### quantization

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 $g_{\mu \nu}$ which is supposed to satisfy the Einstein equation, relating the curvature of the metric to a source field $T_{\mu \nu}$ 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 $S=\frac{1}{G} \int \sqrt{det(g)}R$, where $G$ is the gravitational constant and $R$ is the Ricci scalar that summarizes the curvature of $g$.  This runs into problems: things diverge in various calculations, and since the coupling constant $G$ 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 $g_{kl}$ and some canonically conjugate “momentum” variables $\pi_{kl}$ 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 $g$ are always zero.  Then the program is to get a Poisson algebra giving commutators of the $\pi$ and $g$ 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 $L^2(SU(2))$.  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 $A_e$ for edges and $A_f$ for faces in a generic partition function:

$Z = \sum_{J,I} \prod_{faces} A_f(j_f) \prod_{edges}A_e(i_l)$

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 $A$ 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 $x : \mathbb{R} \rightarrow M^{(D)}$ from a “generic” worldline into a ($D$-dimensional) manifold $M^{(D)}$.  This $M^{(D)}$ is generally taken to be like physical spacetime, which in this context means that it has a metric $g$ with signature $(-1,1,\dots,1)$ (that is, locally there’s a basis for tangent spaces with one timelike vector and $D-1$ 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 $S[x] = m \int d\tau \sqrt{x*g}$, where $x*g$ means the pullback of the metric along the map $x$, $\tau$ is some parameter along the generic worldline, and $m$, 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 $x$, we now have an embedding of a “generic string worldsheet” – let’s say $\Sigma^{(2)} \cong S^1 \times \mathbb{R}$ into spacetime: $x : \Sigma^{(2)} \rightarrow M^{(D)}$.  Then then the analogous action is just $S[x] = \int_{\Sigma^{(2)}} \star_{x*g} 1$.  This is pretty much the same as before: we pull back the metric to get $x*g$, and integrate over the generic worldsheet.  A slight subtlety comes because we’re taking the Hodge dual $\star$.  This is conceptually clean, but expands out to a fairly big integral when you express it in coordinates, where the leading term  involves $\sqrt{det(\partial_{\mu} x^m \partial_{\nu} x^n g_{mn}}$ (the determinant is taken over $(\mu,\nu)$.  Varying this to get the equations of motion produces:

$0 = \partial_{\mu} \partial^{\mu} x^k + \partial_{\mu} x^m \partial^{\mu} x^n \Gamma_{mn}^k$

which is the two-dimensional analog of the geodesic equation for a point particle (the $\Gamma$ 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 $\Sigma^{(2)}$ – in fact, a “maximum”, given the sign of the metric.  For solutions like this, the pullback metric on the worldsheet, $x*g$, 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 $\Sigma^{(2)}$ has a metric of its own, $h$.  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 $h$ says that $x*g \sim h$.  That is, the worldsheet $\Sigma^{(2)}$ will have an image with a shape such that its own metric agrees with the one induced from the spacetime $M^{(D)}$.   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 $\phi(x)$, and a 2-form field $B_{mn}(x)$ (here we’re using the physics convention where $x$ represents both the function, and its values at particular points, in this case, values of parameters $(\sigma_0,\sigma_1)$ on $\Sigma^{(2)}$).

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 $\phi$ 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 $\phi_0$.  (In such a case, the action includes a term that looks like $\phi_0$ times the Euler characteristic of the surface $\Sigma^{(2)}$.)

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 $M^{(D)}$, or by changing the metric on $\Sigma^{(2)}$ by some conformal transformation $h_{\mu \nu} \mapsto exp(2 \omega(\sigma^0,\sigma^1)) h_{\mu \nu}$ (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:

$Z = \sum_{topologies} dx dh \frac{exp(-S[x,h])}{V_{diff} V_{weyl}}$

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 $x : \Sigma^{(2)} \rightarrow M^{(D)}$ and metrics on $\Sigma^{(2)}$.  (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 $S$ 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).

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.

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 $(C,\oplus,\otimes)$ whose Burnside ring (the ring of isomorphism classes of objects, with algebraic operations from $\oplus$ and $\otimes$) is $R$, or more generally has a “valuation” valued in $R$ that is surjective and gets along with the algebra operations.

The category chosen to describe a deformation of $R$ is then the category of functors from $FinSet_0^k$ into $C$.  The main point is then to find a noncommutative product operation $\star$, in place of the obvious one derived from $\otimes$, 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 $\mathfrak{sl}_n$, 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 $sl_2$ 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 $k$-dimensional subspaces of some fixed $V$. In general, flag varieties are spaces whose points consist of a nested sequence of subspaces $V_0 \subset V_1 \subset \dots \subset V_k = V$ (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 $\mathfrak{sl}_2$.

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 $V_0 \subset \dots \subset V_i \subset \dots \subset V_k$, we can project to a flag with one fewer entries just by omitting $V_i$.  So the various flag varieties associated to $V$ are connected by a bunch of projections.  Taking two different projections (dropping, say $V_i$ and $V_j$), 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 $Ind: R^{\lambda'}-mod \rightarrow R^{\lambda}-mod$ finitely many times.

Here, the $R^{\lambda}$ are  rings of functions invariant under $S_{\lambda}$, which is the subgroup of the permutation group $S_n$ which respects a particular composition $\lambda$ of $n$ (like a partition, but with order – compositions also specify flag varieties, by specifying the codimensions at each inclusion).  The point is that, if $S_{\lambda'} < S_{\lambda}$, 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 $(R^{\lambda'},R^{\lambda})$ 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 $q$ 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 $t$ in the Alexander polynomial; for each $k$, the coefficient of $t^k$ 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 $\mathfrak{sl}_3$.

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 $U_q(\mathfrak{sl}_n)$ is the standard one based on an n-vertex (type-A) Dynkin diagram: basically, $n$ dots in a row.  There’s a generator $e_i$ for the $i^{th}$ vertex; the generators for non-adjacent vertices all commute, and for adjacent generators, we have $(q + q^{-1}) e_i e_j e_i = e_j e_i e_j$.  (The factor involving $q$ is the quantum integer $[2]_q$, 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 $R$, 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 $q$.  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 $U_q(\mathfrak{sl}_n)$ and the Grothendieck ring $K_0(R)$, 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.

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 $Z$ for a quantum field theory associates a number to a space $M$, 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 $Z$ for an $n$-dimensional TQFT can be thought of as assigning, not just numbers to $n$-dimensional manifolds $M$, but something more which reduces to this in a special case. Specifically, $Z$ assigns a Hilbert space to any codimension-1 submanifold of $M$, 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 $Z$ 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 $Z$ can be interpreted physically as saying that the number $Z(M)$ given by the partition function isn’t affected by how we chop up the manifold $M$ to analyse it. The fact that $Z$ 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 $\mathbb{C}$, which is the unit for Hilbert spaces with respect to the tensor product $\otimes$. 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 $\mathbb{C}$ to $\mathbb{C}$ – 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 $Z(\Sigma)$ for a (codimension-1) manifold $\Sigma$ to be $L^2(\mathcal{A}_0(\Sigma)/\mathcal{G}(\Sigma))$ – the space of square-integrable functions on the quotient of the space $\mathcal{A}_0(\Sigma)$ of flat $G$-connections on $M$ by the action of the group of gauge transformations $\mathcal{G}(\Sigma)$.

Given a manifold $M$ with boundary components $\Sigma$ and $\Sigma '$, the standard quantum field theory formalism to describe the map $Z(M) : Z(\Sigma) \rightarrow Z(\Sigma ')$ 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 $M$. So then:

$\langle \psi | Z(M) | \phi \rangle = \int_{\mathcal{A}_0(M)/\mathcal{G}} \mathcal{D}A \overline{\psi(A|_{\Sigma '})} e^{i S([A])} \phi(A|_{\Sigma})$

The point being, a flat connection $A$ has some action on it, which depends only on its gauge equivalence class $[A]$ (“the Lagrangian has gauge symmetry”), and it restricts to give flat connections on $\Sigma$ and $\Sigma '$, on which the $L^2$-functions $\psi$ and $\phi$ act, to give something we can integrate. The measure $\mathcal{D}[A]$ 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 $S$, 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 $\Sigma$ has boundary, and connects “inner” codimension-2 boundary component $B_1$ to “outer” boundary component $B_2$. Then, say, $B_1$ gets assigned the category of all gauge-equivariant “bundles” of Hilbert spaces on $\mathcal{A}_0(B_1)$, 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 $G$, 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 $\mathcal{H}$ and $\mathcal{K}$ on the spaces $\mathcal{A}_0(B_1)$ and $\mathcal{A}_0(B_2)$ respectively, we can define what $Z(\Sigma)$ is by:

$\langle \mathcal{K} | Z(M) | \mathcal{H} \rangle = \int_{\mathcal{A}_0(M)/\mathcal{G}} \mathcal{D}A \mathcal{K}(A|_{B_2}) \otimes T_A \otimes \mathcal{H}(A|_{B_1})$

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 $S([A])$ seems to have gone missing, to be replaced by some Hilbert space $T_A$. If we’re using the trivial action $S \cong 0$, this is fine, but otherwise, how exactly $S$ affects the direct integral would take some explaining. For now, let’s just say that we should think of $S([A])$ as being folded into either the inner product on $T_A$, or into the measure $\mathcal{D}A$: 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), $\mathbb{C}$-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 $G$-connections on the manifold $M$.

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 $A$) on the left, and the target algebra ($B$) 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 $H$ 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…

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 $Span(Gpd)$. 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 $n$-particle state is acted on by the permutation group on $n$ 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 $S$? It’s a map into $S$ from the terminal element in $\mathbf{Sets}$, which is “the” one-element set – or, likewise, in $\mathbf{Gpd}$, 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, $A \stackrel{s}{\leftarrow} X \stackrel{t}{\rightarrow} B$ (which by abuse of notation sometimes gets called $X$ but more formally $(X,s,t)$). 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 $X$.

Then we don’t have a terminal object anymore, but the same object $1$ 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 $1 \stackrel{!}{\rightarrow} X \stackrel{\Phi}{\rightarrow} B$. Since the map on the left is unique, by definition of “terminal”, this really just given by the functor $\Phi$, the target map. This is a fibration over $B$, called here $\Phi$ for “phi”-bration, but this is appropriate, since it corresponds to what’s usually thought of as a wavefunction $\phi$.

This correspondence is what groupoidification is all about – it has to do with taking the groupoid cardinality of fibres, where a “phi”bre of $\Phi$ is the essential preimage of an object $b \in B$ – everything whose image is isomorphic to $b$. This gives an equivariant function on $B$ – 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 $B$ is the groupoid of finite sets and bijections (sometimes called $\mathbf{FinSet_0}$), 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 $U: C \rightarrow \mathbf{FinSet_0}$ (giving “underlying sets” for objects in a category $C$) is really supposed to be faithful (so that $C$-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 $X$ is a groupoid of “structured sets”, with an underlying set functor (actually, species are usually described in terms of the reverse, fibre-selecting functor $\mathbf{FinSet_0} \rightarrow \mathbf{Sets}$, where the image of a finite set consists of the set of all “$\Phi$-structured” sets (such as: “graphs on set $S$“, or “trees on $S$“, 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 $\Phi$ 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 $B$-object”. These generalized stuff types are “states” for a system with a configuration groupoid, in $Span(\mathbf{Gpd})$. 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 $1$), 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 $X$.

2-State as Representation

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

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

(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 $\mathcal{A}$ of operators on a Hilbert space $H$, 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 $\mathcal{A}$ 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 $\mathcal{A}$. 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 $\mathbb{Rep}(\mathcal{A})$ – 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 $\mathcal{A}$ acts on this bundle fibre-wise is just a consequence of the fact that the total $H$ 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 $\mathbb{Rep}(X)$ for some groupoid $X$ describing the current system (or its algebra), a 2-state $\Phi$ is a representation of the of the resulting system). A 2-state-vector is a particular representation. The algebra $\mathcal{A}$ can naturally be seen as a subalgebra of the automorphisms of $\Phi$.

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.

I’ve been looking over the last little bit at quantum groupoids, and how they can be used to deform the 2-linearization 2-functor $\Lambda : Span(Gpd) \rightarrow 2Vect$ (or into $2Hilb$) which I’ve discussed in here.

First a little motivation: that functor was part of the way I constructed extended TQFT’s. The inclusion $nCob_2 \rightarrow CoSpan_2(Man)$ realized cobordisms (with corners) in terms of spans of manifolds. Looking at fundamental groupoids using the 2-functor $[\Pi_1(-),G]$ allows us to think about these in terms of the bicategory $Span(Gpd)$, and then applying $\Lambda$ gave 2-Hilbert spaces and 2-linear maps for manifolds and cobordisms (and then natural transformations for cobordisms with corners). Since I made the claim that, with gauge group $G=SU(2)$ – and a suitably infinitary version of $\Lambda$, the extended TQFT gives a theory equivalent to the Ponzano-Regge model of quantum gravity, a reasonable question is: what about the Turaev-Viro model? The PR model is based on labelling edges of a triangulation with representations of $SU(2)$, and the TV model, with representations of $SU_q(2)$.

Now, the groupoids that show up in the above – groupoids of $G$-connections on a manifold, modulo gauge transformations – are quite closely related to this. In particular, the groupoid of connections for a circle (the basic 1-dimensional manifold that the 3-dimensional theory builds from) is $G//Ad G$, the transformation groupoid produced from the action of $G$ on itself by conjugation. (That is: the objects are elements of $G$, and the morphisms are all the conjugacy relations.) Applying $\Lambda$ gives the representation category of this, namely $hom(G // Ad G , Vect)$, so in particular, at the identity of $G$, one has $Rep(SU(2))$ as a sub-2-vector space. (The “states” in the 2-Hilbert space for the circle in the ETQFT are labelled by “masses and spins” – the mass=0 case is what gives the representations of $SU(2)$, and for nonzero mass, one has $Rep(U(1))$.)

More broadly: one can describe the state space of a gauge theory – or many other kinds of theory, in terms of transformation groupoids given by symmetries (gauge transformations, say) acting on states (connections, in that case). Is there a way of doing the same for systems whose symmetries are described by quantum groups? If so, then instead of getting 2-vector spaces which are representation categories of groupoids, we should get some which are representation categories of quantum groupoids.

This paper by Ping Xu describes quantum groupoids – or rather, quantum universal enveloping algebras. They’re described here as a “unification of quantum groups and star products” (star products being the partially-defined composition found in groupoids). This paper by Nikshych and Vainerman describes finite quantum groupoids and some applications – in particular, quantum transformation groupoids, which is the immediately relevant application.

First off, quantum groups: these are Hopf algebras, which in particular are bialgebras – they have both a product

$m : H \otimes H \rightarrow H$

and “coproduct”

$\Delta : H \rightarrow H \otimes H$.

This is because the point here is that we’re following the pattern in which spaces are replaced by algebras: in some simple examples, these are the algebras of functions on a space. The point of noncommutative geometry is that there’s a (contravariant) equivalence between the category of locally compact Hausdorff spaces and the category of commutative algebras, so generalizing to noncommutative algebras (and taking the opposite category) gives a generalization of “locally compact Hausdorff space”. Topological groups like Lie groups are group objects in this category of spaces – and quantum groups are group objects in $Alg^{op}$. So in particular, the group operation shows up as the coproduct $\Delta$, and the inverse operation is the antipode

$S : H \rightarrow H$.

Of course there are also the unit

$\eta : k \rightarrow H$

and co-unit

$\epsilon : H \rightarrow k$

(where $k$ is the base field, say $\mathbb{C}$). The co-unit is of course the “unit” map for the group object. These maps all satisfy some obvious relations.

Now what about quantum groupoids? These are “groupoid objects” – or rather, models of the theory of groupoids – in $Alg^{op}$. We can’t quite say “groupoid objects”, since a groupoid internal to a category $C$ consists of two objects in $C$. For example, a Lie groupoid is a groupoid in $Man$, the category of manifolds. It has a base manifold $B$ and a total manifold $M$, and two maps $s,t : M \rightarrow B$, and so forth. The interpretation is that there is a set (or manifold, or what-have-you) of objects, and a set (etc.) of morphisms. There is a (partially-defined) composition operation allowing morphisms to be composed if the source of one is the target of the other, and so forth.

So (a slightly tweaked version of) the definition of a quantum groupoid given by Xu has it consisting of $(H, R, \alpha, \beta, m, \Delta, \epsilon, S)$. These unpack in pretty natural ways: it helps to compare to both the definition of, say, a Lie groupoid, and a quantum group. $H$ is the “total algebra$and $R$ the “base algebra”, and they correspond to the “noncommutative spaces” of morphisms and objects of a groupoid, respectively. Just as a group can be seen as a groupoid with just one object, a quantum group would be a quantum groupoid where the base algebra $R$ is just the base field $k$. But then, if $R$ is not $k$, we need some nontrivial $\alpha, \beta : R \rightarrow H$ – the source and target maps respectively, which replace the unit map to $k$. Notice they go from the base $R$ to the total algebra $H$, not the other way around, because everything works as usual in $Alg^{op}$. The other maps are likewise dual to those in the definition of a groupoid. The major difference is that we need the equivalent of a partially defined multiplication/composition $m$ and the dual “co-multiplication”/”co-composition” $\Delta$. This works because using $\alpha$ and $\beta$, we get left and right actions of the base $R$ on $H$, which is thus an $(R,R)$-bimodule, hence we can form the bimodule product $H \otimes_R H$, and thus: $m : H \otimes_R H \rightarrow H$ and $\Delta : H \rightarrow H \otimes_R H$ The obvious analog of the unit $\eta : R \rightarrow H$ we had for quantum groups is hidden in Xu’s definition (it seems like it should take the place of the requirement that $H$ be unital), but the co-unit $\epsilon : H \rightarrow R$ is the dual way of describing the “identity” function $x \mapsto 1_x$. The antipode $S : H \rightarrow H$ plays the role of the inverse map for morphisms $g \mapsto g^{-1}$ in groupoids. All these maps have to satisfy various identities which are implied by saying this is a model of the theory of groupoids – check out either of the above papers to see them all explicitly. (A final observation about the definition: a groupoid is a category which has an inverse map from morphisms to morphisms. If we relax the assumption that we have an antipode $S$, we end up with just the definition of a bialgebroid (having $S$ makes it a “Hopf” algebroid). So “bialgebroid” would seem to be the natural “quantum” version of the concept of a general category…) So how might one construct such a “quantum action groupoid”? This is addressed (at least in the finite case) in the paper by Nikshych and Vainerman, in their section 2.6. This is generalizing the action groupoid arising from a group acting on a set. The set $S$ is replaced by an algebra $B$ (which must be separable, for them – the equivalent of a finite set – and thought of as a “quantum space”). The group $G$ is replaced by a quantum group (or, generally, Hopf algebra) $H$. The equivalent of having action of the group on the set is that $B$ is a (right) $H$-module. Now, the action groupoid for a $G$ action on $S$ has for objects the elements of $S$, and for morphisms, all relations $g(s) = s'$, which we can write as morphisms $g_s$, with source $s$ and target $s' = g(s)$. The action quantum groupoid associated to the $H$-module $B$ is the double crossed product $B^{op} \lhd H \rhd B$, with multiplication, co-multiplication, etc. defined in fairly natural ways. (Note: those triangles should be semidirect products, but I can’t seem to make that symbol appear here.) So finally, I seem to be claiming that a such a quantum groupoid, let’s call it $Q=(H,R,\alpha,\beta,m,\Delta,\epsilon,S)$ is the right “classical” state space (if that’s not too blatant a contradiction in terminology) for a theory having quantum-group symmetry – at least in the categorified picture. No doubt in many cases there is additional structure, capturing the equivalent of, say, symplectic structure, that should also be included (such things certainly can be found in NCG, but I’m still absorbing how exactly). Then the 2-vector space for the quantized version of such a theory is the category $Rep(Q)$, and a “2-state” just an object in here – a representation of $Q$. One thing that’s not quite clear to me just now is how this relates to the usual idea of “state” in NCG – a state for a “quantum space” (which is an algebra) being a linear functional on that algebra. Not necessarily a character (i.e. a homomorphism into $\mathbb{C}$), mind you – that would be a 1-dimensional representation, but just a functional. I’m going up to Ottawa for a few days, in part to talk about spans and groupoids (basically, some cross section of the material in these posts here) at a conference put on by the Ottawa U math department, primarily for grad students and postdocs in the general vicinity. This is nice – gives me a chance to visit my parents and friends there (the fraction of my life I lived in Ottawa is now creeping down toward a mere third, but it probably has as strong a claim to “home” as anywhere). May is also one of the most tolerable months to be there. One of the grad students in our department is also going. Enxin Wu recently decided to start working with Dan Christensen too, so probably in future we’ll have various things to talk about. Last week, he gave a seminar talk on algebra deformation that was a long version of the one he’ll be giving in Ottawa. Enxin is one of those guys who seems to really understand – it’s tempting to say grok– algebra, which I always find impressive. I’m a predominantly visual thinker, and the kind of symbolic computations common in algebra always seem a little mysterious to me at first until I can find a picture, or at least practice them a lot. Lie groups, for instance, make some sense to me – you can picture rotation groups, or at least keep a geometric picture of a manifold in mind. Lie algebras, being infinitesimal versions of Lie groups, are also not so hard to visualize. General associative algebras? Harder. The talk was about associative algebras, to give some background on deformation, but the things whose deformations Enxin has been thinking about are $A_{\infty}$-algebras (see this brief intro, for instance), an “invention” of Stasheff. The talk was about deformation of these algebras – the kind of deformation that pertains to deformation quantization. This has been studied by Kontsevich. Deformation quantization has to do with replacing things valued in some algebra $A$ by new things, valued in the bigger algebra $A[[t]]$ of formal power series in $t$ with coefficients in $A$, so that the original structure you started with is just the constant part that appears when you set $t=0$. (The term “quantization” applies when you consider algebras of functions on a manifold, with a Poisson bracket – in other words, algebras of observables of a physical system). Some of the main results have to do with the Hochschild cohomology for some complex associated to the algebra you start with, and the fact that this cohomology classifies obstructions to the deformation. I expected to get lost in a maze of notation – and there certainly is a lot – but as it turns out, I had some mental pictures to attach to these things, because related things came up a few years ago in the quantum gravity seminar at UCR (week 8 on that page especially), which provides a few pictures that helped a lot. Diagrammatic notation makes algebra a lot more comprehensible to me. So let’s get more specific. The point is to replace a multiplication operator $m : A \otimes A \rightarrow A$ with a power series whose coefficients are “multiplication” operators. That is, a deformation of an associative algebra $(A,m)$ (where $m : A \otimes A \rightarrow A$ is the multiplication for $A$) is $(A[[t]],m_t)$, where the new multiplication $m_t$ is defined (by linearity) by its action on elements of $A$, which works like this: $m_t(a,b) = \sum_{i=0}^{\infty} {\alpha_i}(a,b){t^i}$ for some operators $\alpha_i : A \otimes A \rightarrow A$. Then there are a bunch of conditions on the $\alpha$ that are needed to make $m_t$ associative. There’s one condition for each power of $t$, since the coefficients in the associator should be zero: $\sum_{i+j=n\\i,j>0} \alpha_i( (\alpha_j \otimes 1) - (1 \otimes \alpha_j)) = 0$ The $n=0$ condition just says that $\alpha_0$ is associative – so it’s the $m$ from the original algebra, which you get back when $t=0$. Then given an algebra $A$, you can create the deformation category $\mathcal{D}$ of $A$ whose objects are its deformations. The morphisms are continuous algebra homomorphisms that get along with the multiplication operations. It turns out that since formal power series with nonzero $n=0$ term are invertible (a consequence of the Lagrange theorem) this $\mathcal{D}$ is actually a groupoid. Then the question is to classify the isomorphism classes of deformations – that is, $\Pi_0(\mathcal{D})$. One can easily imagine that there might be no nontrivial deformations of some algebra – that is, every one is isomorphic to the deformation where all the $\alpha_i$ are trivial except $\alpha_0 = m$. So when does this happen? More generally, how can one classify the deformations up to isomorphism? The answer has to do with Hochschild cohomology, which is related to a complex you can make from $A$. Taking $C^n(A) = hom(A^{\otimes n},A)$, the space of $n$-ary multilinear operations on $A$, you build this complex: $0 \stackrel{d_0}{\longrightarrow} C^0(A) \stackrel{d_1}{\longrightarrow} C^1(A) \stackrel{d_2}{\longrightarrow} \dots$ where the differential maps are $d_n : C^n(A) \rightarrow C^{n+1}(A)$ defined by an alternating sum: $d(f)(a_1, \dots, a_n) = a_1 f(a_2, \dots, a_{n+1}) + \sum_{i=1}^{n} (-1)^i f(a_1, \dots, a_i a_{i+1}, \dots, a_{n+1}) + (-1)^{n+1} f(a_1, \dots,a_n) a_{n+1}$ (Intuitively: there are too many arguments, so you start with the extra one on the left, push it into the middle as a “lump under the rug” where two arguments are combined, and push the lump all the way to the right. To ensure that $d^2 = 0$, you do this with alternating signs. This kind of algebraic manipulation is the kind of thing I can do, and clearly works, but I don’t exactly grok.) Then you take the Hochschild cohomology groups in the standard cohomology way: $HH^i = \frac{ker(d_{i+1})}{Im(d_i)}$. A cohomology class in one of these groups is a class of multilinear maps from $n$ copies of $A$ to $A$ (up to a factor which is $d_n$ of something). As usual with cohomology, they describe obstructions to something – to exactness. Exactness, in this setting, would mean that $A$ has no interesting deformations at the $n^{th}$ level. What does “level” mean here? Well, for example, at level 2 we’re talking about maps $A \otimes A \rightarrow A$, such as the multiplication map. In fact, we have $d_3(m) = 0$ for an associative algebra – you can check that $d(m)$ is twice the associator $a_1(a_2a_3) - (a_1a_2)a_3$, which is zero. So $m$ is a cochain. Is it a coboundary? Sure – it’s $d_2(1)$. So $m$ is in the trivial class in $HH^2(A)$. The point then is that it turns out that if this is the only class – if $HH^2(A) = 0$ – then there are no interesting deformations of the multiplication of $A$ in the sense described above. The groupoid$\mathcal{D}$has just one object. (One thing that occurs to me is that this makes it a group – which group is something Enxin didn’t discuss. My algebra instincts aren’t quite up to answering that off the top of my head.) For example, if $A = \mathbb{C}$ (as an algebra over $\mathbb{R}$), there are no nontrivial deformations: $HH^2(\mathbb{C}) = 0$. What do the other levels mean? Really, this is where you’d want to look at the generalization from associative algebras to $A_{\infty}$-algebras. Whereas for an associative algebra $A$, the associator$a(x,y,z) = x(yz) – (xy)z\$ is zero, in general an $A_{\infty}$-algebra will have an associator map $a : A^{\otimes 3} \rightarrow A$ (that is, $a \in C^3$ in the complex above), which might not be zero, but which is $d_3(m)$.

This is the beginning of a story relating $A_{\infty}$-algebras to weak $\infty$-categories: a bicategory, for example, has an associator for composition of morphisms. In a bicategory, you expect the associator to satisfy a certain identity – the Pentagon identity – but in general you’d just ask for a “pentagonator” (something in $C^4$), and so on (this is where those seminar notes above help me think in pictures, by the way). An $A_{\infty}$-algebra is a vector space equipped with maps at all these levels – described by Stasheff’s associahedra – satisfying some relations. The general story of deformation relates the Hochschild cohomology groups at different levels to deformations of $A_{\infty}$-algebras. Enxin didn’t go into this in his talk, but he did say a little something about the next level:

An infinitesimal deformation of $A$ is a deformation not in $A[[t]]$, but in the quotient $A[[t]]/(t^2=0)$. This only needs two maps, $\alpha_0 , \alpha_1$. The third Hochschild cohomology measures obstructions to extending an infinitesimal deformation to a full deformation in $A[[t]]$ – if $HH^3(A) = 0$, then any infinitesimal deformation can be extended to a full deformation.

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

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