Recent Talk - Alejandro Adem, “Commuting n-tuples & Spaces of Homomorphisms”

A recent colloquium talk here at UWO caught my attention because it ties in quite directly to some of the things I’ve been talking about here. Alejandro Adem, from UBC (also the PIMS head-to-be) was talking about commuting n-tuples and spaces of homomorphisms. In particular, spaces of homomorphisms where is a discrete group and is a Lie group. If you take to be , then this is a space of -tuples of elements of which all commute (since is abelian).

In particular this turns up when you want to talk about the moduli space of flat -bundles on a manifold , which you do in the area of TQFT’s. Flat -bundles are determined by specifying holonomies in around any loop - the effect of doing transport around . If you take the discrete group , the fundamental group of , then this is an example of the kind of space Adem was talking about. In particular, speaking of commuting -tuples, that is the even more special case when is an -dimensional torus. However, it’s a tricky enough special case in its own right, as it turns out. Adem spent a fair amount of time on some of these.

In geometry, you’re perhaps more likely to be interested in the moduli space of flat bundles up to gauge equivalence - which amounts to saying that if you conjugate all your holonomies by , you have an equivalent bundle. The same thing happens with spaces - since acts on them by conjugation, you can take the quotient under this action. If you started with a finite group , the space was a manifold, but the quotient may not be. However, you do have a bundle , so that each point in the base space is a gauge equivalence class of connections, and the fibre over each point consists of all the gauge-equivalent connections in that class.

(Throughout the talk, I found myself trying to categorify things - in building an extended TQFT, rather than a TQFT, one uses the case where \Gamma = \pi_1(M)$). However, there you take a weak quotient, where instead of forcing gauge-equivalent objects to be equal, you just insert isomorphisms between them, getting a groupoid I’ll call . The bundle picture is related to but different from the groupoid picture. The groupoid is equivalent to its skeleton, where the objects are just the points in . The morphisms at object are the group - the points in the fibre over in the bundle are all stabilized by - it’s a coset space.

Also, when you include the morphisms, instead of looking at functions from this space into, say, , or - its cohomology - you tend to look at functors from the groupoid. The category of functors from it into is exactly the 2-vector space of states it gets in the extended TQFT picture I partially described back here and here. So this is a categorified version of a cohomology module - the non-categorified version being what a regular TQFT based on gauge group would assign to . I’m not sure quite how all the rest of the talk fits into this picture.)

First, though, he described some tools for dealing with such spaces. To start with, you use the classifying spaces and (where is a space whose fundamental group is and which has no other interesting homotopy groups). Since “taking the classifying space” is a functor, homomorphisms turn into continuous maps . (Even better is when for some Riemann surface (i.e. a torus of some genus ), then effectively is the classifying space: ). This correspondence may not be one-to-one, but the point is they tell us something about the shape of the moduli space we were interested in. Looking at homotopy classes of such , which form a space , we get information about the components of the moduli space - there’s a map

which we can try to understand. Alejandro Adem then went on to use this idea to look at spaces of commuting -tuples in a Lie group , namely . Since the image of generates an Abelian subgroup of , one basic result is that if every maximal such subgroup is path-connected, then so is - there’s just one component (since any tuple can be deformed into any other). This can be extended to groups “built from” Abelian subgroups (in various ways he left undefined for this talk).

The other important tool for looking at the geometry/topology of the moduli spaces which he spoke about was (Poincaré-)Alexander-Lefschetz duality, which provides information about the topology of one space embedded in another from the topology of its complement. In particular, it gives an isomorphism between the cohomology of a space and the of its complement, where is -dimensional. In particular, the spaces of commuting -tuples of elements of are subspaces of the manifold , which is much easier to understand.

So finally, among a number of other examples of how these tools come into play, the one Adem described that I was most interested in was the space , and particularly , the space of connections on a torus. The complement in is an open set in a manifold - hence it’s a manifold itself - and in fact it turns out to be equivalent to . You can get partway to seeing this by noting that the projection map turns into a bundle over - the projection never hits the centre of . This centre happens to be just two points, 1 and -1, leaving the base space homotopic to a sphere . The fibre over each point is , the whole group minus the centralizer of (i.e. everything which doesn’t commute with ). The centralizer of any point is just a circle, and the remaining set is homotopic to a circle itself.

So the complement of the moduli space, within , is homotopic to a bundle of circles over a 2-sphere. There are a few of these, and it takes a little more to find out that it happens to be the 3-sphere with the Hopf fibration, but that’s what it is. Then, to find out what the moduli space itself looks like, you have to use the Alexander-Lefschetz duality. Adem didn’t show all the details, so I’m not exactly sure how, but it seems that it turns out you have a space homotopic to the one-point union of three spaces:

Now, as I said before, this is telling us information about the objects of the groupoid (also known as the moduli stack of connections), and while the morphisms shouldn’t be too hard to work out in this case, it might be nice to have a more general picture. When I raised this, Rick Jardine suggested that looking at the maps in should help - the classifying spaces are simplicial sets, and so is the collection of maps between them, and the above is only talking about vertex information. There should be a way of looking at as an infinity-category - and in this case, it should be trivial above the level of morphisms. But I don’t quite know how this works yet.

 

Visiting Perimeter Institute

Once again, I keep meaning to write some less math-heavy posts, if for no other reason than to keep in the habit of thinking up things to write in here. Now is a good occasion to do this, since I’m visiting at the Perimeter Institute in Waterloo to give a talk called “Extended Topological Quantum Field Theories and Quantum Gravity” at the quantum gravity seminar on Thursday (the 28th). This is basically an updated and refined version of the talk I gave for my thesis defense, in which I’ve tried to make more of the link to physics - in particular, to BF theory, and to 3D quantum gravity. This turns out to be hard to do in an hour-long talk and still cover things adequately. Still, I find it worthwhile to get the point of view of real physicists on these apparently physics-related ideas, after thinking about them as a mathematician for some time.

After I arrived, I had lunch with a bunch of the quantum gravity people here. The conversation ranged from hunting for jobs, through cultural differences between Europe, Canada, and the US (a standard conversation to be had anywhere in Canada at the drop of a hat), all the way over to “Why is spacetime 4-dimensional?” Lee Smolin put this last one to me when I was describing how categorification is related to considering higher co-dimensions of spacetime/space/surfaces in space. It’s a reasonable question, though not one I have any answer to. But when you cook up a theory - like this ETQFT stuff - which in principle works in any number of dimensions, and you want it to be physical, you’re left wondering “why so few dimensions?”

Okay - it’s not the main point of what I’m doing here, but it’s a nice light question to blog about, since I don’t pretend to have even a good guess at the answer.

It takes a certain mentality to think that 4 dimensions is astonishingly few - however, I have that mentality, as do many mathematicians. You can work with infinite-dimensional spaces in mathematics - why should “real”, “physical” space only have four? Actually, the segue into this had to do with the question of why all the Lie groups that turn up in physical gauge theories are so tiny - , , - rather than, say, , which describes rotations in a 745 (complex) dimensional space. Again: gauge theory makes just as much sense with big gauge groups as small ones - so what’s special about the low dimensions?

Well, I don’t know the answer - but it’s the kind of question mathematicians probably should be asked more often. We’re perfectly happy to deal with a 745 dimensional space and not worry about the fact that it’s non-physical. But if mathematics really underlies physics in any deep way, there should be some good mathematics in the answer.

There were some possibilities tossed around: what if the exceptional group really does turn out to be important in fundamental physics, and the real gauge group of the right physical theory has to lie inside it somewhere? Then there’s an upper bound on how many dimensions you can have - though, unfortunately, is 248-dimensional, so the upper bound is a bit high. (Mind you, the symmetries of 4D space is, in itself, a 10-dimensional group, so things are not quite as bad as they appear - but still worse than they should be). There’s also no obvious reason why should have such a special role.

A more physics-y answer is that in 5D and higher, you don’t get confinement - quarks and gluons just fly around like a dilute gas, and there would be no matter in the sense we know it. This is a great concise description of why we should be happy to live in a 4D spacetime. The objection to this is that it’s basically an appeal to the anthropic principle: “If space weren’t 4D, we wouldn’t be here to wonder why.” If you’ve read Lee Smolin’s most recent book, you’ll know he doesn’t care for appeals to the anthropic principle. Neither do I, for that matter. If you assume that every possible universe actually exists (which is at least metaphysically parsimonious - no need for two separate categories of “possible” and “actual”), the anthropic principle is undeniable. The problem is, it doesn’t predict very much until you work out enough about what universes are possible that you might as well just try to answer the question for its own sake. Still, maybe it’s just true that there are a huge number of actual universes, and some of them are no good for intelligent life. But that just means the question has no answer, so you might as well give up. It doesn’t take you anywhere. So suppose there’s a reason: what could it be?

In 3 and 4 dimensions, there are regular polyhedra - or, equivalently, discrete subgroups of the rotation group - that don’t correspond to the series which always exists. In 2D, there are infinitely many regular polygons, and in all dimensons, there are simplexes, cubes, and duals of cubes… but in 3 and 4D there are some extras, all of which boil down to the icosahedron, its dual, or things you can construct from it in 4D. Why this should make any difference, I have no idea.

And there are a couple of other special things in low dimensions, which are no more obviously relevant, but seem compelling to me, perhaps because I’m a mathematician…

In 4 dimensions, but no other dimensionality, there are “exotic” which are homeomorphic but not diffeomorphic to the usual . The heuristic explanation for why (which is as much as I really grasp) is that 4D is “big enough” for complicated twisty things to exist, but “too small” for there to always be room to untangle them - so only in 4D can “things be complicated”. Which is suggestive, but hardly a full answer.

4 dimensions is the only case where the classification of manifolds is not understood (now that the Poincaré conjecture has been settled - there were still some lingering doubts last I heard, but they seem to be evaporating day by day). in 2D, manifolds are basically just toruses with some genus; in 3D manifolds can be cut up into pieces each of which can be geometrized (a la Thurston). In 5D and higher, you can classify (in principle) manifolds by constructing them via surgeries. The reason this doesn’t work in 4D is that surgeries building new manifolds correspond to cobordisms between the input and output manifolds, and in 5 or more dimensions, cobordisms are rather trivial (actually, this only refers to cobordisms where the inclusions of the source and target manifolds are homotopy equivalences, which isn’t totally general).

This last bit seems the most intriguing to me, since I’ve been thinking about TQFT’s and ETQFT’s, which are field theories living on cobordisms. But that still doesn’t add up to an answer to the physical question. It would be nice to understand, for instance, whether the above fact means anything helpful in terms of the physics of such a theory.

Anyway, I’ll try to write up something about those theories from a physical point of view after I’ve had a chance to chit-chat about them with some physicists after my talk. It probably won’t answer this rather vague and (perhaps?) unanswerable question, but there seem to be some interesting things to say. Maybe before then (but after I’ve had a chance to give my talk, no doubt!) I’ll also give a little write-up of the colloquium talk by Robert Spekkens I attended today about foundations of quantum mechanics.

 

n-group Representation Theory - part 3: Rep(Poinc)

I’m going to be giving a talk on extended TQFT stuff and quantum gravity at Perimeter Institute next thursday, and then in mid-March I’ll be heading to UC Davis to give the same/similar talk for the String Theory and Quantum Gravity seminar being run by Derek Wise. So I have a bunch of things on my mind right now. However, before heading to Davis, I wanted to go back and look at some of the stuff Derek has done having to do with Cartan geometry, which I was following somewhat at the time, and blog about it a bit here. Before that, I’d like to wrap up this presentation of the talks I gave here about representation theory of the Poincaré 2-group, .

As a side note, thanks to Dan for pointing out these notes on representations of the (normal, uncategorified) Poincaré group, including some general comments on representations of semidirect products. It’s interesting to consider how this relates to the more general picture of 2-group representations - but I won’t do so here and now.

…

In Part 1 I talked about what representations 2-categories of 2-groups are like in general, and in Part 2 a fairly concrete description of . Here I’ll wrap up by summarizing the results of Crane and Sheppeard about what looks like concretely.

It has three parts: the objects are representations (also known as functors from as a 2-category with one object, into ); the morphisms are 1-intertwiners (a.k.a. natural transformations) between reps; and the 2-morphisms are 2-intertwiners (a.k.a. modifications) between 1-intertwiners.

1) Representations: A functor

will pick out some measurable space for the lone object of the 2-group - or rather, , the 2-vector space of all measurable fields of Hilbert spaces on . (This is a matter of taste since to know the one is to know the other.) Then for the morphisms and 2-morphisms of we get, respectively, 2-linear maps from to itself, and natural transformations between them.

The morphisms of are just the group in the crossed-module picture I described in Part 2. For the usual Poincaré 2-group, this is . For each such element, we’re supposed to get an invertible 2-linear map from to itself - that is, a measurable field of Hilbert spaces on (together with measures to do “matrix multiplication” with by direct integrals). This can only be invertible if the only Hilbert spaces which appear are 1-dimensional (since these maps compose by a “matrix multiplication” involving direct sums of tensor products of the components - and the discreteness of dimensions means that if any dimension is higher than 1, you’ll never get back the identity).

So any representation turns out to give what amounts to an action of on - the component is if and 0 otherwise. An irreducible representation gives an with a transitive action (otherwise, you can decompose it into orbits, each of which corresponds to a subrepresentation). Crane and Sheppeard classify several kinds of these, associated to various subgroups of , but an easy example would be a mass shell in Minkowski space - a sphere or hyperboloid (depending on ) that is the full orbit of some point under rotations and boosts (a “mass shell” because it gives all the possible momenta for a particle of a given mass, as seen by an observer in some inertial frame).

The 2-morphism part of gives a homomorphism from at each of these points. Now, one-by-one matrices of complex numbers are just complex numbers, so what we have here is a character of - at each point on . To be functorial, this has to be done in an equivariant way (so that acting on the point by affects the character by acting on by the same ).

2) 1-Intertwiners:

If representations and correspond to actions of on spaces and respectively, with characters , then what is a 1-intertwiner ? Remember from Part 1 that it’s a natural transformation: to the object of it assigns a specific 2-linear map

To each (object of it gives a transformation

This is a specified map which replaces the naturality square in the old definition of an intertwiner. It has to make a certain “pillow” diagram commute (Part 1).

Now, back in the posts on 2-Hilbert spaces, I explained that a 2-linear map is given by some field of Hilbert spaces on (a “matrix” of Hilbert spaces, though of course needn’t be finite), along with a family of measures on indexed by (which allow us to do integration when doing the sum in “matrix multiplication”). The transformations also can be written in components, so that

(Note this uses the two actions given by on - one forward, and one backward. This is the current form of what, in uncategorified representation theory, would be a naturality condition.)

What does this all amount to? One way to think of it is as a representation of itself! In particular, it’s a representation on the direct sum of all the Hilbert spaces which appear as components of . This is since the maps given by the have to satisfy a condition which says that composition is preserved (as long as you’re careful about indexing things):

To get a representation of the group, we can say that elements shuffle vector spaces over points in by the action of and then act within vector spaces by . So then has both intertwiner-like and representation-like properties.

The “intertwiner-ness” of has to do with how it interpolates between two actions on by turning them into an action on the product - but it also has some “representation-ness”, by giving this action of a (semidirect product) group on a big vector space.

3) 2-intertwiners

If a 1-intertwiner can be thought of as a representation of , it shouldn’t be too surprising that a 2-intertwiner between 1-intertwiners ends up being an intertwiner between the associated representations. If 1-intertwiners have some qualities of both reps and intertwiners, the 2-intertwiners are more single-minded.

In particular, a 2-intertwiner assigns to the only object of a 2-morphism in (that is, a field of linear maps between the vector spaces which are the components of ), which satisfies some “pillow” diagram. When we form the big rep. by taking a direct integral of all those spaces, the field of linear maps turns into one big linear map, and the diagram it satisfies just collapses into the condition that it be an intertwiner.

So the representation theory of this interesting 2-group looks a lot like the representation theory of the group of 2-morphisms. The extra structure involving actions on measurable spaces by would be mostly invisible if you just thought about irreducible reps of the group, since the space would be just a single point.

This phenomenon where a lower-order structure turns up in some form at the top level of morphisms of its categorified version has cropped up before in this blog - namely, when extended TQFT’s turn out to contain normal TQFT’s in individual components. In these examples, categorification is less a matter of building more floors “on top” of structures we already know, as “higher morphisms” suggests, but excavating additional floors of subbasement - interpreting what were objects as morphisms.

 

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…

 

TQFT and Gravity - pt 1

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)

 

Recent Talk - Michael Misamore/Galois Theory

Right now I’m making some polishing-up edits to my thesis before posting it on the archive. In the meantime, here, as I suggested, are some comments on one of the talks I’ve seen in the last week or so at UWO. This is partly to help me remember them and have something to look back on besides my dubious notes. Naturally, all the following is my understanding of what went down, so anything wrong is presumably my fault - if you notice, tell me!

Michael Misamore’s talk last week was called “Galois Theory”, though the name “Grothendieck” came up much more than that of poor Evariste Galois. It was interesting to me because it looked at this classical subject from the point of view of schemes. I remember enjoying an algebraic geometry course I took when I was at McGill on the subject of schemes, but I haven’t thought about them much since then.

If you don’t know the idea of a scheme, it’s simple enough, though the details get tricky fairly fast - it’s a (locally ringed topological) space which looks, locally, like the spectrum of some commutative ring. This is sort of like how a manifold is a space which looks locally like (or , if it’s a complex manifold). The spectrum of the ring is a space whose points are the prime ideals of the ring. Case in point would be if the ring is , the polynomial ring in variables. Then the spectrum looks a lot like (with the Zariski topology and some extra “generic points” for each algebraic variety). Schemes also come with, for each open set, a ring of functions on it - all of these together make up the “structure sheaf” of the scheme.

(Philosophical aside: One reason I like the idea of schemes, despite not having thought about them in a while, is that the concept of generating a space as the spectrum of a ring is intrinsically satisfying if you believe that “space” should be a derivative concept anyway. There are various reasons (another coming attraction, maybe!) for thinking this should be the case in fundamental physics. So spectra of commutative rings are nice because they suggest that “spaces” are secondary concepts, just used for classifying information about a ring. Schemes generalize this the same way manifolds generalize Euclidean space. Noncommutative geometry generalizes in an orthogonal direction - taking noncommutative rings and applying intuitions from the study of spectra. Apparently there’s even a concept of noncommutative schemes. Now, I don’t know much about any attempts to use any of these ideas in physics - and I can more easily conceive of cases where these objects fill in for configuration spaces, rather than space, per se - but they do give some reassurance that at least space doesn’t have to be fundamental.)

Anyway, Michael’s talk was ostensibly about Galois theory. Classically, this has to do with field extensions, and the Galois groups of field extensions (i.e. groups automorphisms from the extended field to itself which fix the base field). The basic point, though, seems to be that a field is just a special kind of commutative ring, which has only the one ideal - namely the whole field, since you can divide by everything. So the spectrum of a field is a single point (in fact, this motivates the idea of a “geometric point”: if a point in a scheme is given by a map for a field , then a “geometric point” is given by a map for any ring ).

So: you can look at a field extension as a one single-point scheme sitting over another, in a way that has some group of automorphisms associated to it. That’s not so interesting (though I find it a bit hard to visualize what automorphisms of a point mean - presumably something to do with the structure sheaf). More interesting is to have a covering map - actually a <i>finite etale cover</i> - from one scheme to another (”base”) scheme, and the group of covering transformations (”Deck transformations”) of the covering scheme - that is, the ones that can’t be detected after you apply the covering map. This is a more general analog of the Galois group of a field extension.

You can then see pretty clearly an analog of one of the well-known issues in Galois theory - namely, the problem of finding the absolute Galois group of a field , which is the Galois group of the separable closure of , (some nice subfield of the algebraic closure) over … This corresponds to finding the group of covering transformations of the universal cover of a scheme. The problem is, there may not be a universal cover. An example of a lack of a universal cover (in a category where maps are by definition algebraic maps) would be the punctured complex line . Covers of this are given by maps , whose group of covering transformations is the cyclic group . A universal cover should have infinitely many sheets (since the fundamental group of the punctured line is the integers), but there is no covering map which does this. (The map is analytic, but not algebraic).

So with that setup, Michael went on to explain about pro-groups, pro-representable functors, and how they address this issue. For standard covering spaces over a space , there’s a representable functor which takes a covering space over and gives the fibre over a point. You can represent this as a functor since the points in the fibre over of the universal cover are reached by liftings of distinct paths from to itself.

In the case of schemes, you don’t have a universal cover, necessarily, but you do have a category of covers - in fact, a pro-object in the category of objects over , which is a nice sort of diagram. If there were a universal cover, it would be a limit of this diagram - a universal object with maps into every object in the diagram.

(”Pro-object” and “geometric point”, by the way, are both examples of a common stragegy: replacing a singular gizmo - an object or a point - by a suitable map into the place where the gizmo would live. A pro-object in is a map from some nice small category, giving the “shape” of the diagram, into ; an geometric point in is a map from , giving the “shape” of the “point”, into .)

The fact that there isn’t a limit for this pro-object in the category of schemes over is inconvenient, but the philosophy seems to be that one should just use the pro-object anyway. On top of this, there’s a concept of a “pro-representable” functor, which can be described in terms of a function from a pro-object.

So there’s an analog of the representablity of the functor which gives fibres from covers. It involves a functor , where is a category of “finite etale covers” of a scheme (apparently this is the right notion, though I don’t quite grok it), which when applied to a cover gives the set of (geometric) points in over the (geometric) point .

The theorem says that it’s representable by some pro-object in the category of covers. Namely, , which by definition is the limit . Since each of these is a set, you get a pro-object in Sets: and over HERE, the limit exists! The same sort of thing happens when you look at the Galois groups - i.e. the (finite, since the covers are finite) groups of covering transformations form a pro-object in - a pro-group. Again, you can take the limit, and get a profinite group.

One of the main lessons in all this seems to be that if something doesn’t exist, you approach it in the limit. When there’s no limit, you can just take the whole net of things which are approximating it, and deal with that directly. When you start mapping that net into various other realms (as when we map in to to look at fibres, or to look at Galois groups), sometimes the resulting diagram will have a limit, and you can then look at that, if you like. Somehow it reminds me of compactification…

Anyway, it’s just as well I went through all this stuff, because the next talk was: Joshua Nichols-Barrer - “Intro to Quasicategories”. These turned out to have a lot to do with stacks - and once again we’re into algebraic geometry and Grothendieck’s turf… Today, because I wanted to learn more about stacks for reasons of my own (coming attraction?) I had a somewhat lengthy meeting with Josh, which helped a lot, even if it didn’t much explain quasicategories…

More on that later.