QGQG Nottingham

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

Two Kinds of Categorification And Other Stuff

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

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

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

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

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

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

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

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

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

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

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

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

 

Ottawa

I recently got back to London, Ontario from a trip to Ottawa, the first purpose of which was to attend the Ottawa Mathematics Conference. The other purpose was to visit family and friends, many of whom happen to be located there, which is one reason it’s taken me a week or so to get around to writing about the trip. Now, the OMC was a general-purpose conference, mainly for grad students, and some postdocs, to give short talks (plus a couple of invited faculty from Ottawa’s two universities - the University of Ottawa, and Carleton University - who gave lengthier talks in the mornings). This is not a type of conference I’ve been to before, so I wasn’t sure what to expect.

From one, fairly goal-oriented, point of view, the style of the conference seemed a little scattered. There was no particular topic of focus, for instance. On the other hand, for someone just starting out in mathematical research, this type of thing has some up sides. It gives a chance to talk about new work, see what’s being done across a range of subjects, and meet people in the region (in this case, mainly Ottawa, but also elsewhere across Eastern and Southern Ontario). The only other general-purpose mathematics conference I’ve been to so far was the joint meeting of the AMS in New Orleans in 2007, which had 5000 people and anyone attending talks would pick special sessions suiting their interests. I do think it’s worthwhile to find ways of circumventing the various pressures toward specialization in research - it may be useful in some ways, but balance is also good. Particularly for Ph.D. students, for whom specialization is the name of the game.

One useful thing - again, particularly for students - is the reminder that the world of mathematics is broader than just one’s own department, which almost certainly has its own specialties and peculiarities. For example, whereas here at UWO “Applied” mathematics (mostly involving computer modelling) is done in a separate department, this isn’t so everywhere. Or, again, while my interactions in the UWO department focus a lot on geometry and topology (there are active groups in homotopy theory and noncommutative geometry, for example), it’s been a while since I saw anyone talk about combinatorics, or differential equations. Since I actually did a major in combinatorics at U of Waterloo, it was kind of refreshing to see some of that material again.

There were a couple of invited talks by faculty. Monica Nevins from U of Ottawa gave a broad and enthusiastic survey of representation theory for graduate students. Brett Stevens from Carleton talked about “software testing”, which surprised me by actually being about combinatorial designs. Basically, it’s about the problem of how, if you have many variables with many possible values each, to design a minimal collection of “settings” for those variables which tests all possible combinations of, say, two variables (or three, etc.). One imagines the variables representing circumstances software might have to cope with - combinations of inputs, peripherals, and so on - so the combinatorial problem is if there are 10 variables with 10 possible values each, you can’t possibly test all 10 billion combinations - but you might be able to test all possible settings of any given PAIR of variables, and much more efficiently than just an exhaustive search, by combining some tests together.

Among the other talks were several combinatorial ones - error correcting codes using groups, path ideals in simplicial trees (which I understand to be a sort of generalization to simplicial sets of what trees are for graphs), heuristic algorithms for finding minimal cost collections of edges in weighted graphs that leave the graph with at least a given connectivity, and so on. Charles Starling from U of O gave an interesting talk about how to associate a topological space to an aperiodic tiling (roughly, any finite-size region in an aperiodic tiling is repeated infinitely many times - so the points of the space are translations, and two translations are within of one another if they produce matching regions about the origin of size - then the thing is to study cohomology of such spaces, and so forth).

The talk immediately following mine was by Mehmetcik Pamuk about homotopy self-equivalences of 4-manifolds, which used a certain braid of exact sequences of groups of automorphisms (among other things). I expected this to be very interesting, and it was certainly intriguing, but I can’t adequately summarize it - whatever he was saying, it proved to be hard to pick up from just a 25 minute talk. I did like something he said in his introduction, though: nowadays, if a topologist says they’re doing “low-dimensional” topology, they mean dimension 3, and “high-dimensional” means dimension 4. This is a glib but indicative way to point out that topology of manifolds in dimensions 1 and 2 is well understood (the connected components are, respectively, circles and n-holed tori), and in dimension 5 and above have been straightened out more recently thanks to Smale.

There were some quite applied talks which I missed, though I did catch one on “gravity waves”, which turn out not to be gravitational waves, but the kind of waves produced in fluids of varying density acted on by gravity. (In particular, due to layers of temperature and pressure in the atmosphere, sometimes denser air sits above less dense air, and gravity is trying to reverse this, producing waves. This produces those long rippling patterns you sometimes see in high-altitude clouds. Lidia Nikitina told us about some work modelling these in situations where the ground topography matters, such as near mountains - and had some really nice pictures to illustrate both the theory and the practice.)

On the second day there were quite a few talks of an algebraic or algebra-geometric flavour - about rings of algebraic invariants, about enumerating lines in special “blow-up” varieties, function fields associated to hyperelliptic curves, and so on - but although this is interesting, I had a harder time extracting informative things to say about these, so I’ll gloss over them glibly. However, I did appreciate the chance to gradually absorb a little more of this area of math by osmosis.

The flip side of seeing what many other people are doing was getting a chance to see what other people had to say about my own talk - about groupoids, spans, and 2-vector spaces. One of the things I find is that, while here at UWO the language of category theory is widely used (at least by the homotopy theorists and noncommutative geometry people I’ve been talking to), it’s not as familiar in other places. This seems to have been going on for some time - since the 1970’s if I understand the stories correctly. After MacLane and Eilenberg introduced categories in the 1940’s, the concept had significant effects in algebraic geometry/topology, homological algebra, and spread out from there. There was some deep enthusiasm - possibly well-founded, though I won’t claim so - that category theory was a viable replacement for set theory as a “foundation” for mathematics. True or not, that idea seemed to be one of those which was picked up by mathematicans who didn’t otherwise know much about category theory, and it seems to be one that’s still remembered. So maybe it had something to do with the apparent fall from fashion of category theory. I’ve heard that theory suggested before: roughly, that many mathematicians thought category theory was supposed to be a new foundation for mathematics, couldn’t see the point, and lost interest.

Now, my view of foundations is roughly suggested in my explanation of the title of this blog. I tend to think that our understanding of the world comes in bits and pieces, which we refine, then try to stick together into larger and more inclusive bits and pieces - the “Atlas” of charts of the title. This isn’t really just about the physical world, but the mathematical world as well (in fact I’m not really a Platonist who believes in a separate “world” of mathematical objects - though that’s a different conversation). This is really just a view of epistemology - namely, empirical methods work best because we don’t know things for sure, not being infinitely smart. So the “idealist”-style program of coming up with some foundational axioms (say, for set theory), and deriving all of mathematics from them without further reference to the outside doesn’t seem like the end of the story. It’s useful as a way of generating predictions in physics, but not of testing them. In mathematics, it generates many correct theorems, but doesn’t help identify interesting, or useful, ones.

So could category theory be used in foundations of mathematics? Maybe - but you could also say that mathematics consists of manipulating strings in a formal language, and strings are just words in a free monoid, so actually all of mathematics is the theory of monoids with some extra structure (giving rules of inference in the formal language). Yet monoid theory - indeed, algebra generally - is not mainly interesting as foundations, and probably neither is category theory.

On the whole, it was an interesting step out of the usual routine.

 

Recent Talk - Enxin Wu on Algebra Deformations

I’m going up to Ottawa for a few days, in part to talk about spans and groupoids (basically, some cross section of the material in these posts here) at a conference put on by the Ottawa U math department, primarily for grad students and postdocs in the general vicinity. This is nice - gives me a chance to visit my parents and friends there (the fraction of my life I lived in Ottawa is now creeping down toward a mere third, but it probably has as strong a claim to “home” as anywhere). May is also one of the most tolerable months to be there. One of the grad students in our department is also going. Enxin Wu recently decided to start working with Dan Christensen too, so probably in future we’ll have various things to talk about. Last week, he gave a seminar talk on algebra deformation that was a long version of the one he’ll be giving in Ottawa.

Enxin is one of those guys who seems to really understand - it’s tempting to say grok- algebra, which I always find impressive. I’m a predominantly visual thinker, and the kind of symbolic computations common in algebra always seem a little mysterious to me at first until I can find a picture, or at least practice them a lot. Lie groups, for instance, make some sense to me - you can picture rotation groups, or at least keep a geometric picture of a manifold in mind. Lie algebras, being infinitesimal versions of Lie groups, are also not so hard to visualize. General associative algebras? Harder.

The talk was about associative algebras, to give some background on deformation, but the things whose deformations Enxin has been thinking about are -algebras (see this brief intro, for instance), an “invention” of Stasheff. The talk was about deformation of these algebras - the kind of deformation that pertains to deformation quantization. This has been studied by Kontsevich. Deformation quantization has to do with replacing things valued in some algebra by new things, valued in the bigger algebra of formal power series in with coefficients in , so that the original structure you started with is just the constant part that appears when you set . (The term “quantization” applies when you consider algebras of functions on a manifold, with a Poisson bracket - in other words, algebras of observables of a physical system).

Some of the main results have to do with the Hochschild cohomology for some complex associated to the algebra you start with, and the fact that this cohomology classifies obstructions to the deformation. I expected to get lost in a maze of notation - and there certainly is a lot - but as it turns out, I had some mental pictures to attach to these things, because related things came up a few years ago in the quantum gravity seminar at UCR (week 8 on that page especially), which provides a few pictures that helped a lot. Diagrammatic notation makes algebra a lot more comprehensible to me.

So let’s get more specific.

The point is to replace a multiplication operator with a power series whose coefficients are “multiplication” operators. That is, a deformation of an associative algebra (where is the multiplication for ) is , where the new multiplication is defined (by linearity) by its action on elements of , which works like this:

for some operators . Then there are a bunch of conditions on the that are needed to make associative. There’s one condition for each power of , since the coefficients in the associator should be zero:

The condition just says that is associative - so it’s the from the original algebra, which you get back when .

Then given an algebra , you can create the deformation category of whose objects are its deformations. The morphisms are continuous algebra homomorphisms that get along with the multiplication operations. It turns out that since formal power series with nonzero term are invertible (a consequence of the Lagrange theorem) this is actually a groupoid. Then the question is to classify the isomorphism classes of deformations - that is, . One can easily imagine that there might be no nontrivial deformations of some algebra - that is, every one is isomorphic to the deformation where all the are trivial except . So when does this happen? More generally, how can one classify the deformations up to isomorphism?

The answer has to do with Hochschild cohomology, which is related to a complex you can make from . Taking , the space of -ary multilinear operations on , you build this complex:

where the differential maps are defined by an alternating sum:

(Intuitively: there are too many arguments, so you start with the extra one on the left, push it into the middle as a “lump under the rug” where two arguments are combined, and push the lump all the way to the right. To ensure that , you do this with alternating signs. This kind of algebraic manipulation is the kind of thing I can do, and clearly works, but I don’t exactly grok.)

Then you take the Hochschild cohomology groups in the standard cohomology way: . A cohomology class in one of these groups is a class of multilinear maps from copies of to (up to a factor which is of something). As usual with cohomology, they describe obstructions to something - to exactness. Exactness, in this setting, would mean that has no interesting deformations at the level.

What does “level” mean here? Well, for example, at level 2 we’re talking about maps , such as the multiplication map. In fact, we have for an associative algebra - you can check that is twice the associator , which is zero. So is a cochain. Is it a coboundary? Sure - it’s . So is in the trivial class in . The point then is that it turns out that if this is the only class - if - then there are no interesting deformations of the multiplication of in the sense described above. The groupoid $\mathcal{D}$ has just one object. (One thing that occurs to me is that this makes it a group - which group is something Enxin didn’t discuss. My algebra instincts aren’t quite up to answering that off the top of my head.) For example, if (as an algebra over ), there are no nontrivial deformations: .

What do the other levels mean? Really, this is where you’d want to look at the generalization from associative algebras to -algebras. Whereas for an associative algebra , the associator $a(x,y,z) = x(yz) - (xy)z$ is zero, in general an -algebra will have an associator map (that is, in the complex above), which might not be zero, but which is .

This is the beginning of a story relating -algebras to weak -categories: a bicategory, for example, has an associator for composition of morphisms. In a bicategory, you expect the associator to satisfy a certain identity - the Pentagon identity - but in general you’d just ask for a “pentagonator” (something in ), and so on (this is where those seminar notes above help me think in pictures, by the way). An -algebra is a vector space equipped with maps at all these levels - described by Stasheff’s associahedra - satisfying some relations. The general story of deformation relates the Hochschild cohomology groups at different levels to deformations of -algebras. Enxin didn’t go into this in his talk, but he did say a little something about the next level:

An infinitesimal deformation of is a deformation not in , but in the quotient . This only needs two maps, . The third Hochschild cohomology measures obstructions to extending an infinitesimal deformation to a full deformation in - if , then any infinitesimal deformation can be extended to a full deformation.

All in all, I thought the talk was interesting - it tied in much more closely to things I already knew about TQFTs and higher categories than I’d expected. I’ll be really impressed if he can condense it into a 25-minute version…

 

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 PI - Colloquium by Robert Spekkens

One of the first things I did after arriving at PI on Wednesday (and having lunch) was to attend the colloquium talk which was being given by Robert Spekkens. It was called “Why the Quantum?”, but as he described it, the real point of the talk was to take a close look at the features of quantum physics that are commonly considered “weird” or “mysterious” and see what’s really innovative in the departure from classical physics. For the most part, “physics” here means “mechanics”, but he also touched on optics, theory of computation, and briefly on electromagnetism and gravity in a more speculative way.

The main message of his talk is that very few of the things about quantum physics which seem strange are really all that innovative. He showed this by describing a kind of classical theory that has many of them - interference, noncommuting observables, entanglement, “wavefunction collapse”, wave-particle duality, teleportation and a no-cloning theorem, superposition of states, and so forth. All of these, he told us, will show up in a model based on a classical mechanical system, where the “quantum” theory is a theory of probability distributions (or, equivalently, of the knowledge of observers about a classical system) subject to a restriction about what distributions are allowed.

The point is to start with some classical system: let’s say it’s a mechanical system of some moving particles. Then there’s a configuration space of all the possible (classical) configurations of the system - one point in this space for each configuration. Classical mechanics is then about defining a “flow” on this space, which tells you where a point will move over time (how the system will go from one configuration to another). Then Liouville mechanics is about probability distributions in this space: you might not know exactly which configuration the system is in, but you have a way of estimating the probabilities. Then you impose the restriction that the only allowed probability distributions are ones for which the products of the variances for conjugate variables are at least Planck’s constant. (Actually, I think Spekkens formulated this differently, but that’s about what it amounts to, as I understand it.) The result is equivalent to “Gaussian quantum mechanics” - one where probability distributions are all Gaussians.

This also puts limits on what the rule for evolving states can be: any rule for how individual states evolve over time also gives a result for how probability distributions evolve over time. (Picture a cloud of ink, with varying density, flowing along in moving water - knowing the flow lines tells you where the cloud goes.) If there are restrictions on what kind of probability distributions can be set up, these have to be preserved over time - otherwise, you could set up an allowed distribution, and then wait until it evolves into a disallowed one. In particular, for Gaussian quantum mechanics, he told us that systems with a quadratic Hamiltonian will satisfy this condition.

The important fact here is that this is a “realist” interpretation. It says the quantum mechanical uncertainty reflects that QM is a theory about your knowledge of the state of the system, which, however, really exists. Often in quantum mechanics, one defines a “wave function” as a function living on configuration space (complex-valued, not real-valued like a probability density, but a function nonetheless). However, it’s now pretty standard to think of this wave function as the “real” state of the system - the view that it represents a state of knowledge was popular for a while, but ran into various problems in the form of experiments that are hard to account for, such as Bell inequality violations. The point of the talk was to see just how many of the “strange” features of quantum mechanics are genuine problems for this view, and to show the answer is “not many”.

The features he claimed are really mysterious from this point of view are fairly few: Bell inequality violations, some no-go theorems for models of physics involving local hidden variables such as the Kochen-Specker Theorem, and a few others. So Spekkens’ suggestion was that this concept of quantum mechanics as a theory of probability with an “epistemic” restriction (i.e. limits on what’s knowable) might be salvaged if the underlying classical theory were non-local - and perhaps had some other odd features yet to be precisely delineated - to begin with. However, it might not have to be terribly strange apart from that, since quantum mechanical features like interference and superposition of states all show up in the restricted statistical picture.

The gist of his argument then seemed to be that to really straighten out some foundational issues in quantum physics, one approach would be: (a) come up with a well-founded justification for the assumption about restrictions on possible probability distributions, and (b) come up with at least one (and as few as possible) other principles to account for the remaining mysterious things - he also suggested they all seem to have something to do with “contextuality”. As I understand it, this last is the idea that an observable might have definite, but multiple, values - and that which values are seen depend on which groups of observables are measured together. I don’t know what, if anything, to make of that oddball-sounding idea.

However, he did argue that in some cases at least, the restriction can be justified by the observer effect: you have to look at a system using some apparatus, whose state you don’t know completely, and which interferes with the system in order to observe it (for instance, measuring the position of a particle by scattering it off another one, whose state is partly unknown, and imparts an unknown momentum).

My overall reaction to the talk is that it’s interesting to know that realist interpretations of quantum physics (where the “reality” is more or less classical, and quantum effects some kind of afterthought, or epistemic effect) aren’t as dead as they might have seemed. However, the view that says classical physics emerges as some kind of limiting case of quantum effects seems better developed, at least mathematically, than the reverse. As for his claim that we “understand” the classical picture “physically”, whereas it’s not so for the quantum picture - I personally can only agree that’s true for me, but I don’t entirely see what you can conclude from that.

The bottom line seems to be that there are still problems in epistemology. I suspected as much already - though I’m not sure if I “knew” it, whatever that means.

 

2-Hilbert Spaces - pt 1

So one of the missing pieces in some of what I’ve been posting about recently is a discussion of 2-Hilbert spaces, and particularly the kind that categorify infinite dimensional Hilbert spaces. Part of the issue with these is that there are a number of ways of looking at them, and how these all fit together isn’t quite as clearly developed as with mere finite-dimensional 2-vector spaces.

I gave a little talk about this to our group at UWO, leading up to representation theory on 2-Hilbert spaces, which touches - potentially - on some of the stuff with spin foams that Wade and Igor are working on especially. It’s also a part of the project of trying to work out how the approach to extended TQFT’s I’ve described a bit should work for an infinite gauge group - in particular, a Lie group. The descriptions of these theories which I’ve given describe functors valued in , the 2-category of Kapranov-Voevodsky 2-vector spaces. These had a basis indexed by conjugacy classes in , and representations of their stabilizer subgroups - and for, say, , this is infinite.

Furthermore, a good categorification of a quantum field theory should use something deserving the name 2-Hilbert space (unless you prefer the -algebra approach to quantum theory which doesn’t pick a specific representation on a Hilbert space - this would also be interesting to categorify). So it needs some structure analogous to an inner product, complex conjugation, and so no. There are a number of concepts that stab in this direction, so in my talk I tried to summarize the main points.

An early reference on this is HDA II by John Baez, which defines 2-Hilbert spaces in a nice axiomatic way - abelian categories, enriched over , with a *-structure, satisfying some properties, etc. There are a few provisos: the version of things are presumed to be enriched over includes only finite dimensional Hilbert spaces. On the other hand, there’s no assumption - as there is for Kapranov-Voevodsky 2-vector spaces - that the category itself is finitely generated by simple objects. In other words, there’s no assumption of finite dimensionality for the 2-Hilbert space itself, but there is for its component Hilbert spaces. Then there’s a classification theorem which says what 2-Hilbert spaces in this sense are like. If they are finitely generated, then in particular they happen to be KV 2-vector spaces. But there is more structure, corresponding to two features of Hilbert spaces: the inner product, and complex conjugation.

The interesting thing about the inner product is that every KV 2-vector space is automatically equipped with one. Since it needs to be a map , the obvious choice is , which takes a pair of 2-vectors and gives a vector space - namely, the one containing all morphisms between these two objects. In , the components of a 2-vector are themselves inner-product spaces, so we have a little extra structure. It turns out this has all the properties it needs to be a categorified inner product. As for the equivalent of complex conjugation, the categorified version is just adjunction - it leaves objects as they are, but turns morphisms into their (componentwise, vector-space) adjoints. This process has some important properties, such as being an involution (like conjugation) and so on. This makes into a *-category.

There’s another possible approach to the subject, or a closely related subject, is described by David Yetter in this paper on measurable categories, and which Crane and Yetter use to support representations of 2-groups, the way Hilbert spaces can support representations of groups. This is a more concrete, constructive approach - like describing spaces of complex functions on a topological space, rather than giving an axiomatic definition of a Hilbert space. Actually, what they describe is more like the space of measurable functions on a space. These are measurable fields of Hilbert spaces on a measurable space - such a field defines (a) a Hilbert space at each point, and (b) a space of “measurable sections”, namely ways of picking a vector in the space at each point which are considered measurable. (There are some properties, like the fact that the function giving the local norm of these vectors at each point is measurable, plus some closure-type properties.)

Well, that’s measurable. Given a measure, so you can do integration, you can define something like spaces. Integration works by means of the direct integal, which produces not a scalar, but a Hilbert space; in this kind of categorification, takes the role of . The way this works is that the direct integral

as a vector space is just the whole space of measurable sections. The inner product of sections is

So integrability of a measurable field means not finiteness, per se (which we think of as saying that an inner product gives a well defined map ), but that this direct integral gives an object of (so the inner product integral should be finite, for instance, but also the space of measurable sections needs to be complete in the norm from this inner product). There is clearly a relationship between this way of describing an inner product and the way of describing it as a “hom-space”.

Some things are less clear… This gives a construction for how to get an “infinite dimensional 2-Hilbert space”. There doesn’t seem to be a known classification theorem here analogous to the one for KV 2-vector space, saying that this construction describes all “2-Hilbert spaces”. In fact, a general abstract definition of this concept seems to be a bit trickier than in the finite-dimensional case, and Crane and Yetter don’t really address it in their papers. One would hope that given a nice infinite-dimensional version of the usual definition of a 2-Hilbert space, this type would turn out to be generic.

Another question I’d like the answer to is - can one get one of these 2-Hilbert spaces from a (smooth, let’s say compact, probably) infinite groupoid, the way one can get 2-vector spaces (and, in particular, ones which can be made easily into finite-dimensional 2-Hilbert spaces) from an essentially finite groupoid? I think so - but there are some analysis issues to work out.  Assuming it works, this would be the right setting to support extended TQFT based on topological gauge theory with a Lie group like as gauge group.  (For some analysis reasons I may talk about later on, I only see reason to think this works with a compact group - but happily, that’s one right there!)

However, the question I’ll actually address in the second talk, which I’m giving on Friday, is how these are used for representation theory of 2-groups, since I’ve thought about that some, and some work has already been done with it - by, e.g. Crane, Yetter, Sheppeard, and also in some discussions I had the chance to participate in with John Baez, Laurent Freidel, Derek Wise, and Aristide Baratin (they are putting a paper together on the subject - as far as I know, not released yet).

 

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.

 

Morita Equivalence etc.

A recent talk in the noncommutative geometry seminar here, was by Farzad Fathizadeh. He was talking about a few ideas - the main part of the talk being about how to construct the Dixmier-Douady invariant, which is related to the question of whether or not you can put a spin structure on some manifold. It’s also related to a lof other things I want to figure out anyway for longstanding reasons. Indeed, Dixmier is one of the big early names behind the theory of fields of Hilbert spaces, which are used in Crane and Yetter’s “Measurable Categories”, which are a sort of infinite dimensional analog of the 2-vector spaces I’ve been talking about. (Actually, 2-Hilbert spaces, since that structure starts to look more important there).

Since I’ve started thinking about infinite dimensional 2-Hilbert spaces again I thought I’d check it out. It turned out to be somewhat related, but not very deeply. However, precisely because it’s related to things I’ve yet to figure out, I’m going to give a superficial gloss here, and later maybe try to say something more detailed. I should be giving a talk to our group soon about various aspects of 2-Hilbert spaces, so I’ll post more when I get to that.

…

The second part of the talk had a nice exposition of Morita equivalence, which was a notion that got a lot of use in the things people were talking about at Groupoidfest. I had heard about this concept before, but never quite got the hang of it until now, so here’s a quick little explanation for the record. There are two ways of describing Morita equivalence, and the content of Morita’s theorem is that the two definitions amount to the same thing.

One definition says that two algebras and are equivalent if the categories of modules over them, and are equivalent as categories. The other says that and are equivalent if there is an -bimodule, , and a -bimodule, with the properties that:

(as an -bimodule)

and

(as a -bimodule)

Where, if you’re unclear, an -bimodule is a set where the algebra acts on the left, and the algebra acts on the right, with a compatibility condition that looks like associativity: .

It shouldn’t be too hard to see that the second definition implies the first: given a (left) -module , you can turn it into a (left) module by taking , and given a -module, you turn it into an -module by similarly tensoring over on the left with . Doing both operations gives you . The assumptions on $\latex \mathcal{F}$ and $\mathcal{G}$ mean that this is equivalent to . The same (switching , \mathcal{G}$ and ) goes for a -module. So this is the equivalence from the first definition.

The hard part of Morita’s theorem is that any time you have an equivalence, you can represent it in terms of some bimodules and .

(This is related to the fact that there’s a bicategory structure for algebras in which the morphisms from to are -bimodules, and the 2-morphisms are bimodule homomorphisms. Composition works by exactly the kind of tensoring above. In fact, there’s a pseudocategory of rings, which has