Theoretical Atlas

Well, I was out of town for a weekend, and then had a miserable cold that went away but only after sleeping about 4 extra hours per day for a few days. So it’s been a while since I continued the story here.

To recap: I first explained how to turn a span of sets into a linear operator between the free vector spaces on those sets. Then I described the “free” 2-vector space on a groupoid - namely, the category of functors from to . So now the problem is to describe how to turn a span of groupoids into a 2-linear map. Here’s a span of groupoids:

Here we have a span , of groupoids. In fact, they’re skeletal groupoids: there’s only one object in each isomorphism class, so they’re completely described, up to isomorphism, by the automorphism groups of each object. The object , for instance, has automorphism group , and the object has automorphism group . This diagram shows the object maps of the “source” and “target” functors and explicitly, but note that with each arrow indicated in the diagram, there is a group homomorphism. So, since the object map for sends to , that strand must be labelled with a group homomorphism . (We’re leaving these out of the diagram for clarity).

So, we want to know how to transport a -valued functor - along this span. We know that such a functor attaches to each a representation of on some vector space . As with spans of sets, the first stage is easy: we have the composable pair of functors , so “pulling back” to gives .

What about the other leg of the span? Remember back in Part 1 what happened when we pushed down a function (not a functor) along the second leg of a span. To find the value of the pushed-forward function on an element , we took a sum of the complex values on every element of the preimage . For vector-space-valued functors, we expect to use a direct sum of some terms. Since we’re dealing with functors, things are a little more complex than before, but there should still be a contribution from each object in the preimage (or, if we’re not talking about skeletal groupoids, the essential preimage) of the object we look at.

However, we have to deal with the fact that there are morphisms. Instead of adding scalars, we have to combine vector spaces using the fact that they are given as representation spaces for some particular groups.

To see what needs to be done, consider the situation of groupoids with just one object, so the only important information is the homomorphism of groups. These can be seen as one-object groupoids, which we can just call and . A functor between them is given by the single group homomorphism .

Now suppose we have a representation of the group on (so that and ). Then somehow we need to get a representation of which is “induced” by the homomorphism , :

This diagram shows “the answer” - but how does it work? Essentially, we use the fact that there’s a nice, convenient representation of any group , namely the regular representation of on the group algebra . Elements of are just complex linear combinations of elemenst of , which are acted on by by left multiplication. The group also has regular representation, on . These are the most easily available building blocks with which to build the “push-forward” of onto .

To see how, we use the fact that has a right-action of , and hence , by way of . An element acts on by right-multiplication by - and this extends linearly to . So we can combine this with the left action of on (also extended linearly from ) by taking a tensor product of with over . This lets us “mod out” by the actions of which are not detected in . The result, called the induced representation of , in turn gives us back a left-action of on . I’ll call this .

(Note that usually this name refers to the situation where is a subgroup of , but in fact this can be defined for any homomorphism.)

This tells us what to do for single-object groupoids. As we remarked earlier, if more than one object is sent to the same , we should get a direct sum of all their contributions. So I want to describe the 2-linear map, which I’ll now call which we get from the span above, thought of as in . Here and (where I’m now being more explicit that this whole process is a functor in some reasonable sense).

I have to say what does to a given 2-vector (what it does to morphisms between 2-vectors is straightforward to work out, since every operation we do is a tensor product or direct sum). Suppose we have is one. Then . We can now say what this works out to. At some object , we get (still assuming everything is skeletal for simplicity):

And this is a direct sum of a bunch of such expressions where is a basis 2-vector - i.e. assigns an irreducible representation to some one object, and the trivial rep on the zero vector space to every other. That allows this to be written as a matrix with vector-space components, just like any 2-linear map.

So the 2-linear map has a matrix representation. The indices of the matrix are the simple objects in and , which consist of a choice of (a) object in or (which we assume are skeletal - otherwise it’s a choice of isomorphism class), and (b) irreducible representation of the automorphism group of that object. Given a choice of index on each side, the corresponding coefficient in the matrix is a vector space. Namely the direct sum, over all the objects that restrict down to our chosen pair, of a bunch of terms like . This is just a quotient space of the one group algebra by the image of the other.

Next up: a quick finisher about what happens at the 2-morphism level, then back to TQFT and gravity!

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.

In the last couple of posts, I described how an extended TQFT gives a 2-vector space, with generators corresponding to particular states of matter, for each boundary of space (mostly talking about 1-D uboundaries of 2D space in 3D spacetime). I was starting to build up to talking about how cobordisms give rise to “spin network states” on space with given boundary conditions. Before I can do that, it’s probably helpful to talk about something a little more general. Since the general thing in question is something I’m developing a talk on to give in Iowa, this is helpful for me anyway.

A slightly more general thing has to do with spans of groupoids, and how to get 2-linear maps from them. A span in a category is a diagram like this:

Now, as for spans, let me first give a couple of link-outs (the blathyspherian version of a shout-out) to a couple of guys named John… Given a category with pullbacks, there is a (bi)category , where spans are composed using pullbacks. John Armstrong recently posted about spans, describing , which has the same objects as , and morphisms which are spans in .

In fact, it also has 2-morphisms, which are span maps - given two spans with central objects and , a span map is a map from to which makes the resulting diagram commute. It turns out these make into a bicategory - one of the classic examples, in fact, which goes back to Jean Benabou’s “Introduction to Bicategories” (1967) in which the concept was introduced. However, one can ignore these, and just think of it as a category, by taking spans only up to isomorphism.

John Baez recently posted some slides for a talk about spans in quantum mechanics, which gives a nice overview of the context that makes this stuff relevant to this discussion of TQFT. A key concept is summarized in the abstract:

Many features of quantum theory — quantum teleportation, violations of Bell’s inequality, the no-cloning theorem and so on — become less puzzling when we realize that quantum processes more closely resemble pieces of spacetime than functions between sets.

And the point both of them make is that cobordisms can be seen as spans (actually, cospans, although a cospan in is by definition a span in ). This is an important idea when thinking of TQFTs as functors, since and (or ) are symmetric monoidal categories with duals. A TQFT is a functor , which respects exactly this structure. So it’s important that quantum processes are “like” these “pieces of spacetime”. And “pieces of spacetime” (cobordisms) have these properties is that, any time you start off with a cartesian category with pullbacks, like , then taking spans in it gives you a symmetric monoidal category with duals.

What we’re really talking about are properties of (a) spans, and (b) certain free functors. In particular, free functors taking sets to vector spaces, groupoids to 2-vector spaces, and (potentially) so on. Both of these have something to do with how to go from a cartesian category like , or (really a 2-category), to a monoidal category with duals (”dagger compact”), like , or (also a 2-category) - but also like or … I’ll describe what happens for sets, to keep things simple for this installment.

One example of going from a cartesian category to a dagger compact one is by the “free vector space” functor , taking a set to , the free vector space on , and set maps to linear maps that just permute basis elements. Another is the process of taking and building $\mathbf{Span(C)}$. The point is that these two can be related in a rather interesting way. In particular, there’s a functor

which acts on the objects of $\mathbf{Span(Sets)}$ (which are sets) just like the free-vector-space functor. That is, given a set , it gives , the space of functions from into . (For simplicity, I’ll assume all my sets are finite).

But it does something rather special on morphisms in . These are spans of sets and therefore they have two morphisms in them. If we think of the span as a morphism in , then the two arrows in the span are distinguished as first a “backwards” arrow, then a “forwards” arrow. The point is to take a vector in - a complex-valued function on , through the span.

So the question is, if I have a complex-valued function , how do I get a complex-valued function on ? Well, first, of course, I have to get one on . Since I have a function , the obvious candidate is . Each element of just gets the same complex number as its image down in . That’s easy: we’ve “pulled back” the function along .

Now we have to transport this function down to , which is a little less obvious. A given object in may have several different objects in which map down to it, and no reason why they should all have the same function value under . What can we do with a bunch of complex numbers? The two things which are most obvious are: add them up, or multiply them. The one we pick is to add them up (it may help to remember that the preimage of some object in is the union, or coproduct, of a bunch of elements - and coproducts are like sums, just as products are like… well… products). The result is that we’ve “pushed forward” the function along , and the result is called .

How do I know the process of taking a function - that is, a vector in , and finding the vector in is a linear map? Well, it’s not too hard to check that it’s represented by a matrix, and the summation over the preimage of an object in was the sum in the matrix multiplication. (Go ahead!) This works out very nicely because is cartesian, so any span between and factors through the product . In fact, corresponds to an integer matrix, whose component is the number of elements of that project down to both and . (To get a general matrix, you’d have to give labels to the elements of , which is something I talk about in this paper - the thing I like about which is that it gives lots of pictures which make “matrix mechanics” seem pretty natural - to me, anyway.)

It turns out this gives you a functor which represents inside $\latex \mathbf{Vect}$. In fact, to really get the bigger picture, instead of in everything I’ve said here, you should replace , and for you should replace . I’ll say something about that in the next installment - but “morally speaking” it’s much the same as what I’ve talked about here.

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…

With my thesis available on the arxiv, I thought I should see what I can say about the, as it were, dangling participle of that particular snapshot of this research project. That is, back when I had to declare a title for the thing, quite a long while before I had to finish it, I called it “Extended TQFT’s and Quantum Gravity”, thinking that this would be an accurate title, because it pretty well described the subject of the weekly conversations I’d been having with John while working on it.

However, one thing that gradually becomes clearer as I go further into the process of research is that it’s hard to predict exactly what that process is going to produce. (”Prediction is hard - especially when it comes to the future”, as Yogi Berra said - though possibly it was someone else, since accurate information about the past doesn’t exactly grow on trees either). It turned out that a lot of what I really did was proving some well known folklore theorems about 2-vector spaces; spending a few weeks trying to get a good proof that the weak 2-functor I constructed was actually a weak 2-functor (I still have a kind of unenlightening calculation for a proof); and lots of similarly technical stuff. All of which is - I hope - good mathematics, or at least correct mathematics. But is it physics?  All the references to the physical applications were left to the last section, a kind of sketch of where I expect the project to go.

I think the project does indeed have some nice intimate relations to quantum gravity (at least in 3 dimensions), it just didn’t turn out that there was a lot of material about those relations in the document. Instead, there’s a rather impressionistic sketch of how it ought to work. But you might not get the impression that Derek Wise and I started off working on the same project, though we did. Derek’s thesis is not available online in its entirety yet (though part of it appears in this paper on MacDowell-Mansouri gravity and Cartan geometry), but if you check out this this paper by Derek, John, and Alissa Crans, you see a little overlap.

What is the overlap? The physics of it is rooted in a fairly old idea ususally attributed to Wheeler, called “matter without matter” (John cites a number of references on this in week 208 of “This Week’s Finds”). There are several variants of this idea, but all of them in some way contain the key ingredient that matter should somehow be an expression of the shape of spacetime itself. Some older versions hold that elementary particles should be seen as the mouths of little wormholes. More recent ideas, based on spin networks (originally introduced by Roger Penrose in this paper, and much developed since) represent space as a kind of (labelled, directed) graph with edges connecting nodes - and these recent ideas suggest that a stray edge in a spin network will act just like a particle with the spin associated to that edge.

An example of a theory that fits this last picture, and the thing that most directly inspired the project described in my thesis, is some work of Laurent Freidel, David Louapre, and Etera Livine - a series of papers on the Ponzano-Regge model (parts I, II, and III) which is a model of 3-dimensional Riemannian quantum gravity. This is pretty unphysical - since the standard picture of gravity in the physical world is in terms of 4-dimensional, Lorentzian gravity (which, unlike the Riemannian picture, distinguishes between spacelike and timelike directions). Nevertheless, most people would accept the Ponzano-Regge model as physics… Anyway, their model describes a world where gravity is described by the Ponzano-Regge model, and is coupled to matter which is represented as stray ends of edges in the spin network. As the networks evolve, the stray edges trace out Feynman diagrams for the matter in question.

I could also mention that Laurent, together with Aristide Baratin, has recently done some work going in the other direction - starting with Feynman diagrams and trying to show how a picture of quantum gravity was already hidden in them, but with the gravitational coupling “turned off”. They have a couple of papers doing this in both three and four dimensions.

In any case, this version of “matter without matter” was a major part of the inspiration for
this project, but I describe things from a somewhat different point of view - or at least a dual point of view. When you describing the geometry of space in terms of a spin network, nodes in the network represent volumes in space, and edges in the network represent boundaries between volumes. This is a Poicaré dual picture - it’s also a picture that depends on a triangulation, or some other way of breaking a manifold apart into cells. I allude to this in the beginning of the thesis, talking about the Fukuma-Hosono-Kawai construction for getting a topological quantum field theory in 2 dimensions. However, one of the nice things about this construction is that it ends up being independent of which triangulation you pick (I have an explanation of this in these slides for a talk I gave last year at the Perimeter Institute). So after a bit, we just end up thinking of matter as living on boundaries of some kind.

The idea is that you have a manifold supporting some sort of geometric structure. The manifold has some “defects” - boundaries where that structure has to stop. It could be a 2D surface with some holes bunched out with a hole-punch - holes with a 1D boundary. Or it could be a 3D space with some 2D surface as the boundary. These could be literal defects - the boundaries describe where a pointlike, or line-like “flaw” in the geometry can live, because part of the manifold is just missing. This is the usual way of thinking about singularities. Or, you can just imagine that the boundary marks out some kind of “system” sitting in space that you might want to observe, and the theory tells you what information about the system on the other side of that boundary can be detected by looking at the geometric structure of the space around it.

Now, if we’re looking at 3D space, then gravity is fairly simple. Up to equivalence (i.e. up to a change of coordinates) the information about matter which we expect to be carried by the geometry of the space it lives in would include its (rest) mass and its momentum - in particular, its angular momentum, or spin. Different types of particles - as far as their effects on gravity allows us to tell them apart - are classified by their masses and spins. Any other information about them doesn’t directly affect the geometry of space. What’s more, in 2-dimensional space, particles look like single points - and all the curvature of space is concentrated at those points, leaving it flat everywhere else. The spin gives information about a “skew” in the geometry of 3D spacetime around the worldlines of such points.

In fact, this is just what this extended TQFT business allows us to recover about - but only because we have information about three levels: “boundaries” (around a system, in which the matter lives), “space”, and “spacetime”. And this is what has to be organized into some kind of 2-category…

(more to come on that in pt 2)