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.

 

Recent Talks: Rick Jardine “Categories, Symmetric Groups, and Spheres”

A recent colloquium talk at UWO was given by Rick Jardine, who is a prominent member of the department, with a lot of graduate students. I’m not sure of all the details of what he works on, but it seems to mostly have to do with homotopy theory, category theory, and related things. His talk was called “Categories, Symmetric Groups, and Spheres”. It was rather involved for me to describe here, tying together as it did a bunch of different topics. However, I thought it was interesting, so I’ll try to give a summary of at least some of what it was about.

The last of the three topics - spheres - had to do with the fact that the end result was to show that some construction turns out to be closely related to sphere spectra. Spectra are sequences of spaces, say , such that there’s a map from the suspension of each space into the next, . A suspension is just a sort of double-cone on a space: to get , add two points, and then connect each point of to each of the two new points. For example, if you start with a circle, the result is a sphere - your original circle was the “equator”, then you added two poles, and drew in the points in between. This example generalizes, so a really simple spectrum is just the sequence of spheres of increasing dimension (then the map is just the identity).

These spectra are important in homotopy theory, and in particular, in stable homotopy theory. As I understand it, stable homotopy talks about those parts of homotopy groups that stay the same when you repeatedly take suspensions - so you pass from homotopy classes of maps from a circle into , to maps from a sphere into the suspension , to maps from a 3-sphere into the suspension , and so on… the only changes that can occur is that you might lose some distinctions, so the groups could get smaller. Eventually, they stabilize - and voila!, stable homotopy groups. So anyway, spectra are important to this subject.

In particular, the theorem Rick was explaining (in, as he said, a “modern exposition”, originally due to Barratt and Priddy) has to do with a space called , whose homotopy groups are the same as the stable homotopy groups of spheres. The theorem says that it has the same homology as the infinite symmetric group. So the idea he was presenting is a construction involving symmetric groups. The point of it is that there’s a basically combinatorial description of everything involved - that is, a description involving just finite sets (which is where the symmetric groups come from).

How does this work? Well, first of all, it uses a construction called the “category of elements” for a functor . This is a category whose objects are pairs where , and whose morphisms are morphisms such that . That is, this makes a new category from all the elements of the sets coming from objects in , where the arrows are compatible with those in - each object is multiplied, and so are the morphisms.

The category of elements we’re talking about is a functor . Here, has finite sets for objects, and 1-1 functions (”injections”, “monomorphisms”, etc.) as morphisms, and is the category of pointed sets. This functor depends on a particular choice of pointed set , or for short. The way it works is that is the set of all functions from into - which is pointed, since the function where everything goes to is distinguished - so this is just . Given an injection , you get a map from the set of functions to , which you get by extending a function so anything in not in the image of just goes to the special point (this is why we needed pointed sets). So the category of elements in question is . The point is that it gives a nice space.

Again: how? Well, this uses the idea of a “nerve”.

Any category has a nerve: this is a simplicial set related to . The way you get an -simplex is to look at any chain of arrows in . The vertices form the edges, the arrows give some of the edges, and the various ways of composing (some of) them give other edges. Each composition of two gives a triangle, and the higher simplices come from various equations. The different simplices are stuck together by various incidence relations that show the structure of the category . This nerve is called , which is a purely combinatorial object. (Ultimately, the simplicial set that’ll show up in this story is from the category of elements above). It becomes a space when you take its geometric realization: replace abstract simplices with actual triangles, tetrahedra, and so forth, taken as topological spaces living in . This space is called , and it’s a topologically nice space - a -complex (being built by gluing simplices together).

Then you have this simplicial set, which can be thought of as a space, - a so-called “gamma space”, which are what correspond to these spectra mentioned up above. In particular, if the pointed set , with 0 the distinguished point, then it turns out that , the disjoint union of the spaces obtained from all the finite symmetric groups. This is because the symmetric group acts on the category of elements .

So part of the point of this part is what was, as Rick pointed out, the first adjoint pair of functors which was seriously studied - a pair of functors going between and (simplicial sets and topological spaces). The geometric realization functor is a left adjoint to a functor , so that , giving a simplicial set for a topological space . And homotopy theory in then has an equivalent in - so there’s a completely combinatorial core of homotopy theory. (Technically - and I admittedly don’t quite grok this concept yet - these two adjoint functors are giving a Quillen equivalence). Now, homotopy doesn’t tell you everything about a space - but it tells a lot, so it’s useful to get the idea that all this information about a space from something very combinatorial, like permutations of finite sets.

I have to admit I find a lot of this stuff is a bit technical for me to fully appreciate what’s clearly a very elegant fact relating spaces and combinatorics, but I find it interesting that a correlation like that exists. The apparent dichotomy between “smooth” or “continuous” things like spaces, and discrete, combinatorial things like integers, finite sets, permutations, etc. - and the various ways this dichotomy gets resolved, overcome, or bridged - is one of the really interesting cores of mathematics to my mind.