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.

 

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.

 

TQFT and Gravity - pt 2

So last time I was describing this “matter without matter” idea and claiming that it has something to do with TQFT and the Ponzano-Regge model of quantum gravity. I’d like to get a little more detailed here.

To describe this in physics terms, it’s easiest to understand the point if, instead of using the (more technically accurate) terms “manifold”, “cobordism between manfolds”, and “cobordism with corners between cobordisms, I name-drop the terms “boundary”, “space”, and “spacetime”. But the caveat here is that these terms really imply a certain geometric structure which I’m not actually assuming is there: a specific geometric structure on these manifolds is a state of the theory. Furthermore, with Ponzano-Regge, we’re talking about Riemannian gravity - there’s no such thing as a “timelike” direction. So using the term “spacetime” is being rather optimistic that everything will work out in more physical settings - but it’s a helpful motivation.

At any rate, the way I describe it in the thesis, in dimensions the typical setup for an extended TQFT in the sense of a weak 2-functor into 2-Vect, one has “boundaries”, which are manifolds of dimension (in 3D, each boundary is some union of a bunch of circles, and in 4D it would be a union of surfaces, each with some genus). These are joined by “spaces” (cobordisms), of dimensions, which are in turn connected by “spacetimes” (with the above caveat). These cobordisms are, in particular, cospans in some category of spaces, and they give rise to spans of groupoids of configurations for a gauge theory.

In any case, how does this relate to gravity? The answer is by way of topological gauge theory: the extended TQFT in question has a lot to do with flat connections on manifolds (or indeed manifolds with boundary or corners), which is what topological gauge theory is about. One way to say what a flat connection is, is to say that it takes a path in the space , and gives an element of the gauge group (this is not the most well-known way to describe a flat connection - more on that in another post, but I’ll cite weeks 8 and 9 of the spring 2005 UCR Quantum Gravity Seminar for now).

If the gauge group represents the symmetries of something we’re transporting around the surface, this tells us how that thing is being transformed as we move it. For gravity, we take the gauge group to be the symmetries of a model spacetime - what spacetime “looks like locally”. For standard special relativity, this is the Lorentz group - the symmetries of Minkowski space. For 3D gravity, it’s (symmetries of Minkowski space with two space and one time dimension). For 3D Riemannian gravity, it’s the group of rotations in 3D. Actually, I lied: each of these has a double cover, and this is the gauge group (which allows for a spin structure. To simplify a lot of things in my thesis, I talk about the case where is some finite group, but eventually I’d like it to be , the double cover of the rotation group .

So we imagine the connection tells us how an observer would be rotated by the act of moving along a path. (There is a kind of trivialization of a bundle lurking behind this glib statement, but I’m putting that off). Now, some connections are physically the same, even though we describe them differently. They are related by gauge transformations, which are symmetries of the connections themselves. These amount to a way of changing the coordinate system in which we describe (say) our rotation: two rotations of 60 degrees around different axes are not “really” different, since the observer can turn one into the other by tilting her head. What’s traditionally done is to “mod out” by gauge transformations: take any two connections related in this way to be just the same, and throw away any information that distinguishes them. Instead, we can organize flat connections into a category - in fact, a groupoid - where the objects are the connections, and the morphisms are the gauge transformations. We can organize this into the category of functors from the fundamental groupoid of a manifold into the gauge group (thought of as a one-object category).

What’s the point - from a physical point of view - of keeping all the extra structure of these morphisms? To make a long story short, they’re what ends up allowing the theory to classify particles as having spins, not just masses. (Incidentally, I notice that Marni Sheppeard made a guest post on another blog arguing that category theory is useful to physics. Here is another example of how this can be so. Morphisms encode information that would be absent without them, and which has a straightforward physical meaning.)

How does this extra information appear? Well, first of all, what is a point particle, in this model? It’s represented as a boundary around a puncture in “space” - a circular boundary in a 2D surface of some shape or other. The fundamental groupoid of the circle has objects which are points of the circle, and morphisms which are (homotopy classes of) paths. There is an equivalence of categories between this and the fundmental group of the circle, which we can think of as a category with just one object (this is because the circle is a connected space).

Then we’re looking at a category of functors between a couple of one-object categories. Since , these are determined by the image of the generating path, “1″. So the groupoid of flat connections on this boundary has objects which correpond just to elements of . But wait! There’s more! You also get natural transformations between these functors! These amount to just conjugations relating elements of (those “coordinate transformations” I mentioned before). So the whole groupoid has objects corresponding to elements of , and morphisms for each such that . We call this whole groupoid by the name - or “ weakly modulo the adjoint action of .

This is also equivalent (as a category) to a smaller category I’ll call - the “skeleton” of , namely, a category with one object for each isomorphism class of objects in (i.e. each conjugacy class in ). Each of these has a group (the original category was a groupoid, so the new one is also) of automorphisms. This will be the same as the group of automorphisms of the corresponding object in - namely, the stabilizer subgroup of that element of , which, if is generically , except for a couple of exceptional points corresponding to 0-degree and 360-degree rotations.

Finally, a 2-vector in the 2-vector space assigned to the circle (which I like to think of as a “2-state”) is a functor from this into . Each such functor is a direct sum of a bunch of irreducible ones, and the irreducible ones assign a nontrivial vector space to just one object - and the group of automorphisms of that object are taken to a group of automorphisms of . That is, is specified by a conjugacy class of , and a representation of its stablizer subgroup. If , this is an angle and a spin. And in 3D gravity, the mass of a particle corresponds to an angle, because Einstein’s equation here says that space is locally flat, except where there is matter - where there is an amount of curvature proportional to the mass. This shows up as an “angle deficit” - an amount by which you end up rotated if you travel around the particle.

So that’s how you can see a “hole” in “space” as a point particle with mass and spin in this kind of extended TQFT. In higher dimensions, something similar happens, but the classification is more complicated, because in general the matter looks like “stringy” loops (this is something Derek Wise has looked at in his thesis). Also, above 3D, a theory of flat connections is no longer a theory of gravity, but rather something called BF theory - although in 4D it happens to be a limit of the theory of gravity as you allow Newton’s constant to approach zero. (That is, it describes the topological sector of the theory of gravity.)

What I haven’t yet explained is how this matter, which so far has the properties we might hope for, also gets to live in a spacetime governed by the Ponzano-Regge model. That means looking at what the extended TQFT does to the morphisms and 2-morphisms of the cobordism category - to “space” and to “spacetime”, and what the “2-linear maps” and “transformations” they give are like. Tune in next installment…

 

TQFT and Gravity - pt 1

With my thesis available on the arxiv, I thought I should see what I can say about the, as it were, dangling participle of that particular snapshot of this research project. That is, back when I had to declare a title for the thing, quite a long while before I had to finish it, I called it “Extended TQFT’s and Quantum Gravity”, thinking that this would be an accurate title, because it pretty well described the subject of the weekly conversations I’d been having with John while working on it.

However, one thing that gradually becomes clearer as I go further into the process of research is that it’s hard to predict exactly what that process is going to produce. (”Prediction is hard - especially when it comes to the future”, as Yogi Berra said - though possibly it was someone else, since accurate information about the past doesn’t exactly grow on trees either). It turned out that a lot of what I really did was proving some well known folklore theorems about 2-vector spaces; spending a few weeks trying to get a good proof that the weak 2-functor I constructed was actually a weak 2-functor (I still have a kind of unenlightening calculation for a proof); and lots of similarly technical stuff. All of which is - I hope - good mathematics, or at least correct mathematics. But is it physics?  All the references to the physical applications were left to the last section, a kind of sketch of where I expect the project to go.

I think the project does indeed have some nice intimate relations to quantum gravity (at least in 3 dimensions), it just didn’t turn out that there was a lot of material about those relations in the document. Instead, there’s a rather impressionistic sketch of how it ought to work. But you might not get the impression that Derek Wise and I started off working on the same project, though we did. Derek’s thesis is not available online in its entirety yet (though part of it appears in this paper on MacDowell-Mansouri gravity and Cartan geometry), but if you check out this this paper by Derek, John, and Alissa Crans, you see a little overlap.

What is the overlap? The physics of it is rooted in a fairly old idea ususally attributed to Wheeler, called “matter without matter” (John cites a number of references on this in week 208 of “This Week’s Finds”). There are several variants of this idea, but all of them in some way contain the key ingredient that matter should somehow be an expression of the shape of spacetime itself. Some older versions hold that elementary particles should be seen as the mouths of little wormholes. More recent ideas, based on spin networks (originally introduced by Roger Penrose in this paper, and much developed since) represent space as a kind of (labelled, directed) graph with edges connecting nodes - and these recent ideas suggest that a stray edge in a spin network will act just like a particle with the spin associated to that edge.

An example of a theory that fits this last picture, and the thing that most directly inspired the project described in my thesis, is some work of Laurent Freidel, David Louapre, and Etera Livine - a series of papers on the Ponzano-Regge model (parts I, II, and III) which is a model of 3-dimensional Riemannian quantum gravity. This is pretty unphysical - since the standard picture of gravity in the physical world is in terms of 4-dimensional, Lorentzian gravity (which, unlike the Riemannian picture, distinguishes between spacelike and timelike directions). Nevertheless, most people would accept the Ponzano-Regge model as physics… Anyway, their model describes a world where gravity is described by the Ponzano-Regge model, and is coupled to matter which is represented as stray ends of edges in the spin network. As the networks evolve, the stray edges trace out Feynman diagrams for the matter in question.

I could also mention that Laurent, together with Aristide Baratin, has recently done some work going in the other direction - starting with Feynman diagrams and trying to show how a picture of quantum gravity was already hidden in them, but with the gravitational coupling “turned off”. They have a couple of papers doing this in both three and four dimensions.

In any case, this version of “matter without matter” was a major part of the inspiration for
this project, but I describe things from a somewhat different point of view - or at least a dual point of view. When you describing the geometry of space in terms of a spin network, nodes in the network represent volumes in space, and edges in the network represent boundaries between volumes. This is a Poicaré dual picture - it’s also a picture that depends on a triangulation, or some other way of breaking a manifold apart into cells. I allude to this in the beginning of the thesis, talking about the Fukuma-Hosono-Kawai construction for getting a topological quantum field theory in 2 dimensions. However, one of the nice things about this construction is that it ends up being independent of which triangulation you pick (I have an explanation of this in these slides for a talk I gave last year at the Perimeter Institute). So after a bit, we just end up thinking of matter as living on boundaries of some kind.

The idea is that you have a manifold supporting some sort of geometric structure. The manifold has some “defects” - boundaries where that structure has to stop. It could be a 2D surface with some holes bunched out with a hole-punch - holes with a 1D boundary. Or it could be a 3D space with some 2D surface as the boundary. These could be literal defects - the boundaries describe where a pointlike, or line-like “flaw” in the geometry can live, because part of the manifold is just missing. This is the usual way of thinking about singularities. Or, you can just imagine that the boundary marks out some kind of “system” sitting in space that you might want to observe, and the theory tells you what information about the system on the other side of that boundary can be detected by looking at the geometric structure of the space around it.

Now, if we’re looking at 3D space, then gravity is fairly simple. Up to equivalence (i.e. up to a change of coordinates) the information about matter which we expect to be carried by the geometry of the space it lives in would include its (rest) mass and its momentum - in particular, its angular momentum, or spin. Different types of particles - as far as their effects on gravity allows us to tell them apart - are classified by their masses and spins. Any other information about them doesn’t directly affect the geometry of space. What’s more, in 2-dimensional space, particles look like single points - and all the curvature of space is concentrated at those points, leaving it flat everywhere else. The spin gives information about a “skew” in the geometry of 3D spacetime around the worldlines of such points.

In fact, this is just what this extended TQFT business allows us to recover about - but only because we have information about three levels: “boundaries” (around a system, in which the matter lives), “space”, and “spacetime”. And this is what has to be organized into some kind of 2-category…

(more to come on that in pt 2)

 

Coming Attractions

Due to the rapid-fire the nature of the blogosphere (or, in deference to John Armstrong, the “Blathysphere”, or maybe “blathyscape”), my blog (”blath”) has been discovered before I expected, and in particular before I’ve had the chance to put anything very interesting in it. So here I’ll just say something about “coming attractions” - a sort of mid-level executive summary of the next batch of things I expect to be working and commenting on. Also possibly later on I should have a math post or two about some talks I saw recently.

Since I graduated at UCR in June, I haven’t had much chance to do any actual work - partly because I broke my wrist in a bike accident, and lost the use of my writing hand for six weeks. Between that and the hassle of moving, I wasn’t able to do much but some reading. Now that the cast is off, I’ve been getting back to work. The first “real” research-related post I expect to make will be an announcement that a (slightly) polished version of my dissertation, “Extended TQFT’s and Quantum Gravity” has been released on the preprint archive - hopefully this week. That in turn should kick off some descriptions of what’s inside as I get more into the process of turning it into some smaller, more digestible papers.

These will fall, at first, into three parts:

1) A paper which has already been posted as math.CT/0611930, describing how to get a “double bicategory” of cobordisms with corners, and from that, a bicategory. Here I explain how cobordisms are cospans of manifolds with boundary, so the new structures are double cospans of manifolds with corners, and how that works.

This may end up being two parts. One is a decription of Dominic Verity’s notion of a “double bicategory”, an aside on how to interpret it as a special case of bicategories internal to , and how to get one from double spans (functors ). Marco Grandis has a pretty thorough description of these in this paper and its sequels, although our approaches are slightly different.

The second part has to do with how to apply this to cobordisms with corners (cobordisms between cobordisms) - also something Grandis discusses in the second paper of that series. I also need to show how to collapse the more complicated structure to a mere bicategory, in order to do what I will want to do in part (3) below.

There’s an issue here I’ll want to think about at some point, related to a question Aaron Lauda raised. The question was this. The category whose objects are 1-D manifolds and whose morphisms are 2D cobordisms between them has a nice abstract description. It is the free symmetric monoidal category with a Frobenius object.

In Aaron’s work with Hendryk Pfeiffer, they likewise described a category of “open closed strings”, which can have either 1-D manifolds or 1-D manifolds with boundary (collections of circles and line segments, basically) as objects, and cobordisms between them as morphisms. They showed this has a similar characterization, but with “Knowledgeable Frobenius” replacing “Frobenius” in the above. These have a nice description in terms of adjunctions, so Aaron was asking me if the same could be done for the double bicategory I talk about. That would need a concept of adjunction in double categories (or cubical n-categories, more generally). I don’t know what the state of understanding is on this.

More generally, it’s strange that “cobordisms of cobordisms” really wants to be a cubical 2-category in some sense, whereas, to do what I want to do with them (see below), I have to convert them into a globular one, to take functors into . I don’t know the best way to deal with this: is there a cubical version of , for example?

2) One part will deal with building 2-vector spaces from groupoids using functors into the category ; and 2-linear maps from spans of groupoids, using the pullback (composition) along an inclusion, and its (two-sided) adjoint. Along the way, it includes some proofs of well-known folklore theorems about 2-vector spaces which are hard to find anywhere. I plan to give a talk based on this at Groupoidfest ‘07 in Iowa City in November.

Soon enough - certainly before the Groupoidfest, I’ll have a bigger post about this stuff (and most likely post slides). The basic idea is that the category of functors from an essentially finite groupoid into is a Kapranov-Voevodsky 2-vector space - that is, a $\mathbbm{C}$-linear additive category which is generated by a finite number of simple objects. (The fact that this definition is equivalent to the one given by Kapranov and Voevodsky is one of those theorems which seems to be well known, but hard to track down). The finite number of simple objects correspond to the equivalence classes of . From a span of groupoids, it is possible to build a linear map between the corresponding 2-vector spaces.

The motivation for building 2-vector spaces on groupoids in the new work is to categorify the quantization of a classical system, but the two ways I’ve looked at are a bit different in how they accomplish it. Ignoring complications like symplectic geometry for the moment, the configuration space of a classical system is described as a set . Each element of the set is one possible state of the system. The corresponding quantum system will have states which live in - in particular, they are complex-valued functions on the set . And instead of being able to read off values like position, momentum, energy, and other features of the system by looking at the value these have at a single point, you need some algebra of operators on , whose eigenvalues are the values you can observe for the observable that corresponds to a given operator. In categorifying this, becomes a groupoid, in which the elements of the set can be related to each other - by “symmetries”. Instead of functions into the complex numbers, we take functors into , and obtain a 2-vector space of what I suppose should be called “2-states”. Given spans of groupoids, it becomes possible to get linear maps from one 2-vector space to another, using “pullback” and “pushforward” of these functors into .

I’ll say more about this later on, but one thing that I find perplexing about this is how (if at all), it relates to some earlier work I did in this paper on the categorified harmonic oscillator, which is heavily based on this paper by John Baez and Jim Dolan, which introduces “stuff types”. Both involve groupoids, and spans of groupoids giving rise to linear operators, as part of a categorification of some elementary quantum theory, but there are significant differences. At some point, I’d like to return to the question of whether they’re related, and if so, how.

3) One part uses the above to build an “extended TQFT”. A TQFT, or topological quantum field theory is a quantum field theory, in that it gives a Hilbert space of states for some field on a specifed “space” (i.e. manifold), and linear maps associated to “spacetimes” (cobordisms) joining them. It is topological, in that its states are topologically invariant - that is, they have no local degrees of freedom, only global ones. These started life in physics, but have fallen by the wayside there, and now mostly find life in the subject of quantum topology, where they give manifold invariants.

A TQFT can be described as a functor from a category of manifolds and cobordisms (see (1)) into . This way of putting it makes it relatively easy to see what to do if one wants to categorify - which we do, in order to get higher codimension (more on this later, I’m sure). The idea is to build a 2-functor from the bicategory of cobordisms with corners (see (1)) into . This can be done using gauge theory. The main idea is to turn a cobordism, seen as a cospan of manifolds (with corners) into a span of groupoids - namely, the groupoids of flat connections on these spaces, with gauge transformations as morphisms, and then build 2-vector spaces and 2-linear maps, etc. as laid out in the program of (2) above. The main theorem proving that such a 2-functor exists and is given by this construction was the organizing theme of my dissertation defense talk. This part is the mathematical core of what I’ve been working on.

4) Finally, this is supposed to be related to quantum gravity somehow. I’ll put off talking about this until I actually put the thesis on the archive.

Until then, I may decide to post a little about some talks I’ve been to recently. UWO has a great department with lots of interesting talks. I recently attended a couple of these by graduate students. One was by Arash Pourkia, about Braided Categories and Hopf Algebras. The second was by Michael Misamore, on Galois Theory - from the point of view of Grothendieck, and could equally well be called “Covering Spaces”… from the point of view of Grothendieck.