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…

 

Categorification and Matter

First, the obligatory excuse found in most sporadic blogs: I haven’t taken the time to write anything here recently. I was busy for a while, between the trip to UC Davis to speak (giving a form of this talk) at the “Strings and Gravity” seminar there, and then catching up on teaching - the end of the term is coming up. There: now that’s out of the way.

Right now I want to say something a bit broader than I have been doing - somewhere between “intuitive justification” and “philosophy”. The motivation is that whenever I talk about ETQFT’s and how to see them as introducing matter into quantum gravity, there’s always some puzzlement about this “categorification” business. To people who think a lot about category theory, it may seem natural, but many of those interested in physical questions don’t fall in this category, and the whole idea of “categorifying” a theory seems like a weird, arbitrary imposition.

So talking to these different audiences has forced me to think about how to give an intuitive account of why this might be a good idea. Ideally this will not be so precise as to be incomprehensible, or so vague as to be useless. In reality, this will be at best a rough sketch of such a justification.

Stuff, Structure, and Properties

One aspect of the relationship which I wanted to comment on, one that almost seems like a pun, is the trichotomy which John Baez and Jim Dolan like to use in describing mathematical, um, widgets (I would use the more standard term “objects”, or maybe “structures”, but both of these words have technical meanings in the following) in categorical terms. This is the distinction between “stuff”, “structure”, and “properties”. (More details here and via subsequent links - some of which shows up in my first paper). Almost any usual mathematical widget can be broken down this way: (1) they consist of some “stuff”, often in the form of some sets; (2) the stuff is equipped with “structure”, often described by some functions; (3) the structure satisfies some “properties”, often expressed as equations.

For example: a group is (1) a set of elements, equipped with (2) a group operation (expressed as a function ), and a special identity element (picked out by a function from the one-element set, ), and an inverse for each element (given by an inverse function . These satisfy (3) the group axioms, which are some equations involving expressing some properties - associativity, the properties of and inverses.

In this case, the structure live inside the category of sets and functions - but similar things could be said in any other category. For instance, in the category of topological spaces and continuous functions, the same setup gives the definition of a topological group, likewise divided into “stuff” (objects, in this case topological spaces), “structure” (some morphisms), and “properties” (equations between morphisms).

Widgets which live in an -category of some kind have more of these layers - such a widget will be specified by one or more objects, equipped with specified morphisms and 2-morphisms, satisfying some equations. A monoidal category, for instance, is this kind of widget: it has a category worth of “elements”, equipped with a monoidal operation given as a functor, equipped in turn with specified 2-isomorphisms such as the “associator”, which satisfies some equations such as the Pentagon identity. There are now FOUR levels to specify. I think it was Jim Dolan who came up with the following way of extending the “stuff/structure/properties” terminology (his explanation).

The highest level - equations - always deserves the name “properties”, since they either hold, or don’t (at least, there’s a truth value associated to them - but let’s not worry about multiple-valued logics). By analogy, this suggests the data for our widget given by the -morphisms in the -category where it lives should be called “structure”. The -morphisms (which are the objects in a 1-category) should be called “stuff”.

For the , , and generally -morphisms, Jim introduces the prefix “eka”, as in “eka-stuff”, which follows Mendeleev’s nomenclature for elements predicted by his form of the periodic table of elements which were heavier than known ones. This nomenclature in turn comes from the Sanskrit “eka”, meaning “one” - the new elements were one level lower on the periodic table.

So specifying a widget in a 2-category involves “eka-stuff/stuff/structure/properties”. This is suggestive, in that it seems as if categorification - adding a new level - is like digging out a new sub-basement beneath a house. First “eka-stuff”, then “eka-eka-stuff”, and so on, to “eka^k-stuff”. Since, in many versions of -category, given two objects and , the totality of morphisms form an -category, this is somewhat correct: there is an -categorical structure describing each .

(The periodic-table analogy, I suppose, is meant to imply that the best-understood layer is the layer of equations - which describe properties. This opposes what is probably the more common intuition people have when first encountering higher categories, that we know what “objects” are, but find “higher morphisms” confusing. But when writing things concretely, it’s the highest-level morphisms which look most familiar, like functions.)

A key point here is that “stuff having structure satisfying properties” is a fairly intuitive framework for talking about things. Categorification gives us a more nuanced layering. It may seem odd to speak of “eka-stuff equipped with stuff equipped with structure satisfying properties” (even worse if you want to be consistent, and say “equipped with” instead of “satisfying”). But now the second layer - stuff, refers to 1-morphisms. Here is a layer which has some aspects we associate with “structure”: it describes relations between the eka-stuff (objects). On the other hand, it also has aspects we associate with “stuff” (it can be equipped with its own structure). When would one want something that is on the one hand something like a relational attribute between things (structure), and on the other hand something like an object in its own right (stuff).

One answer: to describe space. As a good Leibnizian, I prefer to think of space relationally: it describes how objects are situated in terms of structural relationships. On the other hand, General Relativity tells us that if we think about space, rather than spacetime, we need to describe it as having dynamics which satisfy some property. From this point of view, space is like material stuff that changes over time, according to some differential equation (classically, at least).

Matter = Stuff?

Now, part of the point of applying extended TQFT ideas to gravity is that the categorification introduces matter into the formerly empty background of topological gravity - in particular, the state of a bit matter is described by looking at the boundary conditions on a codimension-2 surface in spacetime (or codimension-1 surface in space) surrounding it. The “pun” I alluded to above is the idea that introducing matter amounts to introducing a new layer of “stuff”. Adding matter means adding “stuff”…

The pun isn’t quite dead on, however, because in the ETQFT setup, adding matter is actually adding “eka-stuff”: digging out a sub-basement on which the “stuff” of geometrized space and its dynamics can rest.

So how does the periodic table of stuff/structure/properties relate to an extended TQFT? To start with, consider the case of an ordinary TQFT in 2 dimensions. It’s well known that such TQFT’s correspond to commutative Frobenius algebras (though see e.g. this paper by Aaron Lauda and Hendryk Pfeiffer, where they explain this, and a generalization of it). That is, a TQFT defines an object with (1) Stuff: a vector space, equipped with (2) Structure: unit, counit, multiplication, and comultiplication maps, satisfying (3) Properties: a bunch of axioms, including the Frobenius relation, commutativity, and algebra axioms like associativity.

The key thing is that this correspondence comes from the fact that a 2D TQFT is a functor into from the category , which happens to be a symmetric monoidal category freely generated by one object (the circle), and some morphisms (corresponding to four cobordisms: the cap, cup, “pair of pants”, and “inverted pair of pants”), subject to just the topological relations making the circle with these maps into a “Frobenius object”. (Since the cobordisms are only defined up to diffeomorphism).

Then any actual “physical” setting will look like: a bunch of circles, say of them, connected to another bunch of circles, say of them, by some cobordism. We could call this a “string world sheet” (although not in the sense of string theory, exactly, since over there one typically has conformal structure on the cobordisms too, and talks about a CFT, not a TQFT, living on the sheet). In general, the cobordism will be an -punctured, genus- torus (with orientations that distinguish the inputs from the outputs). So if the dynamics of the “physical” world are described by a TQFT corresponding to Frobenius algebra , this topology will mean the space of states of the world is given by at the beginning and at the end (this is “stuff”). A state evolves through “time” by the morphism (”structure”) corresponding to the cobordism - a particular combination of multiplication and comultiplication maps for the

In a theory of gravity without matter, we can see three levels as well - “slices” of space with some geometric information, connected by spacetimes with geometric information, which satisfy some equations. In particular, the geometric information on spacetime has to satisfy Einstein’s equation, if we’re talking about the classical world, or some sort of Hamiltonian constraint in (some approaches to) quantum gravity. In any case, it must have some property to be admissible. So this suggests the classifications: “space geometry” - stuff; “spacetime geometry” - structure; “dynamical laws” - properties.

Categorification suggests adding to this list: “matter/boundary conditions” - eka-stuff. That is, the eka-stuff in a specific physical setting will be a “2-space of states” for matter as measured at a particular boundary. In a 3D ETQFT, for instance, the boundaries to space will be unions of circles (just as in a 2D TQFT), so this will be generated by a 2-space of states for a circle. The circle could be thought of as the boundary around a single excised particle, but in fact that only covers the irreducible 2-states: in general, it’s a boundary around some region containing a system. Space geometry relates such boundaries to each other: it is “stuff” relating the “eka-stuff”. That stuff (space geometry), in turn, can be equipped with structure - maps associated to a spacetime topology, which describe how it evolves in “time” (though a-priori there’s no special time direction - the “stuff” could equally well describe the world-sheet of the system boundary, and the structure describing how that evolution extends outward spatially).

It seems to me there’s a lot here, but to really say it properly would require being much more technically precise than I’m up to at the moment. So that’s about all I have to say about that.

 

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.

 

Visiting Perimeter Institute

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

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

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

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

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

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

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

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

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

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

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

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

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

 

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

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

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

…

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

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

1) Representations: A functor

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

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

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

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

2) 1-Intertwiners:

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

To each (object of it gives a transformation

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

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

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

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

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

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

3) 2-intertwiners

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

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

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

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

 

n-group Representation Theory - (part 2: the Poincaré 2-group)

It’s been a while since I wrote the last entry, on representation theory of n-groups, partly because I’ve been polishing up a draft of a paper on a different subject. Now that I have it at a plateau where other people are looking at it, I’ll carry on with a more or less concrete description of the situation of a 2-group. For higher values of , describing things concretely would get very elaborate quite quickly, but interesting things already happen for . In particular, the case that I gave the talk about, a while back, was mostly the Poincaré 2-group, since this is the one Crane, Sheppeard, and Yetter talk about, and probably the one most interesting to physicists.  It was first described by John Baez.

So what’s the Poincaré 2-group? To begin with, what’s a 2-group again?

I already said that a 2-group is a 2-category with only one object, and all morphisms and 2-morphisms invertible. That’s all very good for summing up the representation theory of as I described last time, but it’s sometimes more informative to describe the structure of concretely. A good tool for doing this is a crossed module. (A lot more on 2-groups can be found in Baez and Lauda’s HDA V, and there are some more references and information in this page by Ronald Brown, who’s done a lot to popularize crossed modules).

A crossed module has two layers, which correspond to the morphisms and 2-morphisms of . These can be represented as , where is the group of morphisms in , consists of the 2-morphisms ending at the identity of (a group under horizontal composition).

There has to be an action of on (morphisms can be composed “horizontally” with 2-morphisms), and a map (which picks out the source of the 2-morphism). The data have to fit together a certain way, which amounts to giving the axioms for a 2-category.

A handy way to remember the conditions is to realize that the action and the injection give ways for elements of to act on each other and for elements of to act on each other. These amount to doing first and then or vice versa, and both of these must amount to conjugation. That is:

and

Both of these are simplified in the case that maps everything in to the identity of - in this case, can be interpreted as the group of 2-automorphisms of the identity 1-morphism of the sole object of . In this case, by the Eckmann-Hilton argument (the clearest explanation of which that I know being the one in TWF Week 100) it turns out that has to be commutative, so the first condition is trivial since , and the second is trivial since it follows from commutativity. This simpler situation is known as an automorphic 2-group.

In any case, given a 2-group represented as a crossed module, automorphic or not, the collection of all morphisms can be seen as a group in itself - namely the semidirect product , which is to say with the multiplication . “What?” you may ask, or maybe “Why?”

Maybe a concrete example would help, since we’d like one anyway: the Poincaré 2-group, which is an automorphic 2-group. There are versions of various signatures , in which case , and .

The group , then, consists of metric-preserving transformations of Minkowski space with the metric of signature - rotations and boosts (if any). The (abelian) group consists of translations of this space - in fact, being a vector space, it’s just a copy of it. Between them, they cover the basic types of transformation. Thinking of the translations as having a “projection” down to the identity rotation/boost may seem a bit artificial, except insofar as translations “don’t rotate” anything. More obvious is that rotations or boosts act on translations: the same translation can look differently in rotated/boosted coordinate systems - that is, to different observers.

So where does the Poincaré group come in? It’s the group of all metric-preserving transformations of Minkowski space, and is built from these two types: but how?

Well, the vector space is the group of transformations of the identity Lorentz transformation , since the map is trivial. But suppose that there is another copy of over each point in . Then we have the set of points , but notice that to talk about this as a group, we’d want a way to act on an element of one copy of over by another over . The obvious way is to just treat the whole set as a product of groups, but this misses the fundamental relation between and , which is that can act on , just as morphisms can act on 2-morphisms by “whiskering with the identity”. (Via Google books, here is the description of this in MacLane’s Categories for the Working Mathematician).

Concretely, this is the fact that there is a sensible way for both parts of to affect the , so we can say (using additive notation for translations, since they’re abelian). The point is that the first rotation we do, , changes coordinates, and therefore the definition of the translation .

So that’s the construction of the Poincaré group from the Poincaré 2-group. What would be nice would be to have some clear description of some higher analog of Minkowski space where it makes sense to say the Poincaré 2-group acts as a 2-group. I don’t quite know how to set this up, but if anyone has thoughts, it would be interesting to hear them.

One reason is that, when describing representations of the 2-group, there’s an important role for spaces (or at least sets) with an action of the group - which raises questions like whether there’s a role for 2-spaces with 2-group actions in representation theory of higher -groups. Again - I don’t really know the answer to this. However, in Part 3 I’ll describe concretely how this works for 2-groups, and particularly the Poincaré 2-group.

 

n-Group Representation Theory - Part 1

Recently I finished up my series of talks on 2-Hilbert spaces with a description of the basics of 2-group representation theory, and a little about the special case of the Poincaré 2-group. The main sources were a paper by Crane and Yetter describing 2-group representations in general, and another by Crane and Sheppeard. The Poincaré 2-group, so far as I know, was first explicitly mentioned by John Baez in the context of higher gauge theory. It’s an example of a kind of 2-group which can be cooked up from any group and abelian group , and which is related to the semidirect product .

One reason people are starting to take an interest in the representation theory of the Poincaré 2-group is that representations of the Poincaré group (among others) and intertwiners between them play a role in spin foam models for field theories such as BF theory, various models of quantum gravity, and so on. Some of these, turn up naturally when looking at TQFT’s, and generalizations of these, which is how I got here. Extending this to 2-groups gives a richer structure to work with. (Whether the extra richness is useful is another matter).

Before getting into more detail, I first would like to take a look at representation theory for groups from a categorical point of view, and then see what happens when we move to -groups - that is, when we categorify.

To begin with, we can think of a representation of a group as a functor. The group can be thought of as a category with one object and all morphisms invertible - so that the group elements are morphisms, and the group operation is composition. In this case, a representation of the group is just any functor:

since this assigns some one vector space (the representation space, ) to the one object of , and a linear map