Theoretical Atlas

I finally submitted my (slightly) polished thesis to the archive.  The real polishing will have to wait for a few papers which should come out of it.  In the meantime, a version very much like the one I submitted to UCR (with some bugfixes, a somewhat improved introduction, and minus the formatting that was so hard to get right to UCR’s exacting standards, and which renders the document a full 143 pages longer than this version) will be available on the archive as of the next update - Monday afternoon.  When it is, I’ll give a link to that.  For now, however, you can look at the very same version hosted on my site at UWO (here).

I will make a post soon talking about some of the physics motivation behind the project it describes, of which I’ve already given a little executive summary in a previous post.  However, I wanted to make the announcement right away.

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.

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.

You may be wondering about the title: “Theoretical Atlas”. Both words have a double meaning here.

First, Atlas: originally, this was the name of a Titan in Greek mythology, who was condemned by Zeus to stand at the Western edge of the world and hold up the sky on his shoulders forever. The Western edge of the Greek world - the Mediterranean - is indeed where the Atlas mountains are found, in the Maghreb. Also named for him is the Atlantic Ocean (and, therefore, Atlantis, a continent once speculated to be located somewhere in it). You can see a picture of the Atlas mountains in the banner at the top of this blog’s main page.

So one meaning comes from a notion that tends to crop up fairly often when one talks about the project of finding a quantum theory of gravity. This is the prospect of a complete unified theory of physics, a Theory of Everything (TOE), or some such name. People peering into the mist of our limited knowledge sometimes seem to see prospects of a single theory that unifies every aspect of the physical world in one single model - all forms of matter, energy, forces, gravity, etc. The name “M-theory” is popular in some circles for this idea - an as-yet undiscovered theory which might go beyond what string theory can do today. Other prospects have been proposed, but the image I have is of a single, immensely powerful theory, holding up the entire world on the strength of its explanatory power - a theoretical Atlas holding up this enormous burden.

But this great Atlas of a theory has never been written down - alas. For myself, I’m quite skeptical if it even could be: why should there be a short, pithy idea that encodes the whole huge, complex, endlessly surprising universe? Even if we had a theory which accounted for all particles and forces in nature, would that be a theory of everything? The point of a theory, after all, is to help us understand things: we’d still need, at the very least, a theory to explain how chemistry emerges from physics, what life is and how it can come into being - all just to account for even our most basic experience. Then there are whole areas of the world that open up from there. So this great single Atlas of an idea that accounts for the entire world of experience is, as they say, just a theory. It’s a (merely) theoretical Atlas.

(Of course, this use of the phrase “just a theory”, often used to dismiss the insights of Darwin, and much less prominently used any other way, is simply wrong. The meaning of “theory” depends on context, but it always means something more than a mere guess. Still, as I said before, I’m not going to worry TOO much about being wrong now and then - and the more accurate hypothetical Atlas just didn’t sound as good.)

The other meaning of the word “atlas” has to do with maps. The other element of the banner above mentions the Bellman’s map from The Hunting of the Snark. It had no markings on it at all - “purely conventional signs”. But mathematics is all about using purely conventional signs as a reference point in describing the features of the world. The Bellman’s map showed no land - only sea - and so it left out not only the conventional reference points, but also anything definite to refer to.

A “theory” can be seen as a way of taking some standard, pre-existing structure, and trying to “map” it onto the features we see in the real world. In a way, a literal map is an example of a theory: it imposes a regular grid of coordinates on some convoluted shape, which is itself a model of some territory off elsewhere in the world. It’s an artificial imposition - but it allows us to find our way around. Assuming it’s accurate enough, and we know how to read it.

In the case of a literal atlas, we have a collection of - usually flat, generally rectangular - drawings of the surface of a sphere (more or less). Each one is a little bit distorted, because the Earth isn’t flat (no, no, I know - that’s just a theory - but I think it’s accurate enough). In the study of manifolds, these are called “charts” - each one is a map from some open subset of Rⁿ to a subset of the manifold. Generally - and, for instance, on the surface of the Earth - one chart won’t be enough. You need several charts, and an understanding of how they fit together. The collection of charts is an atlas, and one imagines a big book filled with these charts, each one imposing a rectilinear grid of coordinates onto some underlying terrain. “Transition maps” tell you how they fit together to cover the whole surface.

So the other meaning of theoretical atlas is the notion that we may need many theories to properly account for the world. Each one may describe some part of it fairly well - maybe with a bit of distortion, but certainly not so much that it doesn’t help to find our way around. None by itself explains everything - but given enough, and some knowledge of how to manage the transition between the domain of one theory and the domain of another, they can tell us a lot. This is my image of what our researches into physics, and the world in general, are aiming at: an atlas of theories that covers everything.

Mind you, I realize that such an atlas, like the other kind of Atlas, is purely theoretical.

Oh, all right: hypothetical.

Here is an apology - with apologies to the Unapologetic Mathematician…

One inspiration for starting this blog is the fact that Dr. Baez has a great abundance of stuff on the Web. Some of the better-known include the ever-popular This Week’s Finds in Mathematical Physics, and the newer n-Category Café, which is a group venture together with Urs Schreiber and David Corfield. Between the three of them, they write on “math, physics, and philosophy”. That’s more or less what I propose to do here.

Why the redundancy?

The n-Category Café has turned out to be a very productive way of sharing ideas informally over long distances, and without being too confined by a narrow topic or the strictures of publishability. The participants have also adopted the ethic that it’s better to share ideas than keep them secret until they’re perfected. One essential reason is that science, math, and philosophy are cultural products - discussion is like oxygen for culture. This is a lesson that has been learned many times in the past, and, I suspect, will have to be learned many times again in the future. Publication, peer review, giving public talks - the whole essence of research is communicating ideas. Of course, you need to develop good ideas to communicate, but the point is to share and discuss them. One more voice in a conversation like that may be a drop in a bucket, but it’s not redundant.

So I aim for this to be my particular drop in our great collective bucket. I’ll relate things that I’ve been thinking about; explain things I’ve figured out; express confusion over things I haven’t; describe the experience of starting a research career; muse; investigate; and, if possible, not bloviate. And I won’t worry too much about being incomplete, tentative, or even (a little bit) wrong. That’s all part of investigating things.

This is as much “apology” (in the sense of a justification of one’s actions - quite the opposite of what we moderns usually mean by “apologize”) as I suppose the minor nuisance of starting yet another blog really requires.

Hello!

This is Entry #1 of this blog, the “Theoretical Atlas”, in which introductions are made. It is the first of a few “meta” entries, setting out where we’re going.

Introductions

My name is Jeffrey Morton. I am a mathematician interested in physics - and life. I’m starting this blog as I start out on my first postdoctoral position after finishing my Ph.D with John Baez at UCR.

The starting point for my research to date was my interest in efforts to find a quantum theory of gravity. It started innocently enough, learning some differential geometry, which led into General Relativity; that, in turn, led into studying quantum field theory in curved spacetimes. There many limitations on what you can say about that: general relativity and quantum field theory are based in very different mathematical vocabularies (not to mention grammar). Relativity theory describes gravity in terms of the geometry of spacetime - which is quite definite. Quantum field theory, on the other hand, describe matter in a very different way, in which the observed values of physical quantities can be any of the eigenvalues of certain linear operators on some Hilbert space. In particular, it doesn’t always predict a definite, specific value for the concentration of mass in any given location. But it’s mass-energy, in particular places, which supposedly creates gravity. These theories have a hard time talking to each other unless you carefully limit how much they interact, or assume a great deal of symmetry.

I encountered this fact when I was studying the Einstein-Dirac-Maxwell equations for my M.Sc. and an interest in confronting it led me to UCR. What I learned from Dr. Baez once there pushed my interests in several other directions. As one might expect: if the problem is that two theories use very different language, a step to reconciling it is to develop a new language which can handle both of them. This is where category theory entered the story. Category theory is a very general mathematical language, which can be applied to many subjects within mathematics, and thence to their many and various applications. One way to state the essential idea is that it takes both “things” and “relations” between things (in a very general way) as fundamental concepts. So far, I’ve been thinking about various ideas regarding how this can show up in physics.

Now I have introduced myself and two of the the main conceptual characters, what about the blog?

This is the vacuous post.