putting it makes it relatively easy to see what to do if one wants to categorify - which we do, in order to get higher codimension (more on this later, I’m sure). The idea is to build a 2-functor from the bicategory of cobordisms with corners (see (1)) into . This can be done using gauge theory. The main idea is to turn a cobordism, seen as a cospan of manifolds (with corners) into a span of groupoids - namely, the groupoids of flat connections on these spaces, with gauge transformations as morphisms, and then build 2-vector spaces and 2-linear maps, etc. as laid out in the program of (2) above. The main theorem proving that such a 2-functor exists and is given by this construction was the organizing theme of my dissertation defense talk. This part is the mathematical core of what I’ve been working on.

4) Finally, this is supposed to be related to quantum gravity somehow. I’ll put off talking about this until I actually put the thesis on the archive.

Until then, I may decide to post a little about some talks I’ve been to recently. UWO has a great department with lots of interesting talks. I recently attended a couple of these by graduate students. One was by Arash Pourkia, about Braided Categories and Hopf Algebras. The second was by Michael Misamore, on Galois Theory - from the point of view of Grothendieck, and could equally well be called “Covering Spaces”… from the point of view of Grothendieck.