“Observer Space”: Cartan Geometry and Lifting General Relativity

This entry is a by-special-request blog, which Derek Wise invited me to write for the blog associated with the International Loop Quantum Gravity Seminar, and it will appear over there as well.  The ILQGS is a long-running regular seminar which runs as a teleconference, with people joining in from various countries, on various topics which are more or less closely related to Loop Quantum Gravity and the interests of people who work on it.  The custom is that when someone gives a talk, someone else writes up a description of the talk for the ILQGS blog, and Derek invited me to write up a description of his talk.  The audio file of the talk itself is available in .aiff and .wav formats, and the slides are here.

The talk that Derek gave was based on a project of his and Steffen Gielen’s, which has taken written form in a few papers (two shorter ones, “Spontaneously broken Lorentz symmetry for Hamiltonian gravity“, “Linking Covariant and Canonical General Relativity via Local Observers“, and a new, longer one called “Lifting General Relativity to Observer Space“).

The key idea behind this project is the notion of “observer space”, which is exactly what it sounds like: a space of all observers in a given universe.  This is easiest to picture when one has a spacetime – a manifold with a Lorentzian metric, – to begin with.  Then an observer can be specified by choosing a particular point in spacetime, as well as a unit future-directed timelike vector .  This vector is a tangent to the observer’s worldline at .  The observer space is therefore a bundle over , the “future unit tangent bundle”.  However, using the notion of a “Cartan geometry”, one can give a general definition of observer space which makes sense even when there is no underlying .

The result is a surprising, relatively new physical intuition is that “spacetime” is a local and observer-dependent notion, which in some special cases can be extended so that all observers see the same spacetime.  This is somewhat related to the relativity of locality, which I’ve blogged about previously.  Geometrically, it is similar to the fact that a slicing of spacetime into space and time is not unique, and not respected by the full symmetries of the theory of Relativity, even for flat spacetime (much less for the case of General Relativity).  Similarly, we will see a notion of “observer space”, which can sometimes be turned into a bundle over an objective spacetime , but not in all cases.

So, how is this described mathematically?  In particular, what did I mean up there by saying that spacetime becomes observer-dependent?

Cartan Geometry

The answer uses Cartan geometry, which is a framework for differential geometry that is slightly broader than what is commonly used in physics.  Roughly, one can say “Cartan geometry is to Klein geometry as Riemannian geometry is to Euclidean geometry”.  The more familiar direction of generalization here is the fact that, like Riemannian geometry, Cartan is concerned with manifolds which have local models in terms of simple, “flat” geometries, but which have curvature, and fail to be homogeneous.  First let’s remember how Klein geometry works.

Klein’s Erlangen Program, carried out in the mid-19th-century, systematically brought abstract algebra, and specifically the theory of Lie groups, into geometry, by placing the idea of symmetry in the leading role.  It describes “homogeneous spaces”, which are geometries in which every point is indistinguishable from every other point.  This is expressed by the existence of a transitive action of some Lie group of all symmetries on an underlying space.  Any given point will be fixed by some symmetries, and not others, so one also has a subgroup .  This is the “stabilizer subgroup”, consisting of all symmetries which fix .  That the space is homogeneous means that for any two points , the subgroups and are conjugate (by a symmetry taking to ).  Then the homogeneous space, or Klein geometry, associated to is, up to isomorphism, just the same as the quotient space of the obvious action of on .

The advantage of this program is that it has a great many examples, but the most relevant ones for now are:

• -dimensional Euclidean space. the Euclidean group is precisely the group of transformations that leave the data of Euclidean geometry, lengths and angles, invariant.  It acts transitively on .  Any point will be fixed by the group of rotations centred at that point, which is a subgroup of isomorphic to .  Klein’s insight is to reverse this: we may define Euclidean space by .
• -dimensional Minkowski space.  Similarly, we can define this space to be .  The Euclidean group has been replaced by the Poincaré group, and rotations by the Lorentz group (of rotations and boosts), but otherwise the situation is essentially the same.
• de Sitter space.  As a Klein geometry, this is the quotient .  That is, the stabilizer of any point is the Lorentz group – so things look locally rather similar to Minkowski space around any given point.  But the global symmetries of de Sitter space are different.  Even more, it looks like Minkowski space locally in the sense that the Lie algebras give representations and are identical, seen as representations of .  It’s natural to identify them with the tangent space at a point.  de Sitter space as a whole is easiest to visualize as a 4D hyperboloid in .  This is supposed to be seen as a local model of spacetime in a theory in which there is a cosmological constant that gives empty space a constant negative curvature.
• anti-de Sitter space. This is similar, but now the quotient is – in fact, this whole theory goes through for any of the last three examples: Minkowski; de Sitter; and anti-de Sitter, each of which acts as a “local model” for spacetime in General Relativity with the cosmological constant, respectively: zero; positive; and negative.

Now, what does it mean to say that a Cartan geometry has a local model?  Well, just as a Lorentzian or Riemannian manifold is “locally modelled” by Minkowski or Euclidean space, a Cartan geometry is locally modelled by some Klein geometry.  This is best described in terms of a connection on a principal -bundle, and the associated -bundle, over some manifold .  The crucial bundle in a Riemannian or Lorenztian geometry is the frame bundle: the fibre over each point consists of all the ways to isometrically embed a standard Euclidean or Minkowski space into the tangent space.  A connection on this bundle specifies how this embedding should transform as one moves along a path.  It’s determined by a 1-form on , valued in the Lie algebra of .

Given a parametrized path, one can apply this form to the tangent vector at each point, and get a Lie algebra-valued answer.  Integrating along the path, we get a path in the Lie group (which is independent of the parametrization).  This is called a “development” of the path, and by applying the -values to the model space , we see that the connection tells us how to move through a copy of as we move along the path.  The image this suggests is of “rolling without slipping” – think of the case where the model space is a sphere.  The connection describes how the model space “rolls” over the surface of the manifold .  Curvature of the connection measures the failure to commute of the processes of rolling in two different directions.  A connection with zero curvature describes a space which (locally at least) looks exactly like the model space: picture a sphere rolling against its mirror image.  Transporting the sphere-shaped fibre around any closed curve always brings it back to its starting position. Now, curvature is defined in terms of transports of these Klein-geometry fibres.  If curvature is measured by the development of curves, we can think of each homogeneous space as a flat Cartan geometry with itself as a local model.

This idea, that the curvature of a manifold depends on the model geometry being used to measure it, shows up in the way we apply this geometry to physics.

Gravity and Cartan Geometry

MacDowell-Mansouri gravity can be understood as a theory in which General Relativity is modelled by a Cartan geometry.  Of course, a standard way of presenting GR is in terms of the geometry of a Lorentzian manifold.  In the Palatini formalism, the basic fields are a connection and a vierbein (coframe field) called , with dynamics encoded in the Palatini action, which is the integral over of , where is the curvature 2-form for .

This can be derived from a Cartan geometry, whose model geometry is de Sitter space .   Then MacDowell-Mansouri gravity gets and by splitting the Lie algebra as .  This “breaks the full symmetry” at each point.  Then one has a fairly natural action on the -connection:

Here, is the part of the curvature of the big connection.  The splitting of the connection means that , and the action above is rewritten, up to a normalization, as the Palatini action for General Relativity (plus a topological term, which has no effect on the equations of motion we get from the action).  So General Relativity can be written as the theory of a Cartan geometry modelled on de Sitter space.

The cosmological constant in GR shows up because a “flat” connection for a Cartan geometry based on de Sitter space will look (if measured by Minkowski space) as if it has constant curvature which is exactly that of the model Klein geometry.  The way to think of this is to take the fibre bundle of homogeneous model spaces as a replacement for the tangent bundle to the manifold.  The fibre at each point describes the local appearance of spacetime.  If empty spacetime is flat, this local model is Minkowski space, , and one can really speak of tangent “vectors”.  The tangent homogeneous space is not linear.  In these first cases, the fibres are not vector spaces, precisely because the large group of symmetries doesn’t contain a group of translations, but they are Klein geometries constructed in just the same way as Minkowski space. Thus, the local description of the connection in terms of -valued forms can be treated in the same way, regardless of which Klein geometry occurs in the fibres.  In particular, General Relativity, formulated in terms of Cartan geometry, always says that, in the absence of matter, the geometry of space is flat, and the cosmological constant is included naturally by the choice of which Klein geometry is the local model of spacetime.

Observer Space

The idea in defining an observer space is to combine two symmetry reductions into one.  The reduction from to gives de Sitter space, as a model Klein geometry, which reflects the “symmetry breaking” that happens when choosing one particular point in spacetime, or event.  Then, the reduction of to similarly reflects the symmetry breaking that occurs when one chooses a specific time direction (a future-directed unit timelike vector).  These are the tangent vectors to the worldline of an observer at the chosen point, so the model Klein geometry, is the space of such possible observers.  The stabilizer subgroup for a point in this space consists of just the rotations of space around the corresponding observer – the boosts in translate between observers.  So locally, choosing an observer amounts to a splitting of the model spacetime at the point into a product of space and time. If we combine both reductions at once, we get the 7-dimensional Klein geometry .  This is just the future unit tangent bundle of de Sitter space, which we think of as a homogeneous model for the “space of observers”

A general observer space , however, is just a Cartan geometry modelled on .  This is a 7-dimensional manifold, equipped with the structure of a Cartan geometry.  One class of examples are exactly the future unit tangent bundles to 4-dimensional Lorentzian spacetimes.  In these cases, observer space is naturally a contact manifold: that is, it’s an odd-dimensional manifold equipped with a 1-form , the contact form, which is such that the top-dimensional form is nowhere zero.  This is the odd-dimensional analog of a symplectic manifold.  Contact manifolds are, intuitively, configuration spaces of systems which involve “rolling without slipping” – for instance, a sphere rolling on a plane.  In this case, it’s better to think of the local space of observers which “rolls without slipping” on a spacetime manifold .

Now, Minkowski space has a slicing into space and time – in fact, one for each observer, who defines the time direction, but the time coordinate does not transform in any meaningful way under the symmetries of the theory, and different observers will choose different ones.  In just the same way, the homogeneous model of observer space can naturally be written as a bundle .  But a general observer space may or may not be a bundle over an ordinary spacetime manifold, .  Every Cartan geometry gives rise to an observer space as the bundle of future-directed timelike vectors, but not every Cartan geometry is of this form, in any natural way. Indeed, without a further condition, we can’t even reconstruct observer space as such a bundle in an open neighborhood of a given observer.

This may be intuitively surprising: it gives a perfectly concrete geometric model in which “spacetime” is relative and observer-dependent, and perhaps only locally meaningful, in just the same way as the distinction between “space” and “time” in General Relativity. It may be impossible, that is, to determine objectively whether two observers are located at the same base event or not. This is a kind of “Relativity of Locality” which is geometrically much like the by-now more familiar Relativity of Simultaneity. Each observer will reach certain conclusions as to which observers share the same base event, but different observers may not agree.  The coincident observers according to a given observer are those reached by a good class of geodesics in moving only in directions that observer sees as boosts.

When one can reconstruct , two observers will agree whether or not they are coincident.  This extra condition which makes this possible is an integrability constraint on the action of the Lie algebra (in our main example, ) on the observer space .  In this case, the fibres of the bundle are the orbits of this action, and we have the familiar world of Relativity, where simultaneity may be relative, but locality is absolute.

Lifting Gravity to Observer Space

Apart from describing this model of relative spacetime, another motivation for describing observer space is that one can formulate canonical (Hamiltonian) GR locally near each point in such an observer space.  The goal is to make a link between covariant and canonical quantization of gravity.  Covariant quantization treats the geometry of spacetime all at once, by means of a Lagrangian action functional.  This is mathematically appealing, since it respects the symmetry of General Relativity, namely its diffeomorphism-invariance.  On the other hand, it is remote from the canonical (Hamiltonian) approach to quantization of physical systems, in which the concept of time is fundamental. In the canonical approach, one gets a Hilbert space by quantizing the space of states of a system at a given point in time, and the Hamiltonian for the theory describes its evolution.  This is problematic for diffeomorphism-, or even Lorentz-invariance, since coordinate time depends on a choice of observer.  The point of observer space is that we consider all these choices at once.  Describing GR in is both covariant, and based on (local) choices of time direction.

This is easiest to describe in the case of a bundle .  Then a “field of observers” to be a section of the bundle: a choice, at each base event in , of an observer based at that event.  A field of observers may or may not correspond to a particular decomposition of spacetime into space evolving in time, but locally, at each point in , it always looks like one.  The resulting theory describes the dynamics of space-geometry over time, as seen locally by a given observer.  In this case, a Cartan connection on observer space is described by to a -valued form.  This decomposes into four Lie-algebra valued forms, interpreted as infinitesimal transformations of the model observer by: (1) spatial rotations; (2) boosts; (3) spatial translations; (4) time translation.  The four-fold division is based on two distinctions: first, between the base event at which the observer lives, and the choice of observer (i.e. the reduction of to , which symmetry breaking entails choosing a point); and second, between space and time (i.e. the reduction of to , which symmetry breaking entails choosing a time direction).

This splitting, along the same lines as the one in MacDowell-Mansouri gravity described above, suggests that one could lift GR to a theory on an observer space .  This amount to describing fields on and an action functional, so that the splitting of the fields gives back the usual fields of GR on spacetime, and the action gives back the usual action.  This part of the project is still under development, but this lifting has been described.  In the case when there is no “objective” spacetime, the result includes some surprising new fields which it’s not clear how to deal with, but when there is an objective spacetime, the resulting theory looks just like GR.

 

Higher Structures in China

Since the last post, I’ve been busily attending some conferences, as well as moving to my new job at the University of Hamburg, in the Graduiertenkolleg 1670, “Mathematics Inspired by String Theory and Quantum Field Theory”.  The week before I started, I was already here in Hamburg, at the conference they were organizing “New Perspectives in Topological Quantum Field Theory“.  But since I last posted, I was also at the 20th Oporto Meeting on Geometry, Topology, and Physics, as well as the third Higher Structures in China workshop, at Jilin University in Changchun.  Right now, I’d like to say a few things about some of the highlights of that workshop.

Higher Structures in China III

So last year I had a bunch of discussions I had with Chenchang Zhu and Weiwei Pan, who at the time were both in Göttingen, about my work with Jamie Vicary, which I wrote about last time when the paper was posted to the arXiv.  In that, we showed how the Baez-Dolan groupoidification of the Heisenberg algebra can be seen as a representation of Khovanov’s categorification.  Chenchang and Weiwei and I had been talking about how these ideas might extend to other examples, in particular to give nice groupoidifications of categorified Lie algebras and quantum groups.

That is still under development, but I was invited to give a couple of talks on the subject at the workshop.  It was a long trip: from Lisbon, the farthest-west of the main cities of (continental) Eurasia all the way to one of the furthest-East.   (Not quite the furthest, but Changchun is in the northeast of China, just a few hours north of Korea, and it took just about exactly 24 hours including stopovers to get there).  It was a long way to go for a three day workshop, but as there were also three days of a big excursion to Changbai Mountain, just on the border with North Korea, for hiking and general touring around.  So that was a sort of holiday, with 11 other mathematicians.  Here is me with Dany Majard, in a national park along the way to the mountains:

Here’s me with Alex Hoffnung, on Changbai Mountain (in the background is China):

And finally, here’s me a little to the left of the previous picture, where you can see into the volcanic crater.  The lake at the bottom is cut out of the picture, but you can see the crater rim, of which this particular part is in North Korea, as seen from China:

Well, that was fun!

Anyway, the format of the workshop involved some talks from foreigners and some from locals, with a fairly big local audience including a good many graduate students from Jilin University.  So they got a chance to see some new work being done elsewhere – mostly in categorification of one kind or another.  We got a chance to see a little of what’s being done in China, although not as much as we might have. I gather that not much is being done yet that fit the theme of the workshop, which was part of the reason to organize the workshop, and especially for having a session aimed specially at the graduate students.

Categorified Algebra

This is a sort of broad term, but certainly would include my own talk.  The essential point is to show how the groupoidification of the Heisenberg algebra is a representation of Khovanov’s categorification of the same algebra, in a particular 2-category.  The emphasis here is on the fact that it’s a representation in a 2-category whose objects are groupoids, but whose morphisms aren’t just functors, but spans of functors – that is, composites of functors and co-functors.  This is a pretty conservative weakening of “representations on categories” – but it lets one build really simple combinatorial examples.  I’ve discussed this general subject in recent posts, so I won’t elaborate too much.  The lecture notes are here, if you like, though – they have more detail than my previous post, but are less technical than the paper with Jamie Vicary.

Aaron Lauda gave a nice introduction to the program of categorifying quantum groups, mainly through the example of the special case , somewhat along the same lines as in his introductory paper on the subject.  The story which gives the motivation is nice: one has knot invariants such as the Jones polynomial, based on representations of groups and quantum groups.  The Jones polynomial can be categorified to give Khovanov homology (which assigns a complex to a knot, whose graded Euler characteristic is the Jones polynomial) – but also assigns maps of complexes to cobordisms of knots.  One then wants to categorify the representation theory behind it – to describe actions of, for instance, quantum on categories.  This starting point is nice, because it can work by just mimicking the construction of and representations in terms of weight spaces: one gets categories which correspond to the “weight spaces” (usually just vector spaces), and the and operators give functors between them, and so forth.

Finding examples of categories and functors with this structure, and satisfying the right relations, gives “categorified representations” of the algebra – the monoidal categories of diagrams which are the “categorifications of the algebra” then are seen as the abstraction of exactly which relations these are supposed to satisfy.  One such example involves flag varieties.  A flag, as one might eventually guess from the name, is a nested collection of subspaces in some -dimensional space.  A simple example is the Grassmannian , which is the space of all 1-dimensional subspaces of (i.e. the projective space ), which is of course an algebraic variety.  Likewise, , the space of all -dimensional subspaces of is a variety.  The flag variety consists of all pairs , of a -dimensional subspace of , inside a -dimensional subspace (the case calls to mind the reason for the name: a plane intersecting a given line resembles a flag stuck to a flagpole).  This collection is again a variety.  One can go all the way up to the variety of “complete flags”, (where is -dimenisonal), any point of which picks out a subspace of each dimension, each inside the next.

The way this relates to representations is by way of geometric representation theory. One can see those flag varieties of the form as relating the Grassmanians: there are projections and , which act by just ignoring one or the other of the two subspaces of a flag.  This pair of maps, by way of pulling-back and pushing-forward functions, gives maps between the cohomology rings of these spaces.  So one gets a sequence , and maps between the adjacent ones.  This becomes a representation of the Lie algebra.  Categorifying this, one replaces the cohomology rings with derived categories of sheaves on the flag varieties – then the same sort of “pull-push” operation through (derived categories of sheaves on) the flag varieties defines functors between those categories.  So one gets a categorified representation.

Heather Russell‘s talk, based on this paper with Aaron Lauda, built on the idea that categorified algebras were motivated by Khovanov homology.  The point is that there are really two different kinds of Khovanov homology – the usual kind, and an Odd Khovanov Homology, which is mainly different in that the role played in Khovanov homology by a symmetric algebra is instead played by an exterior (antisymmetric) algebra.  The two look the same over a field of characteristic 2, but otherwise different.  The idea is then that there should be “odd” versions of various structures that show up in the categorifications of (and other algebras) mentioned above.

One example is the fact that, in the “even” form of those categorifications, there is a natural action of the Nil Hecke algebra on composites of the generators.  This is an algebra which can be seen to act on the space of polynomials in commuting variables, , generated by the multiplication operators , and the “divided difference operators” based on the swapping of two adjacent variables.  The Hecke algebra is defined in terms of “swap” generators, which satisfy some -deformed variation of the relations that define the symmetric group (and hence its group algebra).   The Nil Hecke algebra is so called since the “swap” (i.e. the divided difference) is nilpotent: the square of the swap is zero.  The way this acts on the objects of the diagrammatic category is reflected by morphisms drawn as crossings of strands, which are then formally forced to satisfy the relations of the Nil Hecke algebra.

The ODD Nil Hecke algebra, on the other hand, is an analogue of this, but the are anti-commuting, and one has different relations satisfied by the generators (they differ by a sign, because of the anti-commutation).  This sort of “oddification” is then supposed to happen all over.  The main point of the talk was to to describe the “odd” version of the categorified representation defined using flag varieties.  Then the odd Nil Hecke algebra acts on that, analogously to the even case above.

Marco Mackaay gave a couple of talks about the web algebra, describing the results of this paper with Weiwei Pan and Daniel Tubbenhauer.  This is the analog of the above, for , describing a diagram calculus which accounts for representations of the quantum group.  The “web algebra” was introduced by Greg Kuperberg – it’s an algebra built from diagrams which can now include some trivalent vertices, along with rules imposing relations on these.  When categorifying, one gets a calculus of “foams” between such diagrams.  Since this is obviously fairly diagram-heavy, I won’t try here to reproduce what’s in the paper – but an important part of is the correspondence between webs and Young Tableaux, since these are labels in the representation theory of the quantum group – so there is some interesting combinatorics here as well.

Algebraic Structures

Some of the talks were about structures in algebra in a more conventional sense.

Jiang-Hua Lu: On a class of iterated Poisson polynomial algebras.  The starting point of this talk was to look at Poisson brackets on certain spaces and see that they can be found in terms of “semiclassical limits” of some associative product.  That is, the associative product of two elements gives a power series in some parameter (which one should think of as something like Planck’s constant in a quantum setting).  The “classical” limit is the constant term of the power series, and the “semiclassical” limit is the first-order term.  This gives a Poisson bracket (or rather, the commutator of the associative product does).  In the examples, the spaces where these things are defined are all spaces of polynomials (which makes a lot of explicit computer-driven calculations more convenient). The talk gives a way of constructing a big class of Poisson brackets (having some nice properties: they are “iterated Poisson brackets”) coming from quantum groups as semiclassical limits.  The construction uses words in the generating reflections for the Weyl group of a Lie group .

Li Guo: Successors and Duplicators of Operads – first described a whole range of different algebra-like structures which have come up in various settings, from physics and dynamical systems, through quantum field theory, to Hopf algebras, combinatorics, and so on.  Each of them is some sort of set (or vector space, etc.) with some number of operations satisfying some conditions – in some cases, lots of operations, and even more conditions.  In the slides you can find several examples – pre-Lie and post-Lie algebras, dendriform algebras, quadri- and octo-algebras, etc. etc.  Taken as a big pile of definitions of complicated structures, this seems like a terrible mess.  The point of the talk is to point out that it’s less messy than it appears: first, each definition of an algebra-like structure comes from an operad, which is a formal way of summing up a collection of operations with various “arities” (number of inputs), and relations that have to hold.  The second point is that there are some operations, “successor” and “duplicator”, which take one operad and give another, and that many of these complicated structures can be generated from simple structures by just these two operations.  The “successor” operation for an operad introduces a new product related to old ones – for example, the way one can get a Lie bracket from an associative product by taking the commutator.  The “duplicator” operation takes existing products and introduces two new products, whose sum is the previous one, and which satisfy various nice relations.  Combining these two operations in various ways to various starting points yields up a plethora of apparently complicated structures.

Dany Majard gave a talk about algebraic structures which are related to double groupoids, namely double categories where all the morphisms are invertible.  The first part just defined double categories: graphically, one has horizontal and vertical 1-morphisms, and square 2-morphsims, which compose in both directions.  Then there are several special degenerate cases, in the same way that categories have as degenerate cases (a) sets, seen as categories with only identity morphisms, and (b) monoids, seen as one-object categories.  Double categories have ordinary categories (and hence monoids and sets) as degenerate cases.  Other degenerate cases are 2-categories (horizontal and vertical morphisms are the same thing), and therefore their own special cases, monoidal categories and symmetric monoids.  There is also the special degenerate case of a double monoid (and the extra-special case of a double group).  (The slides have nice pictures showing how they’re all degenerate cases).  Dany then talked about some structure of double group(oids) – and gave a list of properties for double groupoids, (such as being “slim” – having at most one 2-cell per boundary configuration – as well as two others) which ensure that they’re equivalent to the semidirect product of an abelian group with the “bicrossed product”  of two groups and (each of which has to act on the other for this to make sense).  He gave the example of the Poincare double group, which breaks down as a triple bicrossed product by the Iwasawa decomposition:

( is certain group of matrices).  So there’s a unique double group which corresponds to it – it has squares labelled by , and the horizontial and vertical morphisms by elements of and respectively.  Dany finished by explaining that there are higher-dimensional analogs of all this – -tuple categories can be defined recursively by internalization (“internal categories in -tuple-Cat”).  There are somewhat more sophisticated versions of the same kind of structure, and finally leading up to a special class of -tuple groups.  The analogous theorem says that a special class of them is just the same as the semidirect product of an abelian group with an -fold iterated bicrossed product of groups.

Also in this category, Alex Hoffnung talked about deformation of formal group laws (based on this paper with various collaborators).  FGL’s are are structures with an algebraic operation which satisfies axioms similar to a group, but which can be expressed in terms of power series.  (So, in particular they have an underlying ring, for this to make sense).  In particular, the talk was about formal group algebras – essentially, parametrized deformations of group algebras – and in particular for Hecke Algebras.  Unfortunately, my notes on this talk are mangled, so I’ll just refer to the paper.

Physics

I’m using the subject-header “physics” to refer to those talks which are most directly inspired by physical ideas, though in fact the talks themselves were mathematical in nature.

Fei Han gave a series of overview talks intorducing “Equivariant Cohomology via Gauged Supersymmetric Field Theory”, explaining the Stolz-Teichner program.  There is more, using tools from differential geometry and cohomology to dig into these theories, but for now a summary will do.  Essentially, the point is that one can look at “fields” as sections of various bundles on manifolds, and these fields are related to cohomology theories.  For instance, the usual cohomology of a space is a quotient of the space of closed forms (so the cohomology, , is a quotient of the space of closed -forms – the quotient being that forms differing by a coboundary are considered the same).  There’s a similar construction for the -theory , which can be modelled as a quotient of the space of vector bundles over .  Fei Han mentioned topological modular forms, modelled by a quotient of the space of “Fredholm bundles” – bundles of Banach spaces with a Fredholm operator around.

The first two of these examples are known to be related to certain supersymmetric topological quantum field theories.  Now, a TFT is a functor into some kind of vector spaces from a category of -dimensional manifolds and -dimensional cobordisms

Intuitively, it gives a vector space of possible fields on the given space and a linear map on a given spacetime.  A supersymmetric field theory is likewise a functor, but one changes the category of “spacetimes” to have both bosonic and fermionic dimension.  A normal smooth manifold is a ringed space , since it comes equipped with a sheaf of rings (each open set has an associated ring of smooth functions, and these glue together nicely).  Supersymmetric theories work with manifolds which change this sheaf – so a -dimensional space has the sheaf of rings where one introduces some new antisymmetric coordinate functions , the “fermionic dimensions”:

Then a supersymmetric TFT is a functor:

(where is the category of supersymmetric topological vector spaces – defined similarly).  The connection to cohomology theories is that the classes of such field theories, up to a notion of equivalence called “concordance”, are classified by various cohomology theories.  Ordinary cohomology corresponds then to -dimensional extended TFT (that is, with 0 bosonic and 1 fermionic dimension), and -theory to a -dimensional extended TFT.  The Stoltz-Teichner Conjecture is that the third example (topological modular forms) is related in the same way to a -dimensional extended TFT – so these are the start of a series of cohomology theories related to various-dimension TFT’s.

Last but not least, Chris Rogers spoke about his ideas on “Higher Geometric Quantization”, on which he’s written a number of papers.  This is intended as a sort of categorification of the usual ways of quantizing symplectic manifolds.  I am still trying to catch up on some of the geometry This is rooted in some ideas that have been discussed by Brylinski, for example.  Roughly, the message here is that “categorification” of a space can be thought of as a way of acting on the loop space of a space.  The point is that, if points in a space are objects and paths are morphisms, then a loop space shifts things by one categorical level: its points are loops in , and its paths are therefore certain 2-morphisms of .  In particular, there is a parallel to the fact that a bundle with connection on a loop space can be thought of as a gerbe on the base space.  Intuitively, one can “parallel transport” things along a path in the loop space, which is a surface given by a path of loops in the original space.  The local description of this situation says that a 1-form (which can give transport along a curve, by integration) on the loop space is associated with a 2-form (giving transport along a surface) on the original space.

Then the idea is that geometric quantization of loop spaces is a sort of higher version of quantization of the original space. This “higher” version is associated with a form of higher degree than the symplectic (2-)form used in geometric quantization of .   The general notion of n-plectic geometry, where the usual symplectic geometry is the case , involves a -form analogous to the usual symplectic form.  Now, there’s a lot more to say here than I properly understand, much less can summarize in a couple of paragraphs.  But the main theorem of the talk gives a relation between n-plectic manifolds (i.e. ones endowed with the right kind of form) and Lie n-algebras built from the complex of forms on the manifold.  An important example (a theorem of Chris’ and John Baez) is that one has a natural example of a 2-plectic manifold in any compact simple Lie group together with a 3-form naturally constructed from its Maurer-Cartan form.

At any rate, this workshop had a great proportion of interesting talks, and overall, including the chance to see a little more of China, was a great experience!

 

Paper on the Categorified Heisenberg Algebra

This blog has been on hiatus for a while, as I’ve been doing various other things, including spending some time in Hamburg getting set up for the move there. Another of these things has been working with Jamie Vicary on our project on the groupoidified Quantum Harmonic Oscillator (QHO for short). We’ve now put the first of two papers on the arXiv – this one is a relatively nonrigorous look at how this relates to categorification of the Heisenberg Algebra. Since John Baez is a high-speed blogging machine, he’s already beaten me to an overview of what the paper says, and there’s been some interesting discussion already. So I’ll try to say some different things about what it means, and let you take a look over there, or read the paper, for details.

I’ve given some talks about this project, but as we’ve been writing it up, it’s expanded considerably, including a lot of category-theoretic details which are going to be in the second paper in this series. But the basic point of this current paper is essentially visual and, in my opinion, fairly simple. The groupoidification of the QHO has a nice visual description, since it is all about the combinatorics of finite sets. This was described originally by Baez and Dolan, and in more detail in my very first paper. The other visual part here is the relation to Khovanov’s categorification of the Heisenberg algebra using a graphical calculus. (I wrote about this back when I first became aware of it.)

As a Representation

The scenario here actually has some common features with my last post. First, we have a monoidal category with duals, let’s say presented in terms of some generators and relations. Then, we find some concrete model of this abstractly-presented monoidal category with duals in a specific setting, namely .

Calling this “concrete” just refers to the fact that the objects in have some particular structure in terms of underlying sets and so on. By a “model” I just mean a functor (“model” and “representation” mean essentially the same thing in this context). In fact, for this to make sense, I think of as a 2-category with one object. Then a model is just some particular choices: a groupoid to represent the unique object, spans of groupoids to represent the generating morphisms, spans of spans to represent the generating 2-morphisms, all chosen so that the defining relations hold.

In my previous post, was a category of cobordisms, but in this case, it’s essentially Khovanov’s monoidal category whose objects are (oriented) dots and whose morphisms are certain classes of diagrams. The nice fact about the particular model we get is that the reasons these relations hold are easy to see in terms of a the combinatorics of sets. This is why our title describes what we got as “a combinatorial representation” Khovanov’s category of diagrams, for which the ring of isomorphism classes of objects is the integral form of the algebra. This uses that is not just a monoidal category: it can be a monoidal 2-category. What’s more, the monoidal category “is” also a 2-category – with one object. The objects of are really the morphisms of this 2-category.

So is in some sense a universal theory (because it’s defined freely in terms of generators and relations) of what a categorification of the Heisenberg algebra must look like. Baez-Dolan groupoidification of the QHO then turns out to be a representation or model of it. In fact, the model is faithful, so that we can even say that it provides a combinatorial interpretation of that category.

The Combinatorial Model

Between the links above, you can find a good summary of the situation, so I’ll be a bit cursory. The model is described in terms of structures on finite sets. This is why our title calls this a “combinatorial representation” of Khovanov’s categorification.

This means that the one object of (as a 2-category) is taken to the groupoid of finite sets and bijections (which we just called in the paper for brevity). This is the “Fock space” object. For simplicity, we can take an equivalent groupoid, which has just one -element set for each .

Now, a groupoid represents a system, whose possible configurations are the objects and whose symmetries are the morphisms. In this case, the possible configurations are the different numbers of “quanta”, and the symmetries (all set-bijections) show that all the quanta are interchangeable. I imagine a box containing some number of ping-pong balls.

A span of groupoids represents a process. It has a groupoid whose objects are histories (and morphisms are symmetries of histories). This groupoid has a pair of maps: to the system the process starts in, and to the system it ends in. In our model, the most important processes (which generate everything else) are the creation and annihilation operators, and – and their categorified equivalents, and . The spans that represent them are very simple: they are processes which put a new ball into the box, or take one out, respectively. (Algebraically, they’re just a way to organize all the inclusions of symmetric groups .)

The “canonical commutation relation“, which we write without subtraction thus:

is already understood in the Baez-Dolan story: it says that there is one more way to remove a ball from a box after putting a new one into it (one more history for the process ) than to remove a ball and then add a new one (histories for ). This is fairly obvious: in the first instance, you have one more to choose from when removing the ball.

But the original Baez-Dolan story has no interesting 2-morphisms (the actual diagrams which are the 1-morphisms in ), whereas these are absolutely the whole point of a categorification in the sense Khovanov gets one, since the 1-morphisms of determine what the isomorphism classes of objects even are.

So this means that we need to figure out what the 2-morphisms in need to be – first in general, and second in our particular representation of .

In general, a 2-morphism in is a span of span-maps. You’ll find other people who take it to be a span-map. This would be a functor between the groupoids of histories: roughly, a map which assigns a history in the source span to a history in the target span (and likewise for symmetries), in a way that respects how they’re histories. But we don’t want just a map: we want a process which has histories of its own. We want to describe a “movie of processes” which change one process into another. These can have many histories of their own.

In fact, they’re not too complicated. Here’s one of Khovanov’s relation in which forms part of how the commutation relation is expressed (shuffled to get rid of negatives, which we constantly need to do in the combinatorial model since we have no negative sets):

We read an upward arrow as “add a ball to the box”, and a downward arrow as “remove a ball”, and read right-to-left.  Both processes begin and end with“add then remove”. The right-hand side just leaves this process alone: it’s the identity.

The left-hand side shows a process-movie whose histories have two different cases. Suppose we begin with a history for which we add and then remove . The first case is that : we remove the same ball we put in. This amounts to doing nothing, so the first part of the movie eliminates all the adding and removing. The second part puts the add-remove pair back in.

The second case ensures that , since it takes the initial history to the history (of a different process!) in which we remove and then add (impossible if , since we can’t remove this ball before adding it). This in turn is taken to the history (of the original process!) where we add and then remove ; so this relates every history to itself, except for the case that . Overall the sum of these relations give the identity on histories, which is the right hand side.

This picture includes several of the new 2-morphisms that we need to add to the Baez-Dolan picture: swapping the order of two generators, and adding or removing a pair of add/remove operations. Finding spans of spans which accomplish this (and showing they satisfy the right relations) is all that’s needed to finish up the combinatorial model.  So, for instance, the span of spans which adds a “remove-then-add” pair is this one:

If this isn’t clear, well, it’s explained in more detail in the paper.  (Do notice, though, that this is a diagram in groupoids: we need to specify that there are identity 2-cells in the span, rather than some other 2-cells.)

So this is basically how the combinatorial model works.

Adjointness

But in fact this description is (as often happens) chronologically backwards: what actually happened was that we had worked out what the 2-morphisms should be for different reasons. While trying to to understand what kind of structure this produced, we realized (thanks to Marco Mackaay) that the result was related to , which in turn shed more light on the 2-morphisms we’d found.

So far so good. But what makes it possible to represent the kind of monoidal category we’re talking about in this setting is adjointness. This is another way of saying what I meant up at the top by saying we start with a monoidal category with duals.  This means morphisms each have a partner – a dual, or adjoint – going in the opposite direction.  The representations of the raising and lowering operators of the Heisenberg algebra on the Hilbert space for the QHO are linear adjoints. Their categorifications also need to be adjoints in the sense of adjoint 1-morphisms in a 2-category.

This is an abstraction of what it means for two functors and to be adjoint. In particular, it means there have to be certain 2-cells such as the unit and counit satisfying some nice relations. In fact, this only makes a left adjoint and a right adjoint – in this situation, we also have another pair which makes a right adjoint and a left one. That is, they should be “ambidextrous adjoints”, or “ambiadjoints” for short. This is crucial if they’re going to represent any graphical calculus of the kind that’s involved here (see the first part of this paper by Aaron Lauda, for instance).

So one of the theorems in the longer paper will show concretely that any 1-morphism in has an ambiadjoint – which happens to look like the same span, but thought of as going in the reverse direction. This is somewhat like how the adjoint of a real linear map, expressed as a matrix relative to well-chosen bases, is just the transpose of the same matrix. In particular, and are adjoints in just this way. The span-of-span-maps I showed above is exactly the unit for one side of this ambi-adjunction – but it is just a special case of something that will work for any span and its adjoint.

Finally, there’s something a little funny here. Since the morphisms of aren’t functors or maps, this combinatorial model is not exactly what people often mean by a “categorified representation”. That would be an action on a category in terms of functors and natural transformations. We do talk about how to get one of these on a 2-vector space out of our groupoidal representation toward the end.

In particular, this amounts to a functor into – the objects of being categories of a particular kind, and the morphisms being functors that preserve all the structure of those categories. As it turns out, the thing about this setting which is good for this purpose is that all those functors have ambiadjoints. The “2-linearization” that takes into is a 2-functor, and this means that all the 2-cells and equations that make two morphisms ambiadjoints carry over. In , it’s very easy for this to happen, since all those ambiadjoints are already present. So getting representations of categorified algebras that are made using these monoidal categories of diagrams on 2-vector spaces is fairly natural – and it agrees with the usual intuition about what “representation” means.

Anything I start to say about this is in danger of ballooning, but since we’re already some 40 pages into the second paper, I’ll save the elaboration for that…

 

Cohomology, Groupoidification, and TQFT

I’ve written here before about building topological quantum field theories using groupoidification, but I haven’t yet gotten around to discussing a refinement of this idea, which is in the most recent version of my paper on the subject.  I also gave a talk about this last year in Erlangen. The main point of the paper is to pull apart some constructions which are already fairly well known into two parts, as part of setting up a category which is nice for supporting models of fairly general physical systems, using an extension of the  concept of groupoidification. So here’s a somewhat lengthy post which tries to unpack this stuff a bit.

Factoring TQFT

The older version of this paper talked about the untwisted version of the Dijkgraaf-Witten (DW for short) model, which is a certain kind of TQFT based on a gauge theory with a finite gauge group.  (Freed and Quinn put it as: “Chern-Simons theory with finite gauge group”).  The new version gets the general – that is, the twisted – form in the same way: factoring the theory into two parts. So, the DW model, which was originally described by Dijkgraaf and Witten in terms of a state-sum, is a functor

The “twisting” is the point of their paper, “Topological Gauge Theories and Group Cohomology”.  The twisting has to do with the action for some physical theory. Now, for a gauge theory involving flat connections, the kind of gauge-theory actions which involve the curvature of a connection make no sense: the curvature is zero.  So one wants an action which reflects purely global features of connections.  The cohomology of the gauge group is where this comes from.

Now, the machinery I describe is based on a point of view which has been described in a famous paper by Freed, Hopkins, Lurie and Teleman (FHLT for short – see further discussion here) in terms in which the two stages are called the “classical field theory” (which has values in groupoids), and the “quantization functor”, which takes one into Hilbert spaces.

Actually, we really want to have an “extended” TQFT: a TQFT gives a Hilbert space for each 2D manifold (“space”), and a linear map for a 3D cobordism (“spacetime”) between them. An extended TQFT will assign (higher) algebraic data to lower-dimension boundaries still.  My paper talks only about the case where we’ve extended down to codimension 2, whereas FHLT talk about extending “down to a point”. The point of this first stopping point is to unpack explicitly and computationally what the factorization into two parts looks like at the first level beyond the usual TQFT.

In the terminology I use, the classical field theory is:

This depends on a cohomology class . The “quantization functor” (which in this case I call “2-linearization”):

The middle stage involves the monoidal 2-category I call .  (In FHLT, they use different terminology, for instance “families” rather than “spans”, but the principle is the same.)

Freed and Quinn looked at the quantization of the “extended” DW model, and got a nice geometric picture. In it, the action is understood as a section of some particular line-bundle over a moduli space. This geometric picture is very elegant once you see how it works, which I found was a little easier in light of a factorization through .

This factorization isolates the geometry of this particular situation in the “classical field theory” – and reveals which of the features of their setup (the line bundle over a moduli space) are really part of some more universal construction.

In particular, this means laying out an explicit definition of both and .

2-Linearization Recalled

While I’ve talked about it before, it’s worth a brief recap of how 2-linearization works with a view to what happens when you twist it via groupoid cohomology. Here we have a 2-category , whose objects are groupoids (, , etc.), whose morphisms are spans of groupoids:

and whose 2-morphisms are spans of span-maps (taken up to isomorphism), which look like so:

(And, by the by: how annoying that WordPress doesn’t appear to support xypic figures…)

These form a (symmetric monoidal) 2-category, where composition of spans works by taking weak pullbacks.  Physically, the idea is that a groupoid has objects which are configurations (in the cause of gauge theory, connections on a manifold), and morphisms which are symmetries (gauge transformations, in this case).  Then a span is a groupoid of histories (connections on a cobordism, thought of as spacetime), and the maps pick out its starting and ending configuration.  That is, is the groupoid of flat -connections on a manifold , and is the groupoid of flat -connections on some cobordism , of which is part of the boundary.  So any such connection can be restricted to the boundary, and this restriction is .

Now 2-linearization is a 2-functor:

It gives a 2-vector space (a nice kind of category) for each groupoid .  Specifically, the category of its representations, .  Then a span turns into a functor which comes from “pulling” back along (the restricted representation where acts by first applying then the representation), then “pushing” forward along (to the induced representation).

What happens to the 2-morphisms is conceptually more complicated, but it depends on the fact that “pulling” and “pushing” are two-sided adjoints. Concretely, it ends up being described as a kind of “sum over histories” (where “histories” are the objects of ), which turns out to be exactly the path integral that occurs in the TQFT.

Or at least, it’s the path integral when the action is trivial! That is, if , so that what’s integrated over paths (“histories”) is just . So one question is: is there a way to factor things in this way if there’s a nontrivial action?

Cohomological Twisting

The answer is by twisting via cohomology. First, let’s remember what that means…

We’re talking about groupoid cohomology for some groupoid (which you can take to be a group, if you like).  “Cochains” will measure how much some nice algebraic fact, such as being a homomorphism, or being associative, “fails to occur”.  “Twisting by a cocycle” is a controlled way to force some such failure to happen.

So, an -cocycle is some function of composable morphisms of (or, if there’s only one object, “group elements”, which amounts to the same thing).  It takes values in some group of coefficients, which for us is always . 

The trivial case where is actually slightly subtle: a 0-cocycle is an invariant function on the objects of a groupoid. (That is, it takes the same value on any two objects related by an (iso)morphism. (Think of the object as a sequence of zero composable morphisms: it tells you where to start, but nothing else.)

The case is maybe a little more obvious. A 1-cochain  can measure how a function on objects might fail to be a 0-cocycle. It is a -valued function of morphisms (or, if you like, group elements).  The natural condition to ask for is that it be a homomorphism:

This condition means that a cochain is a cocycle. They form an abelian group, because functions satisfying the cocycle condition are closed under pointwise multiplication in . It will automatically by satisfied for a coboundary (i.e. if comes from a function on objects as ). But not every cocycle is a coboundary: the first cohomology is the quotient of cocycles by coboundaries. This pattern repeats.

It’s handy to think of this condition in terms of a triangle with edges , , and .  It says that if we go from the source to the target of the sequence with or without composing, and accumulate -values, our gives the same result.  Generally, a cocycle is a cochain satisfying a “coboundary” condition, which can be described in terms of an -simplex, like this triangle. What about a 2-cocycle? This describes how composition might fail to be respected.

So, for instance, a twisted representation of a group is not a representation in the strict sense. That would be a map into , such that .  That is, the group composition rule gets taken directly to the corresponding rule for composition of endomorphisms of the vector space .  A twisted representation only satisfies this up to a phase:

where is a function that captures the way this “representation” fails to respect composition.  Still, we want some nice properties: is a “cocycle” exactly when this twisting still makes respect the associative law:

Working out what this says in terms of , the cocycle condition says that for any composable triple we have:

So – the second group-cohomology group of – consists of exactly these which satisfy this condition, which ensures we have associativity.

Given one of these maps, we get a category of all the -twisted representations of . It behaves just like an ordinary representation category… because in fact it is one! It’s the category of representations of a twisted version of the group algebra of , called . The point is, we can use to twist the convolution product for functions on , and this is still an associative algebra just because satisfies the cocycle condition.

The pattern continues: a 3-cocycle captures how some function of 2 variable may fail to be associative: it specifies an associator map (a function of three variables), which has to satisfy some conditions for any four composable morphisms. A 4-cocycle captures how a map might fail to satisfy this condition, and so on. At each stage, the cocycle condition is automatically satisfied by coboundaries. Cohomology classes are elements of the quotient of cocycles by coboundaries.

So the idea of “twisted 2-linearization” is that we use this sort of data to change 2-linearization.

Twisted 2-Linearization

The idea behind the 2-category is that it contains , but that objects and morphisms also carry information about how to “twist” when applying the 2-linearization .  So in particular, what we have is a (symmetric monoidal) 2-category where:

• Objects consist of , where is a groupoid and $\theta \in Z^2(A,U(1))$
• Morphisms from to consist of a span from to , together with
• 2-Morphisms from to consist of a span from , together with

The cocycles have to satisfy some compatibility conditions (essentially, pullbacks of the cocycles from the source and target of a span should land in the same cohomology class).  One way to see the point of this requirement is to make twisted 2-linearization well-defined.

One can extend the monoidal structure and composition rules to objects with cocycles without too much trouble so that is a subcategory of . The 2-linearization functor extends to :

• On Objects: , the category of -twisted representation of
• On Morphisms: comes by pulling back a twisted representation in to one in , pulling it through the algebra map “multiplication by “, and pushing forward to
• On 2-Morphisms: For a span of span maps, one uses the usual formula (see the paper for details), but a sum over the objects picks up a weight of at each object

When the cocycles are trivial (evaluate to 1 always), we get back the 2-linearization we had before. Now the main point here is that the “sum over histories” that appears in the 2-morphisms now carries a weight.

So the twisted form of 2-linearization uses the same “pull-push” ideas as 2-linearization, but applied now to twisted representations. This twisting (at the object level) uses a 2-cocycle. At the morphism level, we have a “twist” between “pull” and “push” in constructing . What the “twist” actually means depends on which cohomology degree we’re in – in other words, whether it’s applied to objects, morphisms, or 2-morphisms.

The “twisting” by a 0-cocycle just means having a weight for each object – in other words, for each “history”, or connection on spacetime, in a big sum over histories. Physically, the 0-cocycle is playing the role of the Lagrangian functional for the DW model. Part of the point in the FHLT program can be expressed by saying that what Freed and Quinn are doing is showing how the other cocycles are also the Lagrangian – as it’s seen at higher codimension in the more “local” theory.

For a TQFT, the 1-cocycles associated to morphisms describe how to glue together values for the Lagrangian that are associated to histories that live on different parts of spacetime: the action isn’t just a number. It is a number only “locally”, and when we compose 2-morphisms, the 0-cocycle on the composite picks up a factor from the 1-morphism (or 0-morphism, for a horizontal composite) where they’re composed.

This has to do with the fact that connections on bits of spacetime can be glued by particular gauge transformations – that is, morphisms of the groupoid of connections. Just as the gauge transformations tell how to glue connections, the cocycles associated to them tell how to glue the actions. This is how the cohomological twisting captures the geometric insight that the action is a section of a line bundle – not just a function, which is a section of a trivial bundle – over the moduli space of histories.

So this explains how these cocycles can all be seen as parts of the Lagrangian when we quantize: they explain how to glue actions together before using them in a sum-over histories. Gluing them this way is essential to make sure that is actually a functor. But if we’re really going to see all the cocycles as aspects of “the action”, then what is the action really? Where do they come from, that they’re all slices of this bigger thing?

Twisting as Lagrangian

Now the DW model is a 3D theory, whose action is specified by a group-cohomology class . But this is the same thing as a class in the cohomology of the classifying space: . This takes a little unpacking, but certainly it’s helpful to understand that what cohomology classes actually classify are… gerbes. So another way to put a key idea of the FHLT paper, as Urs Schreiber put it to me a while ago, is that “the action is a gerbe on the classifying space for fields“.

What does this mean?

This map is given as a path integral over all connections on the space(-time) , which is actually just a sum, since the gauge group is finite and so all the connections are flat.  The point is that they’re described by assigning group elements to loops in :

But this amounts to the same thing as a map into the classifying space of :

This is essentially the definition of , and it implies various things, such as the fact that is a space whose fundamental group is , and has all other homotopy groups trivial. That is, is the Eilenberg-MacLane space . But the point is that the groupoid of connections and gauge transformations on just corresponds to the mapping space . So the groupoid cohomology classes we get amount to the same thing as cohomology classes on this space. If we’re given , then we can get at these by “transgression” – which is very nicely explained in a paper by Simon Willerton.

The essential idea is that a 3-cocycle (representing the class ) amounts to a nice 3-form on which we can integrate over a 3-dimentional submanifold to get a number. For a -dimensional , we get such a 3-manifold from a -dimensional submanifold of : each point gives a copy of in . Then we get a -cocycle on whose values come from integrating over this image. Here’s a picture I used to illustrate this in my talk:

Now, it turns out that this gives 2-cocycles for 1-manifolds (the objects of , 1-cocycles on 2D cobordisms between them, and 0-cocycles on 3D cobordisms between these cobordisms. The cocycles are for the groupoid of connections and gauge transformations in each case. In fact, because of Stokes’ theorem in , these have to satisfy all the conditions that make them into objects, morphisms, and 2-morphisms of . This is the geometric content of the Lagrangian: all the cocycles are really “reflections” of as seen by transgression: pulling back along the evaluation map from the picture. Then the way you use it in the quantization is described exactly by .

What I like about this is that is a fairly universal sort of thing – so while this example gets its cocycles from the nice geometry of which Freed and Quinn talk about, the insight that an action is a section of a (twisted) line bundle, that actions can be glued together in particular ways, and so on… These presumably can be moved to other contexts.

 

Representations and Categories – Part II (Erlangen)

Well, as promised in the previous post, I’d like to give a summary of some of what was discussed at the conference I attended (quite a while ago now, late last year) in Erlangen, Germany.  I was there also to visit Derek Wise, talking about a project we’ve been working on for some time.

(I’ve also significantly revised this paper about Extended TQFT since then, and it now includes some stuff which was the basis of my talk at Erlangen on cohomological twisting of the category .  I’ll get to that in the next post.  Also coming up, I’ll be describing some new things I’ve given some talks about recently which relate the Baez-Dolan groupoidification program to Khovanov-Lauda categorification of algebras – at least in one example, hopefully in a way which will generalize nicely.)

In the meantime, there were a few themes at the conference which bear on the Extended TQFT project in various ways, so in this post I’ll describe some of them.  (This isn’t an exhaustive description of all the talks: just of a selection of illustrative ones.)

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Categories with Structures

A few talks were mainly about facts regarding the sorts of categories which get used in field theory contexts.  One important type, for instance, are fusion categories is a monoidal category which is enriched in vector spaces, generated by simple objects, and some other properties: essentially, monoidal 2-vector spaces.  The basic example would be categories of representations (of groups, quantum groups, algebras, etc.), but fusion categories are an abstraction of (some of) their properties.  Many of the standard properties are described and proved in this paper by Etingof, Nikshych, and Ostrik, which also poses one of the basic conjectures, the “ENO Conjecture”, which was referred to repeatedly in various talks.  This is the guess that every fusion category can be given a “pivotal” structure: an isomorphism from to .  It generalizes the theorem that there’s always such an isomorphism into .  More on this below.

Hendryk Pfeiffer talked about a combinatorial way to classify fusion categories in terms of certain graphs (see this paper here).  One way I understand this idea is to ask how much this sort of category really does generalize categories of representations, or actually comodules.  One starting point for this is the theorem that there’s a pair of functors between certain monoidal categories and weak Hopf algebras.  Specifically, the monoidal categories are , which consists of monoidal categories equipped with a forgetful functor into .  Then from this one can get (via a coend), a weak Hopf algebra over the base field (in the category ).  From a weak Hopf algebra , one can get back such a category by taking all the modules of .  These two processes form an adjunction: they’re not inverses, but we have maps between the two composites and the identity functors.

The new result Hendryk gave is that if we restrict our categories over to be abelian, and the functors between them to be linear, faithful, and exact (that is, roughly, that we’re talking about concrete monoidal 2-vector spaces), then this adjunction is actually an equivalence: so essentially, all such categories may as well be module categories for weak Hopf algebras.  Then he gave a characterization of these in terms of the “dimension graph” (in fact a quiver) for , where is one of the monoidal generators of .  The vertices of are labelled by the irreducible representations (i.e. set of generators of the category), and there’s a set of edges labelled by a basis of .  Then one can carry on and build a big graded algebra whose -graded part consists of length- paths in .  Then the point is that the weak Hopf algebra of which is (up to isomorphism) the module category will be a certain quotient of (after imposing some natural relations in a systematic way).

The point, then, is that the sort of categories mostly used in this area can be taken to be representation categories, but in general only of these weak Hopf algebras: groups and ordinary algebras are special cases, but they show up naturally for certain kinds of field theory.

Tensor Categories and Field Theories

There were several talks about the relationship between tensor categories of various sorts and particular field theories.  The idea is that local field theories can be broken down in terms of some kind of n-category: -dimensional regions get labelled by categories, -D boundaries between regions, or “defects”, are labelled by functors between the categories (with the idea that this shows how two different kinds of field can couple together at the defect), and so on (I think the highest-dimension that was discussed explicitly involved 3-categories, so one has junctions between defects, and junctions between junctions, which get assigned some higher-morphism data).  Alteratively, there’s the dual picture where categories are assigned to points, functors to 1-manifolds, and so on.  (This is just Poincaré duality in the case where the manifolds come with a decomposition into cells, which they often are if only for convenience).

Victor Ostrik gave a pair of talks giving an overview role tensor categories play in conformal field theory.  There’s too much material here to easily summarize, but the basics go like this: CFTs are field theories defined on cobordisms that have some conformal structure (i.e. notion of angles, but not distance), and on the algebraic side they are associated with vertex algebras (some useful discussion appears on mathoverflow, but in this context they can be understood as vector spaces equipped with exactly the algebraic operations needed to model cobordisms with some local holomorphic structure).

In particular, the irreducible representations of these VOA’s determine the “conformal blocks” of the theory, which tell us about possible correlations between observables (self-adjoint operators).  A VOA is “rational” if the category is semisimple (i.e. generated as finite direct sums of these conformal blocks).  For good VOA’s, will be a modular tensor category (MTC), which is a fusion category with a duality, braiding, and some other strucutre (see this for more).   So describing these gives us a lot of information about what CFT’s are possible.

The full data of a rational CFT are given by a vertex algebra, and a module category : that is, a fusion category is a sort of categorified ring, so it can act on as an ring acts on a module.  It turns out that choosing an is equivalent to finding a certain algebra (i.e. algebra object) , a “Lagrangian algebra” inside the centre of .  The Drinfel’d centre of a monoidal category is a sort of free way to turn a monoidal category into a braided one: but concretely in this case it just looks like .  Knowing the isomorphism class determines a “modular invariant”.  It gets “physics” meaning from how it’s equipped with an algebra structure (which can happen in more than one way), but in any case has an underlying vector space, which becomes the Hilbert space of states for the conformal field theory, which the VOA acts on in the natural way.

Now, that was all conformal field theory.  Christopher Douglas described some work with Chris Schommer-Pries and Noah Snyder about fusion categories and structured topological field theories.  These are functors out of cobordism categories, the most important of which are -categories, where the objects are points, morphisms are 1D cobordisms, and so on up to -morphisms which are -dimensional cobordisms.  To keep things under control, Chris Douglas talked about the case , which is where , and a “local” field theory is a 3-functor for some 3-category .  Now, the (Baez-Dolan) Cobordism Hypothesis, which was proved by Jacob Lurie, says that is, in a suitable sense, the free symmetric monoidal 3-category with duals.  What this amounts to is that a local field theory whose target 3-category is is “just” a dualizable object of .

The handy example which links this up to the above is when has objects which are tensor categories, morphisms which are bimodule categories (i.e. categories acted), 2-morphisms which are functors, and 3-morphisms which are natural transformations.  Then the issue is to classify what kind of tensor categories these objects can be.

The story is trickier if we’re talking about, not just topological cobordisms, but ones equipped with some kind of structure regulated by a structure group (for instance, orientation by , spin structure by its universal cover , and so on).  This means the cobordisms come equipped with a map into .  They take as the starting point, and then consider groups with a map to , and require that the map into is a lift of the map to .  Then one gets that a structured local field theory amounts to a dualizable objects of with a homotopy-fixed point for some -action – and this describes what gets assigned to the point by such a field theory.  What they then show is a correspondence between and classes of categories.  For instance, fusion categories are what one gets by imposing that the cobordisms be oriented.

Liang Kong talked about “Topological Orders and Tensor Categories”, which used the Levin-Wen models, from condensed matter phyiscs.  (Benjamin Balsam also gave a nice talk describing these models and showing how they’re equivalent to the Turaev-Viro and Kitaev models in appropriate cases.  Ingo Runkel gave a related talk about topological field theories with “domain walls”.).  Here, the idea of a “defect” (and topological order) can be understood very graphically: we imagine a 2-dimensional crystal lattice (of atoms, say), and the defect is a 1-dimensional place where the two lattices join together, with the internal symmetry of each breaking down at the boundary.  (For example, a square lattice glued where the edges on one side are offset and meet the squares on the other side in the middle of a face, as you typically see in a row of bricks – the slides linked above have some pictures).  The Levin-Wen models are built using a hexagonal lattice, starting with a tensor category with several properties: spherical (there are dualities satisfying some relations), fusion, and unitary: in fact, historically, these defining properties were rediscovered independently here as the requirement for there to be excitations on the boundary which satisfy physically-inspired consistency conditions.

These abstract the properties of a category of representations.  A generalization of this to “topological orders” in 3D or higher is an extended TFT in the sense mentioned just above: they have a target 3-category of tensor categories, bimodule categories, functors and natural transformations.  The tensor categories (say, , , etc.) get assigned to the bulk regions; to “domain walls” between different regions, namely defects between lattices, we assign bimodule categories (but, for instance, to a line within a region, we get understood as a -bimodule); then to codimension 2 and 3 defects we attach functors and natural transformations.  The algebra for how these combine expresses the ways these topological defects can go together.  On a lattice, this is an abstraction of a spin network model, where typically we have just one tensor category applied to the whole bulk, namely the representations of a Lie group (say, a unitary group).  Then we do calculations by breaking down into bases: on codimension-1 faces, these are simple objects of ; to vertices we assign a Hom space (and label by a basis for intertwiners in the special case); and so on.

Thomas Nickolaus spoke about the same kind of -equivariant Dijkgraaf-Witten models as at our workshop in Lisbon, so I’ll refer you back to my earlier post on that.  However, speaking of equivariance and group actions:

Michael Müger  spoke about “Orbifolds of Rational CFT’s and Braided Crossed -Categories” (see this paper for details).  This starts with that correspondence between rational CFT’s (strictly, rational chiral CFT’s) and modular categories .  (He takes to be the name of the CFT).  Then we consider what happens if some finite group acts on (if we understand as a functor, this is an action by natural transformations; if as an algebra, then ).  This produces an “orbifold theory” (just like a finite group action on a manifold produces an orbifold), which is the “-fixed subtheory” of , by taking -fixed points for every object, and is also a rational CFT.  But that means it corresponds to some other modular category , so one would like to know what category this is.

A natural guess might be that it’s , where is a “weak fixed-point” category that comes from a weak group action on a category .  Objects of are pairs where and is a specified isomorphism.  (This is a weak analog of , the set of fixed points for a group acting on a set).  But this guess is wrong – indeed, it turns out these categories have the wrong dimension (which is defined because the modular category has a trace, which we can sum over generating objects).  Instead, the right answer, denoted by , is the -fixed part of some other category.  It’s a braided crossed -category: one with a grading by , and a -action that gets along with it.  The identity-graded part of is just the original .

State Sum Models

This ties in somewhat with at least some of the models in the previous section.  Some of these were somewhat introductory, since many of the people at the conference were coming from a different background.  So, for instance, to begin the workshop, John Barrett gave a talk about categories and quantum gravity, which started by outlining the historical background, and the development of state-sum models.  He gave a second talk where he began to relate this to diagrams in Gray-categories (something he also talked about here in Lisbon in February, which I wrote about then).  He finished up with some discussion of spherical categories (and in particular the fact that there is a Gray-category of spherical categories, with a bunch of duals in the suitable sense).  This relates back to the kind of structures Chris Douglas spoke about (described above, but chronologically right after John).  Likewise, Winston Fairbairn gave a talk about state sum models in 3D quantum gravity – the Ponzano Regge model and Turaev-Viro model being the focal point, describing how these work and how they’re constructed.  Part of the point is that one would like to see that these fit into the sort of framework described in the section above, which for PR and TV models makes sense, but for the fancier state-sum models in higher dimensions, this becomes more complicated.

Higher Gauge Theory

There wasn’t as much on this topic as at our own workshop in Lisbon (though I have more remarks on higher gauge theory in one post about it), but there were a few entries.  Roger Picken talked about some work with Joao Martins about a cubical formalism for parallel transport based on crossed modules, which consist of a group and abelian group , with a map and an action of on satisfying some axioms.  They can represent categorical groups, namely group objects in (equivalently, categories internal to ), and are “higher” analogs of groups with a set of elements.  Roger’s talk was about how to understand holonomies and parallel transports in this context.  So, a “connection” lets on transport things with -symmetries along paths, and with -symmetries along surfaces.  It’s natural to describe this with squares whose edges are labelled by -elements, and faces labelled by -elements (which are the holonomies).  Then the “cubical approach” means that we can describe gauge transformations, and higher gauge transformations (which in one sense are the point of higher gauge theory) in just the same way: a gauge transformation which assigns -values to edges and -values to vertices can be drawn via the holonomies of a connection on a cube which extends the original square into 3D (so the edges become squares, and so get -values, and so on).  The higher gauge transformations work in a similar way.  This cubical picture gives a good way to understand the algebra of how gauge transformations etc. work: so for instance, gauge transformations look like “conjugation” of a square by four other squares – namely, relating the front and back faces of a cube by means of the remaining faces.  Higher gauge transformations can be described by means of a 4D hypercube in an analogous way, and their algebraic properties have to do with the 2D faces of the hypercube.

Derek Wise gave a short talk outlining his recent paper with John Baez in which they show that it’s possible to construct a higher gauge theory based on the Poincare 2-group which turns out to have fields, and dynamics, which are equivalent to teleparallel gravity, a slightly unusal theory which nevertheless looks in practice just like General Relativity.  I discussed this in a previous post.

So next time I’ll talk about the new additions to my paper on ETQFT which were the basis of my talk, which illustrates a few of the themes above.

 

HGTQGR Workshop – Part IIb (School)

Continuing from the previous post, there are a few more lecture series from the school to talk about.

Higher Gauge Theory

The next was John Huerta’s series on Higher Gauge Theory from the point of view of 2-groups.  John set this in the context of “categorification”, a slightly vague program of replacing set-based mathematical ideas with category-based mathematical ideas.  The general reason for this is to get an extra layer of “maps between things”, or “relations between relations”, etc. which tend to be expressed by natural transformations.  There are various ways to go about this, but one is internalization: given some sort of structure, the relevant example in this case being “groups”, one has a category , and can define a 2-group as a “category internal to “.  So a 2-group has a group of objects, a group of morphisms, and all the usual maps (source and target for morphisms, composition, etc.) which all have to be group homomorphisms.  It should be said that this all produces a “strict 2-group”, since the objects necessarily form a group here.  In particular, satisfies group axioms “on the nose” – which is the only way to satisfy them for a group made of the elements of a set, but for a group made of the elements of a category, one might require only that it commute up to isomorphism.  A weak 2-group might then be described as a “weak model” of the theory of groups in , but this whole approach is much less well-understood than the strict version as one goes to general n-groups.

Now, as mentioned in the previous post, there is a 1-1 correspondence between 2-groups and crossed modules (up to equivalence): given a crossed module , there’s a 2-group whose objects are and whose morphisms are ; given a 2-group with objects , there’s a crossed module .  (The action acts on a morphism in such as way as to act by multiplication on its source and target).  Then, for instance, the Peiffer identity for crossed modules (see previous post) is a consequence of the fact that composition of morphisms is supposed to be a group homomorphism.

Looking at internal categories in [your favourite setting here] isn’t the only way to do categorification, but it does produce some interesting examples.  Baez-Crans 2-vector spaces are defined this way (in ), and built using these are Lie 2-algebras.  Looking for a way to integrate Lie 2-algebras up to Lie 2-groups (which are internal categories in Lie groups) brings us back to the current main point.  This is the use of 2-groups to do higher gauge theory.  This requires the use of “2-bundles”.  To explain these, we can say first of all that a “2-space” is an internal category in (whether that be manifolds, or topological spaces, or what-have-you), and that a (locally trivial) 2-bundle should have a total 2-space , a base 2-space , and a (functorial) projection map , such that there’s some open cover of by neighborhoods where locally the bundle “looks like” , where is the fibre of the bundle.  In the bundle setting, “looks like” means “is isomorphic to” by means of isomorphisms .  With 2-bundles, it’s interpreted as “is equivalent to” in the categorical sense, likewise by maps .

Actually making this precise is a lot of work when is a general 2-space – even defining open covers and setting up all the machinery properly is quite hard.  This has been done, by Toby Bartels in his thesis, but to keep things simple, John restricted his talk to the case where is just an ordinary manifold (thought of as a 2-space which has only identity morphisms).   Then a key point is that there’s an analog to how (principal) -bundles (where as a -set) are classified up to isomorphism by the first Cech cohomology of the manifold, .  This works because one can define transition functions on double overlaps , by .  Then these will automatically satisfy the 1-cocycle condidion ( on the triple overlap ) which means they represent a cohomology class .

A comparable thing can be said for the “transition functors” for a 2-bundle – they’re defined superficially just as above, except that being functors, we can now say there’s a natural isomorphism , and it’s these , defined on triple overlaps, which satisfy a 2-cocycle condition on 4-fold intersections (essentially, the two ways to compose them to collapse into agree).  That is, we have and which fit together nicely.  In particular, we have an element of the second Cech cohomology of : “principal -bundles are classified by second Cech cohomology of “.  This sort of thing ties in to an ongoing theme of the later talks, the relationship between gerbes and higher cohomology – a 2-bundle corresponds to a “gerbe”, or rather a “1-gerbe”.  (The consistent terminology would have called a bundle a “0-gerbe”, but as usual, terminology got settled before the general pattern was understood).

Finally, having defined bundles, one usually defines connections, and so we do the same with 2-bundles.  A connection on a bundle gives a parallel transport operation for paths in , telling how to identify the fibres at points along by means of a functor , thinking of as a category with one object, and where is the path groupoid whose objects are points in and whose morphisms are paths (up to “thin” homotopy). At least, it does so once we trivialize the bundle around , anyway, to think of it as locally – in general we need to get the transition functions involved when we pass into some other local neighborhood.  A connection on a 2-bundle is similar, but tells how to parallel transport fibres not only along paths, but along homotopies of paths, by means of , where is seen as a 2-category with one object, and now has 2-morphisms which are (essentially) homotopies of paths.

Just as connections can be described by 1-forms valued in , which give by integrating, a similar story exists for 2-connections: now we need a 1-form valued in and a 2-form valued in .  These need to satisfy some relations, essentially that the curvature of has to be controlled by .   Moreover, that is related to the -field of string theory, as I mentioned in the post on the pre-school… But really, this is telling us about the Lie 2-algebra associated to , and how to integrate it up to the group!

Infinite Dimensional Lie Theory and Higher Gauge Theory

This series of talks by Christoph Wockel returns us to the question of “integrating up” to a Lie group from a Lie algebra , which is seen as the tangent space of at the identity.  This is a well-understood, well-behaved phenomenon when the Lie algebras happen to be finite dimensional.  Indeed the classification theorem for the classical Lie groups can be got at in just this way: a combinatorial way to characterize Lie algebras using Dynkin diagrams (which describe the structure of some weight lattice), followed by a correspondence between Lie algebras and Lie groups.  But when the Lie algebras are infinite dimensional, this just doesn’t have to work.  It may be impossible to integrate a Lie algebra up to a full Lie group: instead, one can only get a little neighborhood of the identity.  The point of such infinite-dimensional groups, and ultimately their representation theory, is to deal with string groups that have to do with motions of extended objects.  Christoph Wockel was describing a result which says that, going to 2-groups, this problem can be overcome.  (See the relevant paper here.)

The first lecture in the series presented some background on a setting for infinite dimensional manifolds.  There are various approaches, a popular one being Frechet manifolds, but in this context, the somewhat weaker notion of locally convex spaces is sufficient.  These are “locally modelled” by (infinite dimensional) locally convex vector spaces, the way finite dimensonal manifolds are locally modelled by Euclidean space.  Being locally convex is enough to allow them to support a lot of differential calculus: one can find straight-line paths, locally, to define a notion of directional derivative in the direction of a general vector.  Using this, one can build up definitions of differentiable and smooth functions, derivatives, and integrals, just by looking at the restrictions to all such directions.  Then there’s a fundamental theorem of calculus, a chain rule, and so on.

At this point, one has plenty of differential calculus, and it becomes interesting to bring in Lie theory.  A Lie group is defined as a group object in the category of manifolds and smooth maps, just as in the finite-dimensional case.  Some infinite-dimensional Lie groups of interest would include: , the group of diffeomorphisms of some compact manifold ; and the group of smooth functions from into some (finite-dimensional) Lie group (perhaps just ), with the usual pointwise multiplication.  These are certainly groups, and one handy fact about such groups is that, if they have a manifold structure near the identity, on some subset that generates as a group in a nice way, you can extend the manifold structure to the whole group.  And indeed, that happens in these examples.

Well, next we’d like to know if we can, given an infinite dimensional Lie algebra , “integrate up” to a Lie group – that is, find a Lie group for which is the “infinitesimal” version of .  One way this arises is from central extensions.  A central extension of Lie group by is an exact sequence where (the image of) is in the centre of .  The point here is that extends .  This setup makes is a principal -bundle over .

Now, finding central extensions of Lie algebras is comparatively easy, and given a central extension of Lie groups, one always falls out of the induced maps.  There will be an exact sequence of Lie algebras, and now the special condition is that there must exist a continuous section of the second map.  The question is to go the other way: given one of these, get back to an extension of Lie groups.  The problem of finding extensions of by , in particular as a problem of finding a bundle with connection having specified curvature, which brings us back to gauge theory.  One type of extension is the universal cover of , which appears as , so that the fibre is .

In general, whether an extension can exist comes down to a question about a cocycle: that is, if there’s a function which is locally smooth (i.e. in some neighborhood in ), and is a cocyle (so that ), by the same sorts of arguments we’ve already seen a bit of.  For this reason, central extensions are classified by the cohomology group .  The cocycle enables a “twisting” of the multiplication associated to a nontrivial loop in , and is used to construct (by specifying how multiplication on lifts to ).  Given a  2-cocycle at the Lie algebra level (easier to do), one would like to lift that up the Lie group.  It turns out this is possible if the period homomorphism – which takes a chain (with ) to the integral of the original cocycle on it, – lands in a discrete subgroup of . A popular example of this is when is just , and the discrete subgroup is (or, similarly, and respectively).  This business of requiring a cocycle to be integral in this way is sometimes called a “prequantization” problem.

So suppose we wanted to make the “2-connected cover”  as a central extension: since will be abelian, this is conceivable.  If the dimension of is finite, this is trivial (since in finite dimensions), which is why we need some theory  of infinite-dimensional manifolds.  Moreover, though, this may not work in the context of groups: the in the extension above needs to be a “twisting” of associativity, not multiplication, being lifted from .  Such twistings come from the THIRD cohomology of (see here, e.g.), and describe the structure of 2-groups (or crossed modules, whichever you like).  In fact, the solution (go read the paper for more if you like) to define a notion of central extension for 2-groups (essentially the same as the usual definition, but with maps of 2-groups, or crossed modules, everywhere).  Since a group is a trivial kind of 2-group (with only trivial automorphisms of any element), the usual notion of central extension turns out to be a special case.  Then by thinking of and as crossed modules, one can find a central extension which is like the 2-connected cover we wanted – though it doesn’t work as an extension of groups because we think of as the base group of the crossed module, and as the second group in the tower.

The pattern of moving to higher group-like structures, higher cohomology, and obstructions to various constructions ran all through the workshop, and carried on in the next school session…

Higher Spin Structures in String Theory

Hisham Sati gave just one school-lecture in addition to his workshop talk, but it was packed with a lot of material.  This is essentially about cohomology and the structures on manifolds to which cohomology groups describe the obstructions.  The background part of the lecture referenced this book by Fridrich, and the newer parts were describing some of Sati’s own work, in particular a couple of papers with Schreiber and Stasheff (also see this one).

The basic point here is that, for physical reasons, we’re often interested in putting some sort of structure on a manifold, which is really best described in terms of a bundle.  For instance, a connection or spin connection on spacetime lets us transport vectors or spinors, respectively, along paths, which in turn lets us define derivatives.  These two structures really belong on vector bundles or spinor bundles.  Now, if these bundles are trivial, then one can make the connections on them trivial as well by gauge transformation.  So having nontrivial bundles really makes this all more interesting.  However, this isn’t always possible, and so one wants to the obstruction to being able to do it.  This is typically a class in one of the cohomology groups of the manifold – a characteristic class.  There are various examples: Chern classes, Pontrjagin classes, Steifel-Whitney classes, and so on, each of which comes in various degrees .  Each one corresponds to a different coefficient group for the cohomology groups – in these examples, the groups and which are the limits of the unitary and orthogonal groups (such as )

The point is that these classes are obstructions to building certain structures on the manifold – which amounts to finding sections of a bundle.  So for instance, the first Steifel-Whitney classes, of a bundle are related to orientations, coming from cohomology with coefficients in .  Orientations for the manifold can be described in terms of its tangent bundle, which is an -bundle (tangent spaces carry an action of the rotation group).  Consider , where we have actually .  The group has two elements, and there are two types of line bundle on the circle : ones with a nowhere-zero section, like the trivial bundle; and ones without, like the Moebius strip.  The circle is orientable, because its tangent bundle is of the first sort.

Generally, an orientation can be put on if the tangent bundle, as a map , can be lifted to a map – that is, it’s “secretly” an -bundle – the special orthogonal group respects orientation, which is what the determinant measures.  Its two values, , are what’s behind the two classes of bundles.  (In short, this story relates to the exact sequence ; in just the same way we have big groups , , and so forth.)

So spin structures have a story much like the above, but where the exact sequence plays a role – the spin groups are the universal covers (which are all double-sheeted covers) of the special rotation groups.  A spin structure on some bundle , let’s say represented by  is thus, again, a lifting to .  The obstruction to doing this (the thing which must be zero for the lifting to exist) is the second Stiefel-Whitney class, .  Hisham Sati also explained the example of “generalized” spin structures in these terms.  But the main theme is an analogous, but much more general, story for other cohomology groups as obstructions to liftings of some sort of structures on manifolds.  These may be bundles, for the lower-degree cohomology, or they may be gerbes or n-bundles, for higher-degree, but the setup is roughly the same.

The title’s term “higher spin structures” comes from the fact that we’ve so far had a tower of classifying spaces (or groups), , and so on.  Then the problem of putting various sorts of structures on has been turned into the problem of lifting a map up this tower.  At each point, the obstruction to lifting is some cohomology class with coefficients in the groups (, , etc.)  So when are these structures interesting?

This turns out to bring up another theme, which is that of special dimensions – it’s just not true that the same phenomena happen in every dimension.  In this case, this has to do with the homotopy groups  – of and its cousins.  So it turns out that the homotopy group (which is the same as as long as is bigger than ) follows a pattern, where if , and if .  The fact that this pattern repeats mod-8 is one form of the (real) Bott Periodicity theorem.  These homotopy groups reflect that, wherever there’s nontrivial homotopy in some dimension, there’s an obstruction to contracting maps into from such a sphere.

All of this plays into the question of what kinds of nontrivial structures can be put on orthogonal bundles on manifolds of various dimensions.  In the dimensions where these homotopy groups are non-trivial, there’s an obstruction to the lifting, and therefore some interesting structure one can put on which may or may not exist.  Hisham Sati spoke of “killing” various homotopy groups – meaning, as far as I can tell, imposing conditions which get past these obstructions.  In string theory, his application of interest, one talks of “anomaly cancellation” – an anomaly being the obstruction to making these structures.  The first part of the punchline is that, since these are related to nontrivial cohomology groups, we can think of them in terms of defining structures on n-bundles or gerbes.  These structures are, essentially, connections – they tell us how to parallel-transport objects of various dimensions.  It turns out that the homotopy group is related to parallel transport along -dimensional surfaces in , which can be thought of as the world-sheets of -dimensional “particles” (or rather, “branes”).

So, for instance, the fact that is nontrivial means there’s an obstruction to a lifting in the form of a class in , which has to do with spin structure – as above.  “Cancelling” this “anomaly” means that for a theory involving such a spin structure to be well-defined, then this characteristic class for must be zero.  The fact that is nontrivial means there’s an obstruction to a lifting in the form of a class in .  This has to do with “string bundles”, where the string group is a higher analog of in exactly the sense we’ve just described.  If such a lifting exists, then there’s a “string-structure” on which is compatible with the spin structure we lifted (and with the orientation a level below that).  Similarly, being nontrivial, by way of an obstruction in , means there’s an interesting notion of “five-brane” structure, and a group, and so on.  Personally, I think of these as giving a geometric interpretation for what the higher cohomology groups actually mean.

A slight refinement of the above, and actually more directly related to “cancellation” of the anomalies, is that these structures can be defined in a “twisted” way: given a cocycle in the appropriate cohomology group, we can ask that a lifting exist, not on the nose, but as a diagram commuting only up to a higher cell, which is exactly given by the cocycle.  I mentioned, in the previous section, a situation where the cocycle gives an associator, so that instead of being exactly associative, a structure has a “twisted” associativity.  This is similar, except we’re twisting the condition that makes a spin structure (or higher spin structure) well-defined.  So if has the wrong characteristic class, we can only define one of these twisted structures at that level.

This theme of higher cohomology and gerbes, and their geometric interpretation, was another one that turned up throughout the talks in the workshop…

And speaking of that: coming up soon, some descriptions of the actual workshop.

 

HGTQGR Workshop – Part I (Pre-school)

So I had a busy week from Feb 7-13, which was when the workshop Higher Gauge Theory, TQFT, and Quantum Gravity (or HGTQGR) was held here in Lisbon.  It ended up being a full day from 0930h to 1900h pretty much every day, except the last.  We’d tried to arrange it so that there were coffee breaks and discussion periods, but there was also a plethora of talks.  Most of the people there seemed to feel that it ended up pretty well.  Since then I’ve been occupied with other things – family visiting the country, for one, so it’s taken a while to get around to writing it up.  Since there were several parts to the event, I’ll do this in several parts as well, of which this is the first one.

Part of the point of the workshop was to bring together a few related subjects in which category theoretic ideas come into areas of mathematics which play a role in physics, and hopefully to build some bridges toward applications.  While it leaned pretty strongly on the mathematical side of this bridge, I think we did manage to get some interaction at the overlap.  Roger Picken drew a nifty picture on the whiteboard at the end of the workshop summarizing how a lot of the themes of the talks clustered around the three areas mentioned in the title, and suggesting how TQFT really does form something of a bridge between the other two – one reason it’s become a topic of some interest recently.  I’ll try to build this up to a similar punchline.

Pre-School

Before the actual event began, though, we had a bunch of talks at IST for a local audience, to try to explain to mathematicians what the physics part of the workshop was about.  Aleksandr Mikovic gave a two-talk introduction to Quantum Gravity, and Sebastian Guttenberg gave a two-part intro to String Theory.  These are two areas where higher gauge theory (in the form of n-connections and n-bundles, or of n-gerbes) has made an appearance, and were the main physics content of the workshop talks.  They set up the basics to help put those talks in context.

Quantum Gravity

Aleksandr’s first talk set out the basic problem of quantizing the gravitational field (this isn’t the only attitude to what the problem of quantum gravity is, but it’s a good starting point), starting with the basic ingredients.  He summarized how general relativity describes gravity in terms of a metric which is supposed to satisfy the Einstein equation, relating the curvature of the metric to a source field which comes from matter.  Quantization then, starting from a classical picture involving trajectories of particles (or sections of fibre bundles to describe fields), one gets a picture where states are vectors in a Hilbert space, and there’s an algebra of operators including observables (self-adjoint operators) and time-evolution (hermitian ones).   An initial try at quantum gravity was to do this using the metric as the field, using the methods of perturbative QFT: treating the metric in terms of “small” fluctuations from some background metric like the flat Minkowski metric.  This uses the Einstein-Hilbert action , where is the gravitational constant and is the Ricci scalar that summarizes the curvature of .  This runs into problems: things diverge in various calculations, and since the coupling constant has units, one can’t “renormalize” the divergences away.  So one needs a non-perturbative approach,  one of which is “canonical quantization“.

After some choice of coordinates (so-called “lapse” and “shift” functions), this involves describing the action in terms of the (space part of) the metric and some canonically conjugate “momentum” variables which describe its extrinsic curvature.  The Euler-Lagrange equations (found as usual by variational calculus methods) then turn out to give the “Hamiltonian constraint” that certain functions of are always zero.  Then the program is to get a Poisson algebra giving commutators of the and variables, then turn it into an algebra of operators in a standard way.  This also runs into problems because the space of metrics isn’t a Hilbert space.  One solution is to not use the metric, but instead a connection and a “frame field” – the so-called Ashtekar variables for GR.  This works better, and gives the “Loop Quantum Gravity” setup, since observables tend to be expressed as holonomies around loops.

Finally, Aleksandr outlined the spin foam approach to quantizing gravity.  This is based on the idea of a quantum geometry as a network (graph) with edges labelled by spins, i.e. representations of SU(2) (which are labelled by half-integers).  Vertices labelled by intertwining operators (which imposes triangle inequalities, as it happens).  The spin foam approach takes a Hilbert space with a basis given by these spin networks.  These are supposed to be an alternative way of describing geometries given by SU(2)-connections. The representations arise because, as the Peter-Weyl theorem shows, they form a nice basis for .  Then to get operators associated to “foams” that interpolate the spacetime between two such geometries (i.e. linear combinations of spin networks).  These are 2-complexes where faces are labelled with spins, and edges with intertwiners for the spins on the faces incident to them.  The operators arise from  a discrete variant of the Feynman path-integral, where time-evolution comes from integrating an action over a space of (classical) trajectories, which in this case are foams.  This needs an action to integrate – in the discrete world, this corresponds to ways of choosing weights for edges and for faces in a generic partition function:

which is a sum over the labels for representations and intertwiners.  Some of the talks that came later in the conference (e.g. by Benjamin Bahr and Bianca Dittrich) came back to discuss principles behind how these functions could be chosen.  (Aristide Baratin’s talk described a similar but more general kind of model based on 2-groups.)

String Theory

In parallel with these, Sebastian Guttenberg gave us a two-lecture introduction to string theory.  His starting point is the intuition that a lot of classical physics studies particles living on a background of some field.  The field can be understood as an approximate way of talking about a large number of quantum-mechanical particles, rather as the dynamics of a large number of classical particles can be approximated by the equations of state for a fluid or gas (depending on how much they interact with one another, among other things).  In string theory and “string field theory”, we have a similar setup, except we replace the particles with small strings – either open strings (which look like intervals) or closed ones (which look like circles).

To begin with, he introduced the basic tools of “classical” string theory – the analog of classical mechanics of point particles.  This is the string analog of the following: one can describe a moving particle by its worldline – a path from a “generic” worldline into a (-dimensional) manifold .  This is generally taken to be like physical spacetime, which in this context means that it has a metric with signature (that is, locally there’s a basis for tangent spaces with one timelike vector and spacelike ones).  Then one can define an action for a moving particle which is just determined by the length of the line’s image.  The nicest way to say this is , where means the pullback of the metric along the map , is some parameter along the generic worldline, and , the particle’s mass, is a coupling constant which doesn’t happen to affect the result in this simple case, but eventually becomes important.  One can do the usual variational-calculus of the Lagrangian approach here, finding a critical point of the action occurs when the particle is travelling in a geodesic – a straight line, in flat space, or the closest available approximation.  In paritcular, the Euler-Lagrange equations say that the covariant derivative of the path should be zero.

There’s an analogous action for a string, the Nambu-Goto action.  Instead of a single-parameter , we now have an embedding of a “generic string worldsheet” – let’s say into spacetime: .  Then then the analogous action is just .  This is pretty much the same as before: we pull back the metric to get , and integrate over the generic worldsheet.  A slight subtlety comes because we’re taking the Hodge dual .  This is conceptually clean, but expands out to a fairly big integral when you express it in coordinates, where the leading term  involves (the determinant is taken over .  Varying this to get the equations of motion produces:

which is the two-dimensional analog of the geodesic equation for a point particle (the are the Christoffel symbols associated to the connection that goes with the metric).  The two-dimensional analog says we have a critical point for the area of the surface which is the image of – in fact, a “maximum”, given the sign of the metric.  For solutions like this, the pullback metric on the worldsheet, , looks flat.  (Naturally, the metric looks flat along a geodesic, too, but this is stronger in 2 dimensions, where there can be intrinsic curvature.)

A souped up version of the Nambu-Goto action is the Polyakov action, which is a natural variation that comes up when has a metric of its own, .  You can check out the details behind that link, but part of what makes this action nice is that the corresponding Euler-Lagrange equation from varying says that .  That is, the worldsheet will have an image with a shape such that its own metric agrees with the one induced from the spacetime .   This action is called the Polyakov action (even though it was introduced by Deser and Zumino, among others) because Polyakov used it for quantizing the string.

Other variations on this action add additional terms which represent fields which the string might be affected by: a scalar , and a 2-form field (here we’re using the physics convention where represents both the function, and its values at particular points, in this case, values of parameters on ).

That 2-form, the “B-field”, is an important field in string theory, and eventually links up with higher gauge theory, which we’ll get to as we go on: one can interpret the B-field as part of a higher connection, to which the string is coupled (as in Baez and Perez, say).  The scalar field essentially determines how strongly the shape of the string itself affects the action – it’s a “string coupling” term, or string coupling “constant” if it’s chosen to be just a number .  (In such a case, the action includes a term that looks like times the Euler characteristic of the surface .)

Sebastian briefly explained some of the physical intuition for why these are the kinds of couplings which it makes sense to introduce.  Essentially, any coupling one writes in coordinates has to get along with gauge symmetries, changes of coordinates, etc.  That is, there should be no physical difference between the class of solutions one finds in a given set of coordinates, and the coordinates one gets by doing some diffeomorphism on the spacetime , or by changing the metric on by some conformal transformation (that is, scaling by some function of position on the worldsheet – underlying string theory is Conformal Field Theory in that the scale of the generic worldsheet is irrelevant – only the light-cones).  Anything a string couples to should be a field that transforms in a way that respects this.  One important upshot for the quantum theory is that when one quantizes a string coupled to such a field, this makes sure that time evolution is unitary.

How this is done is a bit more complicated than Sebastian wanted to go into in detail (and I got a little lost in the summary) so I won’t attempt to do it justice here.  The end results include a partition function:

Remember: if one is finding amplitudes for various observables, the partition function is a normalizing factor, and finding the value of any observables means squeezing them into a similar-looking integral (and normalizing by this factor).  So this says that they’re found by summing over all the string topologies which go from the input to the output, and integrating over all embeddings and metrics on .  (The denominator in that fraction is dividing out by the volumes of the symmetry groups, as usual is quantum field theory since these symmetries mean one is “overcounting” physically identical situations.)

This is just the beginning of string field theory, of course: just as the dynamics of a free moving particle, or even a particle coupled to a background field, are only the beginning of quantum field theory.  But many later additions can be understood as adding various terms to the action in some such formalism.  These would be analogs of giving a point-particle attributes like charge, spin, “colour” and so forth in the Standard Model: these define how it couples to, hence is affected by, various kinds of fields.  Such fields can be understood in terms of connections (or, in general, higher connections, as we’ll get to later), which define how structures are “parallel-transported” along a path (or higher-dimensional surface).

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Coming up in In Part II… I’ll summarize the School portion of the HGTQGR workshop, including lecture series by: Christopher Schommer-Pries on Classifying 2D Extended TQFT, which among other things explained Chris’ proof of the Cobordism Hypothesis using Cerf theory; Tim Porter on Homotopy QFT and the “Crossed Menagerie”, which describe a general framework for talking about quantum theories on cobordisms with structure; John Huerta on Higher Gauge Theory, which gave an introductory account of 2-groups and 2-bundles with 2-connections; Christoph Wockel on connections between Higher Gauge Theory and Infinite Dimensional Lie Theory, which described how some infinite-dimensional Lie algebras can’t be integrated to Lie groups, but only to 2-groups; and one by Hisham Sati on Higher Spin Structures in String Theory, which among other things described how cohomological obstructions to putting certain kinds of structure on manifolds motivates the use of particular higher dimensions.

 

Visit to IST, and XIX Oporto Meeting (Categorification)

It’s been a while since I posted last, but in there I described some issues related to talks I gave in Portugal recently. I’m beginning a postdoc at the Instituto Superior Tecnico, in Lisbon, in less than a month’s time. In the meantime, I’ve been two weeks in Portugal, including a conference and apartment hunting.  Then, last week, I got married. So not surprisingly, I’ve been a bit slow in updating.

The talks I gave are this one, which I gave at IST and this one at the XIX Oporto Meeting on Geometry, Topology and Physics which was held this year in Faro, which this year was a conference on the theme of Categorification!  These talks also appear on my new website, which I got because my hosting at UWO will expire sooner or later, and I wanted something portable (and a portable email address came with it).

Lisbon

Lisbon is an interesting city.  I’ve visited Europe before for conferences and travel and so on, but never for long, and have only lived in North America, where most urban areas are much newer and ancient history more poorly documented.  This is even more so in the southern parts of Europe that were part of the Roman Empire (and even more so in areas of India I’ve travelled in).  I’m looking forward to getting more familiar with the place, which has an exciting and under-appreciated history.  At least I assume it’s underappreciated, since a majority of people in Canada who ask me where I’m moving have never even heard of Lisbon, which I find surprising.

Human settlement in Portugal actually pre-dates homo sapiens, going back to Neanderthals (we often forget there’ve been a few dozen human species before ours. and our era is unusual in human history for having just the one).  Among Sapiens, there have been various periods, most recently the ancient megalith-building cultures, Phoenecians, Greeks, Carthaginians, Romans, Visigoths, Arabs, and then the kingdom (now a republic) of Portugal, established during the Christian reconquest of Iberia.  Lisbon itself dates back at least to Roman times. The oldest surviving areas of Lisbon date back (in streetplan, if not actual buildings) to the Moorish kingdom, when Iberia was known as al-Andalus, some 800 years ago.  Lisbon’s downtown, immediately below this area, couldn’t be more different, being one of the first urban areas planned on a grid – this followed the original area being destroyed in an earthquake and resulting tsunami in 1755.  As the capital of Europe’s first overseas empire, which had reached Japan and Brazil by well over 400 years ago, Lisbon has been a “global city” for at least that long, with spells of boom and bust, and more recently, dictatorship and revolution.  Its location means it was historically a hub that linked the older Mediterranean trading world and the larger Atlantic and Indian Ocean world.

Here is a picture of the main pavillion on the IST campus:

And here is a picture of the neighborhood where I’ll be living, about 10-15 minutes’ walk or two metro stops away:

As you can also see from these pictures, Lisbon contains a number of hills.  It is occasionally reminiscent of San Francisco in that way, and the style of buildings, which also resembles New Orleans occasionally.  And of course, since this is Europe, public spaces that look like this:

And so on.

Visit at IST

Anyway, in the visit at IST, we also had a little mini-conference on categorification, featuring some people who also spoke at Faro (including me) giving longer and more elaborate versions of our talks.  I already commented on mine, so I’ll mention the others:

Rafael Diaz gave a talk about how to categorify noncommutative or “quantum” algebras, in the sense of algebras of power series in noncommuting variables, using ideas similar to the way commutative polynomial algebras can be “categorified” by Joyal’s species.  This “quantum species” idea is laid out partially in this paper. This leads on into ideas about categorifying deformation quantization.

The basic point is to think of “a categorification of a ring R” as a distributive category whose Burnside ring (the ring of isomorphism classes of objects, with algebraic operations from and ) is , or more generally has a “valuation” valued in that is surjective and gets along with the algebra operations.

The category chosen to describe a deformation of is then the category of functors from into .  The main point is then to find a noncommutative product operation , in place of the obvious one derived from , which gives a categorification of a polynomial ring.  This has to do with sticking structured sets together, where some elements of the set can form “links” between the elements of sets – this uses three-”coloured” sets, where one “colour” denotes elements associated to links.

Yazuyoshi Yonezawa gave a talk about some stuff related to link homology invariants such as Khovanov homology.  Such invariants are a major theme for people interested in categorification these days, for various specific reasons, but in general because tangle categories have some nice universal properties, so doing certain kinds of universal higher-dimensional algebra naturally has applications to studying tangles, hence links, hence knots.  In particular, invariants like the Jones, HOMFLY, and HOMFLYPT polynomials, and Reshitikhin-Turaev invariants.  Yazuyoshi’s talk was about an approach to these things based on – as I understood it – some representation theory of quantum , and a diagrammatic calculus that goes with it, for assigning data to strands and crossings of a knot.  (This sort of thing gives a knot invariant as long as it’s invariant under Reidemeister moves – that is, is unaffected by changing the presentation of the knot.  Many of the knot invariants that come up here arise from treating the knot using some sort of diagrammatic calculus – which is where the category theory comes in.)

Aleksandar Mikovic gave a talk about higher gauge theory in the form of 2-BF theory – also known as BFCG theory, this is sort of the “categorified” equivalent of the theory of a flat connection, now taking values in a Lie 2-group.  Actually, he speaks about these in terms of Lie crossed modules, which is a rather nice language for talking about higher-algebraic group-like gadgets in terms of chains of groups with some extra structure (actions of lower groups on higher, and some other things) – see Tim Porter’s “Crossed Menagerie” for a comprehensive look.  The talk was related to finding gauge invariant actions for theories of this sort – the paper it’s based on is one with Joao Faria Martins.

XIX Oporto Meeting

The Oporto meeting on geometry and physics, specifically devoted this year to categorification, was very interesting, with a range of good speakers. Unfortunately, Faro is not optimal as a conference site: the accomodations are a half-hour bus ride from the campus where the conference is held, and the buses come only about once per hour and as a result (of that, and jet-lag, which could happen anyway), I missed some of the talks. Otherwise, it’s a pleasant town with a nice atmosphere, and it was interesting to see some of the variety of people working on categorification.  In particular, a lot of people are working on categorifying aspects of representation theory, which in turn is interesting to topologists, and knot theorists in particular.

One bunch of ideas about categorical representations which was referred to a lot is due to Chuang and Rouquier, substantially described in a paper from a few years ago – here is a post from the Secret Blogging Seminar a few years back describing some of the ideas a bit more succinctly.  The basis for the most popular program being discussed, and the big idea in recent years, is due to Khovanov and Lauda – see the bottom section of this post.

Now, the main invited speakers each gave a series of three hour-long classes on their topic in the mornings, while in the afternoons the other speakers gave 20-minute talks.  The main speakers were these:

Mikhail Khovanov wasn’t able to attend for personal reasons, but there was a great deal of discussion about work that builds on his categorification of quantum groups with Aaron Lauda, who however was there and gave a nice series of talks introducing the ideas (though I missed some because of the unfortunate bus infrastructure). Aaron collects a bunch of resources on this subject here, and I’ll explain a bit of this below.

Sabin Cautis talked about the categorification of in terms of geometric representation theory; the idea here is that there are certain spaces that carry natural representations.  These are flag varieties – the simplest example being Grassmanians – spaces whose points are the -dimensional subspaces of some fixed . In general, flag varieties are spaces whose points consist of a nested sequence of subspaces (the terminology “flag” suggests a flagpole with a 2D rectangle, suspended from a 1D pole, rooted at a 0D point).  The talk was an overview of how to use this to categorify some representation theory.  Here is a recent related paper by Cautis, including Joel Kamnitzer, (I blogged his talk here at UWO a while ago on a similar subject in some more detail), and Anthony Licata.  The basic point is that categories of sheaves on such spaces carry a categorical representation of .

One thing I found interesting – this time, as with Joel’s talk, is that span constructions turn up in this stuff quite naturally, but there is both a similarity and a difference in how they’re used.  In particular, given a flag , we can project to a flag with one fewer entries just by omitting .  So the various flag varieties associated to are connected by a bunch of projections.  Taking two different projections (dropping, say and ), we get a span of varieties – that is, one object with two maps out of it.  We’re talking about spaces of functions on these varieties, so pushing these through spans is of interest.  Lifting a function (by pre-composition – assign a flag the value of the function at its image) is easy – pushing forward is harder.  This involves taking a sum over the function values over the preimage – all the long flags that project to a given short one (to make sure this is tractable, we consider only constructible functions, with finitely many values).  But this sum is weighted.  In the groupoidification program, something similar happens, but the weight there is the groupoid cardinality of the preimage.  Here, it is the Euler characteristic of the preimage (or rather, for each function value, the part of the preimage taking a given value contributes its Eular char. as the weight for that value).  Since groupoid cardinality is like a multiplicative sort of Euler characteristic, there seems to be a close analog here I’d like to understand better.

Catharina Stroppel talked about how the subject relates to Soergel bimodules, and led up to categorifying 3j-symbols.  Soergel bimodules showed up in several different talks about this stuff.  These are the irreducible summands in the bimodule that comes from applying induction functions between module categories finitely many times.

Here, the are  rings of functions invariant under , which is the subgroup of the permutation group which respects a particular composition of (like a partition, but with order – compositions also specify flag varieties, by specifying the codimensions at each inclusion).  The point is that, if , we get inclusions of the rings of invariant functions, and then we can induce representations along those inclusions.  (Notice, by the way, that the correspondence between compositions and the signature of a flag means that this is actually much the same as the inclusions I just described under Sabin Cautis’ talk).  Then doing a finite chain of such inductions gives a functor between module categories.  This can be described by tensoring with some bimodule – the direct summands in this are the Soergel bimodules.  So these are central in talking about these categorical actions and categorified representation theory.  This in turn ended up, in this series of talks, at a categorification of 3j-symbols (which can be built using representations and intertwiners).

Ben Webster talked about how diagrammatic methods used in the Khovanov-Lauda program can be used to categorify algebra representations, and through that, the Reshitikhin-Turaev invariant; the key diagrammatic element turns out to be marking special “red” lines with special rules allowing strands to “act” on them by concatenation.  I must admit Ben Webster’s talks, which ended up rather technical, went far enough over my head that I’m reluctant to summarize, since I was still catching up on the KL program, and this was carrying it quite a bit further.  I do recall that there was much discussion of cyclotomic quotients (partly because Alex Hoffnung later came back to the matter and I had a chance to talk to him about it briefly) – that is, the quotients imposing the relations forcing something to be a root of unity, which isn’t surprising since quantum groups at a root of unity are important and special.  Luckily for the reader who is more up on this stuff than I, the slides can be found here and here.

Dylan Thurston spoke on Heegard-Floer homology (slides here, here, and here – full of great pictures, by the way), which is a homology theory for 3-manifolds (then an invariant for a closed 4-manifold), due to Oszvath and Szabo.  It’s a bi-graded homology theory (i.e. homology theories give complexes for spaces – this gives a bicomplex, with grading in two directions).  This theory gives back the (Conway-)Alexander polynomial for a knot when you take the Euler characteristic of the bicomplex.  That is: there are two directions this complex is graded in: one (columns, say) will correspond to the degree of the variable in the Alexander polynomial; for each , the coefficient of is the Euler characteristic (alternating sum of dimensions) of the entries in that column.  So this is a categorification of this polynomial, in somewhat the way that Khovanov homology categorifies the Jones polynomial.

HF homology can be defined for a knot can be defined in a combinatorial way: a 3-manifold can be represented by a “Heegard diagram” – a 2D surface marked with (coloured) curves, which is a way of keeping track of how a 3-manifold is built by splitting it into parts.  From this diagram, one gets “grid diagrams”, and by a combinatorial process (see the slides for more details) generates the complex.

Others.  I didn’t manage to attend all the other talks (partly because of aforementioned bus issues, and partly because I was still working on mine, having taken a lot of time in Lisbon doing useless things like finding a place to live), but among those I did, there were several that were based on the Khovanov-Lauda program for categorified quantum groups: Anna Beliakova in particular worked with them on categorifying the Casimir (generator of the centre) of the categorified quantum group; people working with Soergel bimodules and categorified Hecke algebras such as Ben Elias and Nicholas Liebedinski.  Then there were the connections to link homology: Christian Blandet and Geordie Williamson talked about things related to the HOMFLYPT polynomial; Krystof Putyra and Emmanuel Wagner gave talks related to Khovanov homology and link homology.  Alex Hoffnung talked about a combinatorial approach for dealing with categorification of cyclotomic quotients as discussed by Ben Webster.

Categorification of Quantum Groups

The reason for categorifying quantum groups, at least in this context, has to do with the manifold invariants associated to them.  Often these come from categories of representations of groups or quantum groups – more generally ones with similar formal properties, meaning roughly monoidal categories with duals (and possibly more structure).  These give state sum invariants, by assigning data from the category to a triangulation of a manifold – objects on edges and morphisms on triangles, say.   The categorification of quantum groups means we pass from having a monoidal category with duals (of representations), to a monoidal 2-category with duals (of representations).  This would mean the state-sum invariants it’s natural to construct are now for 4-manifolds, rather than 3-manifolds.  This is the premise behind spin foam models in gravity, but also has its own life within quantum topology as tools for classifying manifolds, whether or not it accurately describes anything physical.  Marco Mackaay, one of the conference organizers (among several others), has written a bunch on this – for example, this constructs a state-sum invariant given any “spherical” 2-category (a property of certain monoidal 2-categories – see inside for details), and this gives a specific consstruction using the Khovanov-Lauda categorification of .

The Khovanov-Lauda approach to categorifying quantum groups (in particular, deformations of envelopoing algebras of classical Lie algebras, within the category of Hopf algebras)  is most basically about “categorifying” the presentation of an algebra in terms of generators and relations.  That is, we describe a set with some operations in terms of some elements of the set (generators), and some equations (relations) which they satisfy involving the operations.  The presentation used for is the standard one based on an n-vertex (type-A) Dynkin diagram: basically, dots in a row.  There’s a generator for the vertex; the generators for non-adjacent vertices all commute, and for adjacent generators, we have .  (The factor involving is the quantum integer , and becomes 2 in the limit).

To categorify this, we still give generators, but the equations are replaced with isomorphisms – this means we need to be working in some category , hence one essential task is to describe the morphisms.  So: the objects are just rows of dots, labelled by vertices of the Dynkin diagram.  The morphisms are (linearly generated by) isotopy classes of braids from one row to another.  The essential thing is that we have to carefully define “isotopy” here to ensure we get the categorified version of the relations above.  So for non-adjacent-vertex labels, we have the usual Reidemeister moves (the key ones being: we can slide a strand past a crossing, straighten out two complementary crossings); for adjacent-vertex labels, though, we have to tweak this, imposing some relations on strands involving the factors of .  The relations take up a few slides in the talk, but essentially are chosen so that:

Theorem (Khovanov-Lauda): There is an isomorphism of twisted bialgebras between the positive part of and the Grothendieck ring , where multiplication and comultiplication are given by the image of induction and restriction.

Obviously, much more could be said from a five-day conference, but this seems like a nice punchline.

 

Recent Talk: Derek Wise on ETQFT

So I recently received word that this paper had been accepted for publication by Applied Categorical Structures. Since I’ll shortly be putting out another which uses its main construction to build Extended Topological Quantum Field Theories, it’s nice and appropriate to say something about that. But actually, just at the moment, I want to take a slightly different approach.

Toward the end of February, I went up to Waterloo to the Perimeter Institute, where my friend Derek Wise was visiting with Andy Randono – apparently they’re working on a project together that has something to do with Cartan Geometry, which is a subject that plays a big role in Derek’s thesis.

However, Derek was speaking in their seminar about Extended TQFT (his slides are now up on his website, and there’s also a video of the talk available). Actually, a lot of what he was talking about was work of mine, since we’re working on a project together to constructs ETQFT’s from Lie groups (most likely compact ones at first, since all the usual analytical problems with noncompact groups turn up here). However, I really enjoyed seeing Derek talk about it, because he has a sharper grasp than I do of how this subject appears to physicists, and the way he presented this stuff is very different from the way I usually talk about it (you can see me in the video trying to help deal with a question at the end from Rafael Sorkin and Laurent Freidel, and taking a while to correctly understand what it was, partly because of this jargon gap – I hope to get better).

So, for example, describing a TQFT in the Atiyah/Segal axiomatic formulation is fairly natural to someone who works with category theory, but Derek motivated it as a way of taking a “deeper look at the partition function” for a certain field theory. The idea is that a partition function for a quantum field theory associates a number to a space , satisfying certain rules. It is usually described by some kind of integral. Typically in QFT, these are rather tricky integrals – a topological QFT has the nice feature that, since it has no local degrees of freedom, these integrals are much more tractable. Of course, this is a mathematically nice feature that comes at the expense of physical relevance, but such is life.

Anyway, the idea is that the partition function for an -dimensional TQFT can be thought of as assigning, not just numbers to -dimensional manifolds , but something more which reduces to this in a special case. Specifically, assigns a Hilbert space to any codimension-1 submanifold of , in a particular way which Derek passed over by saying it “satisfies some compatibility conditions”. For an audience of mathematicians, you can gloss over this just as quickly by saying the assignments are “functorial”, or even with more detail saying the conditions make a symmetric monoidal functor.

Part of the point is that these conditions are about as obvious on physical grounds as they are if you’re a category theorist. For example, the fact that composition is preserved by the functor can be interpreted physically as saying that the number given by the partition function isn’t affected by how we chop up the manifold to analyse it. The fact that is a monoidal functor ends up meaning that the “unit” for manifolds under unions (namely, the empty manifold with no points, which you can add to things without affecting them) gets assigned the Hilbert space , which is the unit for Hilbert spaces with respect to the tensor product . The fact that this is so means we can treat a manifold with no boundary as going from one (empty) boundary to another (empty) boundary – it therefore gets assigned a linear map from to – a number. Seeing how this linear map comes from composing pieces of the manifold is what “a deeper look at the partition function” means.

ETQFT does essentially the same thing, at one level deeper. The point is that a TQFT breaks apart a manifold by treating it as a series of pieces – manifolds with boundary, glued together at their boundaries. An ETQFT does the same to these pieces, treating them as composed of pieces – manifolds with corners – which are glued orthogonally to the gluing just mentioned. That is, there are two kinds of composition, so we’re in some sort of 2-category (bi-, or double- depending on how you formulate things). The essential point is that now, to manifolds without boundary, which are of codimension 1, we assign Hilbert spaces – and to top-dimensional manifolds WITH boundary, we assign maps of Hilbert spaces.

An ETQFT attempts to give a “deeper-still look at the partition function” by seeing how the Hilbert space arises from composition of pieces in this new direction, along boundaries of codimension 2. The way Derek describes this for physicists is to say that the ETQFT describes how that Hilbert space is “built from local data”, which he described in the usual physics language of path integrals. First of all, the conventional thing in physics is to take for a (codimension-1) manifold to be – the space of square-integrable functions on the quotient of the space of flat -connections on by the action of the group of gauge transformations .

Given a manifold with boundary components and , the standard quantum field theory formalism to describe the map given by a TQFT is to describe how it interacts with particular state-vectors in the Hilbert spaces for the source and target boundary components of . So then:

The point being, a flat connection has some action on it, which depends only on its gauge equivalence class (“the Lagrangian has gauge symmetry”), and it restricts to give flat connections on and , on which the -functions and act, to give something we can integrate. The measure is a crucial entity here, and in general can be a real puzzle, but at least for discrete groups, it’s just a weighted counting measure which effectively gives us the groupoid cardinality of the quotient space. As for the action , the simplest possible case just says the action of any flat connection is zero – hence this expression is just finding the (groupoid) cardinality, or more generally measuring the (stacky) volume, of the configuration space for flat connections. There are other possible actions, though.

Derek gives an explanation of how to interpret this in terms of the “pull-push” construction, which I’ve talked about elsewhere here, including in the above paper, so right now, I’ll just pass to the next layer of the ETQFT layer cake – codimension-2. Here, there is a similar formula, which also has an interpretation in terms of a “pull-push” construction, but which can be written as a categorified path integral.

So now the has boundary, and connects “inner” codimension-2 boundary component to “outer” boundary component . Then, say, gets assigned the category of all gauge-equivariant “bundles” of Hilbert spaces on , rather than the space of gauge-invariant functions. (Derek carefully avoided using the term “category”, to stay physically motivated – and the term “bundle” is accurate in the case of a discrete gauge group , but in general one has to appeal to the theory of measurable fields of Hilbert spaces, since they needn’t be locally trivial). Then given particular Hilbert bundles and on the spaces and respectively, we can define what is by:

The interpretation is much like the previous formula: now we’re direct-integrating Hilbert spaces, instead of integrating complex functions – and we get a Hilbert space instead of a complex number, but this is in some sense superficial. Something any physicist would notice right away (or anyone comparing this to the previous formula) is that the exponential of the action seems to have gone missing, to be replaced by some Hilbert space . If we’re using the trivial action , this is fine, but otherwise, how exactly affects the direct integral would take some explaining. For now, let’s just say that we should think of as being folded into either the inner product on , or into the measure : it shows up in its effect on the inner product on the Hilbert space that this direct integral produces.

Let me jump to the end of Derek’s talk here, to get at some conceptual aspect of what’s happening here. The axiomatic way of talking about ETQFT, namely Ruth Lawrence’s way, is to say we assign a 2-Hilbert space to the codimension-2 manifolds. But “2-Hilbert space” is an off-putting bit of jargon, so instead the suggestion is to replace it with “von Neumann algebra”.

The point is that 2-Hilbert spaces are thought (according to a paper by Baez, Baratin, Friedel and Wise) to be just categories of representations of vN algebras. Being a 2-Hilbert space means, for instance, that they’re additive (by direct sum), -linear (there is a vector space of intertwiners between any two representations), have duals, and so on. Moreover, they’re monoidal 2-Hilbert spaces, since there is a tensor product. Their idea is that the two ideas correspond exactly. In any case, the way the ETQFT construction in question works actually passes through a von Neumann algebra. This comes from the groupoid algebra that’s associated to a certain group action. Namely, the action of the gauge group on the space of flat -connections on the manifold .

Then the way we can look more closely at the “structure of the partition function” is by seeing the Hilbert space associated to a codimension-1 manifold as actually being a kind of morphism of von Neumann algebras. In particular, it’s a Hilbert bimodule, which is acted on by the source algebra (say ) on the left, and the target algebra () on the right. This is intimately connected with the stuff I was writing about recently about Morita equivalence, and so to the 2-Hilbert space view. In particular, a Hilbert bimodule gives an adjoint pair of linear functors (or “2-linear maps”) between the representation categories of algebras.

So shortly I’ll make a post about some papers coming out, and get back to this point…

 

Some different ideas of “state”, part 2: ensembles, 2-states, and selection sectors

I just posted the slides for “Groupoidification and 2-Linearization”, the colloquium talk I gave at Dalhousie when I was up in Halifax last week. I also gave a seminar talk in which I described the quantum harmonic oscillator and extended TQFT as examples of these processes, which covered similar stuff to the examples in a talk I gave at Ottawa, as well as some more categorical details.

Now, in the previous post, I was talking about different notions of the “state” of a system – all of which are in some sense “dual to observables”, although exactly what sense depends on which notion you’re looking at. Each concept has its own particular “type” of thing which represents a state: an element-of-a-set, a function-on-a-set, a vector-in-(projective)-Hilbert-space, and a functional-on-operators. In light of the above slides, I wanted to continue with this little bestiary of ontologies for “states” and mention the versions suggested by groupoidification.

State as Generalized Stuff Type

This is what groupoidification introduces: the idea of a state in . As I said in the previous post, the key concepts behind this program are state, symmetry, and history. “State” is in some sense a logical primitive here – given a bunch of “pure” states for a system (in the harmonic oscillator, you use the nonnegative integers, representing n-photon energy states of the oscillator), and their local symmetries (the -particle state is acted on by the permutation group on elements), one defines a groupoid.

So at a first approximation, this is like the “element of a set” picture of state, except that I’m now taking a groupoid instead of a set. In a more general language, we might prefer to say we’re talking about a stack, which we can think of as a groupoid up to some kind of equivalence, specifically Morita equivalence. But in any case, the image is still that a state is an object in the groupoid, or point in the stack which is just generalizing an element of a set or point in configuration space.

However, what is an “element” of a set ? It’s a map into from the terminal element in , which is “the” one-element set – or, likewise, in , from the terminal groupoid, which has only one object and its identity morphism. However, this is a category where the arrows are set maps. When we introduce the idea of a “history “, we’re moving into a category where the arrows are spans, (which by abuse of notation sometimes gets called but more formally ). A span represents a set/groupoid/stack of histories, with source and target maps into the sets/groupoids/stacks of states of the system at the beginning and end of the process represented by .

Then we don’t have a terminal object anymore, but the same object is still around – only the morphisms in and out are different. Its new special property is that it’s a monoidal unit. So now a map from the monoidal unit is a span . Since the map on the left is unique, by definition of “terminal”, this really just given by the functor , the target map. This is a fibration over , called here for “phi”-bration, but this is appropriate, since it corresponds to what’s usually thought of as a wavefunction .

This correspondence is what groupoidification is all about – it has to do with taking the groupoid cardinality of fibres, where a “phi”bre of is the essential preimage of an object – everything whose image is isomorphic to . This gives an equivariant function on – really a function of isomorphism classes. (If we were being crude about the symmetries, it would be a function on the quotient space – which is often what you see in real mechanics, when configuration spaces are given by quotients by the action of some symmetry group).

In the case where is the groupoid of finite sets and bijections (sometimes called ), these fibrations are the “stuff types” of Baez and Dolan. This is a groupoid with something of a notion of “underlying set” – although a forgetful functor (giving “underlying sets” for objects in a category ) is really supposed to be faithful (so that -morphisms are determined by their underlying set map). In a fibration, we don’t necessarily have this. The special case corresponds to “structure types” (or combinatorial species), where is a groupoid of “structured sets”, with an underlying set functor (actually, species are usually described in terms of the reverse, fibre-selecting functor , where the image of a finite set consists of the set of all “$\Phi$-structured” sets (such as: “graphs on set “, or “trees on “, etc.) The fibres of a stuff type are sets equipped with “stuff”, which may have its own nontrivial morphisms (for example, we could have the groupoid of pairs of sets, and the “underlying” functor selects the first one).

Over a general groupoid, we have a similar picture, but instead of having an underlying finite set, we just have an “underlying -object”. These generalized stuff types are “states” for a system with a configuration groupoid, in . Notice that the notion of “state” here really depends on what the arrows in the category of states are – histories (i.e. spans), or just plain maps.

Intuitively, such a state is some kind of “ensemble”, in statistical or quantum jargon. It says the state of affairs is some jumble of many configurations (which we apparently should see as histories starting from the vacuous unit ), each of which has some “underlying” pure state (such as energy level, or what-have-you). The cardinality operation turns this into a linear combination of pure states by defining weights for each configuration in the ensemble collected in .

2-State as Representation

A linear combination of pure states is, as I said, an equivariant function on the objects of . It’s one way to “categorify” the view of a state as a vector in a Hilbert space, or map from (i.e. a point in the projective Hilbert space of lines in the Hilbert space ), which is really what’s defined by one of these ensembles.

The idea of 2-linearization is to categorify, not a specific state , but the concept of state. So it should be a 2-vector in a 2-Hilbert space associated to . The Hilbert space was some space of functions into $mathbb{C}$, which we categorify by taking instead of a base field, a base category, namely . A 2-Hilbert space will be a category of functors into – that is, the representation category of the groupoid .

(This is all fine for finite groupoids. In the inifinte case, there are some issues: it seems we really should be thinking of the 2-Hilbert space as category of representations of an algebra. In the finite case, the groupoid algebra is a finite dimensional C*-algebra – that is, just a direct sum (over iso. classes of objects) of matrix algebras, which are the group algebras for the automorphism groups at each object. In the infinite dimensional world, you probable should be looking at the representations of the von Neumann algebra completion of the C*-algebra you get from the groupoid. There are all sorts of analysis issues about measurability that lurk in this area, but they don’t really affect how you interpret “state” in this picture, so I’ll skip it.)

A “2-state”, or 2-vector in this Hilbert space, is a representation of the groupoid(-algebra) associated to the system. The “pure” states are irreducible representations – these generate all the others under the operations of the 2-Hilbert space (“sum”, “scalar product”, etc. in their 2-vector space forms). Now, an irreducible representation of a von Neumann algebra is called a “superselection sector” for a quantum system. It’s playing the role of a pure state here.

There’s an interesting connection here to the concept of state as a functional on a von Neumann algebra. As I described in the last post, the GNS representation associates a representation of the algebra to a state. In fact, the GNS representation is irreducible just when the state is a pure state. But this notion of a superselection sector makes it seem that the concept of 2-state has a place in its own right, not just by this correspondence.

So: if a quantum system is represented by an algebra of operators on a Hilbert space , that representation is a direct sum (or direct integral, as the case may be) of irreducible ones, which are “sectors” of the theory, in that any operator in can’t take a vector out of one of these “sectors”. Physicists often associate them with conserved quantities – though “superselection” sectors are a bit more thorough: a mere “selection sector” is a subspace where the projection onto it commutes with some subalgebra of observables which represent conserved quantities. A superselection sector can equivalently be defined as a subspace whose corresponding projection operator commutes with EVERYTHING in . In this case, it’s because we shouldn’t have thought of the representation as a single Hilbert space: it’s a 2-vector in – but as a direct integral of some Hilbert bundle that lives on the space of irreps. Those projections are just part of the definition of such a bundle. The fact that acts on this bundle fibre-wise is just a consequence of the fact that the total is a space of sections of the “2-state”. These correspond to “states” in usual sense in the physical interpretation.

Now, there are 2-linear maps that intermix these superselection sectors: the ETQFT picture gives nice examples. Such a map, for example, comes up when you think of two particles colliding (drawn in that world as the collision of two circles to form one circle). The superselection sectors for the particles are labelled by (in one special case) mass and spin – anyway, some conserved quantities. But these are, so to say, “rest mass” – so there are many possible outcomes of a collision, depending on the relative motion of the particles. So these 2-maps describe changes in the system (such as two particles becoming one) – but in a particular 2-Hilbert space, say for some groupoid describing the current system (or its algebra), a 2-state is a representation of the of the resulting system). A 2-state-vector is a particular representation. The algebra can naturally be seen as a subalgebra of the automorphisms of .

So anyway, without trying to package up the whole picture – here are two categorified takes on the notion of state, from two different points of view.

I haven’t, here, got to the business about Tomita flows coming from states in the von Neumann algebra sense: maybe that’s to come.