## Hamburg

Since I moved to Hamburg,   Alessandro Valentino and I have been organizing one series of seminar talks whose goal is to bring people (mostly graduate students, and some postdocs and others) up to speed on the tools used in Jacob Lurie’s big paper on the classification of TQFT and proof of the Cobordism Hypothesis.  This is part of the Forschungsseminar (“research seminar”) for the working groups of Christoph Schweigert, Ingo Runkel, and Christoph Wockel.  First, I gave one introducing myself and what I’ve done on Extended TQFT. In our main series We’ve had a series of four so far – two in which Alessandro outlined a sketch of what Lurie’s result is, and another two by Sebastian Novak and Marc Palm that started catching our audience up on the simplicial methods used in the theory of $(\infty,n)$-categories which it uses.  Coming up in the New Year, Nathan Bowler and I will be talking about first $(\infty,1)$-categories, and then $(\infty,n)$-categories.   I’ll do a few posts summarizing the talks around then.

Some people in the group have done some work on quantum field theories with defects, in relation to which, there’s this workshop coming up here in February!  The idea here is that one could have two regions of space where different field theories apply, which are connected along a boundary. We might imagine these are theories which are different approximations to what’s going on physically, with a different approximation useful in each region.  Whatever the intuition, the regions will be labelled by some category, and boundaries between regions are labelled by functors between categories.  Where different boundary walls meet, one can have natural transformations.  There’s a whole theory of how a 3D TQFT can be associated to modular tensor categories, in sort of the same sense that a 2D TQFT is associated to a Frobenius algebra. This whole program is intimately connected with the idea of “extending” a given TQFT, in the sense that it deals with theories that have inputs which are spaces (or, in the case of defects, sub-spaces of given ones) of many different dimensions.  Lurie’s paper describing the n-dimensional cobordism category, is very much related to the input to a theory like this.

## Brno Visit

This time, I’d like to mention something which I began working on with Roger Picken in Lisbon, and talked about for the first time in Brno, Czech Republic, where I was invited to visit at Masaryk University.  I was in Brno for a week or so, and on Thursday, December 13, I gave this talk, called “Higher Gauge Theory and 2-Group Actions”.  But first, some pictures!

This fellow was near the hotel I stayed in:

Since this sculpture is both faceless and hard at work on nonspecific manual labour, I assume he’s a Communist-era artwork, but I don’t really know for sure.

The Christmas market was on in Náměstí Svobody (Freedom Square) in the centre of town.  This four-headed dragon caught my eye:

On the way back from Brno to Hamburg, I met up with my wife to spend a couple of days in Prague.  Here’s the Christmas market in the Old Town Square of Prague:

Anyway, it was a good visit to the Czech Republic.  Now, about the talk!

### Moduli Spaces in Higher Gauge Theory

The motivation which I tried to emphasize is to define a specific, concrete situation in which to explore the concept of “2-Symmetry”.  The situation is supposed to be, if not a realistic physical theory, then at least one which has enough physics-like features to give a good proof of concept argument that such higher symmetries should be meaningful in nature.  The idea is that Higher Gauge theory is a field theory which can be understood as one in which the possible (classical) fields on a space/spacetime manifold consist of maps from that space into some target space $X$.  For the topological theory, they are actually just homotopy classes of maps.  This is somewhat related to Sigma models used in theoretical physics, and mathematically to Homotopy Quantum Field Theory, which considers these maps as geometric structure on a manifold.  An HQFT is a functor taking such structured manifolds and cobordisms into Hilbert spaces and linear maps.  In the paper Roger and I are working on, we don’t talk about this stage of the process: we’re just considering how higher-symmetry appears in the moduli spaces for fields of this kind, which we think of in terms of Higher Gauge Theory.

Ordinary topological gauge theory – the study of flat connections on $G$-bundles for some Lie group $G$, can be looked at this way.  The target space $X = BG$ is the “classifying space” of the Lie group – homotopy classes of maps in $Hom(M,BG)$ are the same as groupoid homomorphisms in $Hom(\Pi_1(M),G)$.  Specifically, the pair of functors $\Pi_1$ and $B$ relating groupoids and topological spaces are adjoints.  Now, this deals with the situation where $X = BG$ is a homotopy 1-type, which is to say that it has a fundamental groupoid $\Pi_1(X) = G$, and no other interesting homotopy groups.  To deal with more general target spaces $X$, one should really deal with infinity-groupoids, which can capture the whole homotopy type of $X$ – in particular, all its higher homotopy groups at once (and various relations between them).  What we’re talking about in this paper is exactly one step in that direction: we deal with 2-groupoids.

We can think of this in terms of maps into a target space $X$ which is a 2-type, with nontrivial fundamental groupoid $\Pi_1(X)$, but also interesting second homotopy group $\pi_2(X)$ (and nothing higher).  These fit together to make a 2-groupoid $\Pi_2(X)$, which is a 2-group if $X$ is connected.  The idea is that $X$ is the classifying space of some 2-group $\mathcal{G}$, which plays the role of the Lie group $G$ in gauge theory.  It is the “gauge 2-group”.  Homotopy classes of maps into $X = B \mathcal{G}$ correspond to flat connections in this 2-group.

For practical purposes, we use the fact that there are several equivalent ways of describing 2-groups.  Two very directly equivalent ways to define them are as group objects internal to $\mathbf{Cat}$, or as categories internal to $\mathbf{Grp}$ – which have a group of objects and a group of morphisms, and group homomorphisms that define source, target, composition, and so on.  This second way is fairly close to the equivalent formulation as crossed modules $(G,H,\rhd,\partial)$.  The definition is in the slides, but essentially the point is that $G$ is the group of objects, and with the action $G \rhd H$, one gets the semidirect product $G \ltimes H$ which is the group of morphisms.  The map $\partial : H \rightarrow G$ makes it possible to speak of $G$ and $H$ acting on each other, and that these actions “look like conjugation” (the precise meaning of which is in the defining properties of the crossed module).

The reason for looking at the crossed-module formulation is that it then becomes fairly easy to understand the geometric nature of the fields we’re talking about.  In ordinary gauge theory, a connection can be described locally as a 1-form with values in $Lie(G)$, the Lie algebra of $G$.  Integrating such forms along curves gives another way to describe the connection, in terms of a rule assigning to every curve a holonomy valued in $G$ which describes how to transport something (generally, a fibre of a bundle) along the curve.  It’s somewhat nontrivial to say how this relates to the classic definition of a connection on a bundle, which can be described locally on “patches” of the manifold via 1-forms together with gluing functions where patches overlap.  The resulting categories are equivalent, though.

In higher gauge theory, we take a similar view. There is a local view of “connections on gerbes“, described by forms and gluing functions (the main difference in higher gauge theory is that the gluing functions related to higher cohomology).  But we will take the equivalent point of view where the connection is described by $G$-valued holonomies along paths, and $H$-valued holonomies over surfaces, for a crossed module $(G,H,\rhd,\partial)$, which satisfy some flatness conditions.  These amount to 2-functors of 2-categories $\Pi_2(M) \rightarrow \mathcal{G}$.

The moduli space of all such 2-connections is only part of the story.  2-functors are related by natural transformations, which are in turn related by “modifications”.  In gauge theory, the natural transformations are called “gauge transformations”, and though the term doesn’t seem to be in common use, the obvious term for the next layer would be “gauge modifications”. It is possible to assemble a 2-groupoid $Hom(\Pi_2(M),\mathcal{G}$, whose space of objects is exactly the moduli space of 2-connections, and whose 1- and 2-morphisms are exactly these gauge transformations and modifications.  So the question is, what is the meaning of the extra information contained in the 2-groupoid which doesn’t appear in the moduli space itself?

Our claim is that this information expresses how the moduli space carries “higher symmetry”.

### 2-Group Actions and the Transformation Double Category

What would it mean to say that something exhibits “higher” symmetry? A rudimentary way to formalize the intuition of “symmetry” is to say that there is a group (of “symmetries”) which acts on some object. One could get more subtle, but this should be enough to begin with. We already noted that “higher” gauge theory uses 2-groups (and beyond into $n$-groups) in the place of ordinary groups.  So in this context, the natural way to interpret it is by saying that there is an action of a 2-group on something.

Just as there are several equivalent ways to define a 2-group, there are different ways to say what it means for it to have an action on something.  One definition of a 2-group is to say that it’s a 2-category with one object and all morphisms and 2-morphisms invertible.  This definition makes it clear that a 2-group has to act on an object of some 2-category $\mathcal{C}$. For our purposes, just as we normally think of group actions on sets, we will focus on 2-group actions on categories, so that $\mathcal{C} = \mathbf{Cat}$ is the 2-category of interest. Then an action is just a map:

$\Phi : \mathcal{G} \rightarrow \mathbf{Cat}$

The unique object of $\mathcal{G}$ – let’s call it $\star$, gets taken to some object $\mathbf{C} = \Phi(\star) \in \mathbf{Cat}$.  This object $\mathbf{C}$ is the thing being “acted on” by $\mathcal{G}$.  The existence of the action implies that there are automorphisms $\Phi(g) : \mathbf{C} \rightarrow \mathbf{C}$ for every morphism in $\mathbf{G}$ (which correspond to the elements of the group $G$ of the crossed module).  This would be enough to describe ordinary symmetry, but the higher symmetry is also expressed in the images of 2-morphisms $\Phi( \eta : g \rightarrow g') = \Phi(\eta) : \Phi(g) \rightarrow \Phi(g')$, which we might call 2-symmetries relating 1-symmetries.

What we want to do in our paper, which the talk summarizes, is to show how this sort of 2-group action gives rise to a 2-groupoid (actually, just a 2-category when the $\mathbf{C}$ being acted on is a general category).  Then we claim that the 2-groupoid of connections can be seen as one that shows up in exactly this way.  (In the following, I have to give some credit to Dany Majard for talking this out and helping to find a better formalism.)

To make sense of this, we use the fact that there is a diagrammatic way to describe the transformation groupoid associated to the action of a group $G$ on a set $S$.  The set of morphisms is built as a pullback of the action map, $\rhd : (g,s) \mapsto g(s)$.

This means that morphisms are pairs $(g,s)$, thought of as going from $s$ to $g(s)$.  The rule for composing these is another pullback.  The diagram which shows how it’s done appears in the slides.  The whole construction ends up giving a cubical diagram in $\mathbf{Sets}$, whose top and bottom faces are mere commuting diagrams, and whose four other faces are all pullback squares.

To construct a 2-category from a 2-group action is similar. For now we assume that the 2-group action is strict (rather than being given by $\Phi$ a weak 2-functor).  In this case, it’s enough to think of our 2-group $\mathcal{G}$ not as a 2-category, but as a group-object in $\mathbf{Cat}$ – the same way that a 1-group, as well as being a category, can be seen as a group object in $\mathbf{Set}$.  The set of objects of this category is the group $G$ of morphisms of the 2-category, and the morphisms make up the group $G \ltimes H$ of 2-morphisms.  Being a group object is the same as having all the extra structure making up a 2-group.

To describe a strict action of such a $\mathcal{G}$ on $\mathbf{C}$, we just reproduce in $\mathbf{Cat}$ the diagram that defines an action in $\mathbf{Sets}$:

The fact that $\rhd$ is an action just means this commutes. In principle, we could define a weak action, which would mean that this commutes up to isomorphism, but we won’t be looking at that here.

Constructing the same diagram which describes the structure of a transformation groupoid (p29 in the slides for the talk), we get a structure with a “category of objects” and a “category of morphisms”.  The construction in $\mathbf{Set}$ gives us directly a set of morphisms, while $S$ itself is the set of objects. Similarly, in $\mathbf{Cat}$, the category of objects is just $\mathbf{C}$, while the construction gives a category of morphisms.

The two together make a category internal to $\mathbf{Cat}$, which is to say a double category.  By analogy with $S / \!\! / G$, we call this double category $\mathbf{C} / \!\! / \mathcal{G}$.

We take $\mathbf{C}$ as the category of objects, as the “horizontal category”, whose morphisms are the horizontal arrows of the double category. The category of morphisms of $\mathbf{C} /\!\!/ \mathcal{G}$ shows up by letting its objects be the vertical arrows of the double category, and its morphisms be the squares.  These look like this:

The vertical arrows are given by pairs of objects $(\gamma, x)$, and just like the transformation 1-groupoid, each corresponds to the fact that the action of $\gamma$ takes $x$ to $\gamma \rhd x$. Each square (morphism in the category of morphisms) is given by a pair $( (\gamma, \eta), f)$ of morphisms, one from $\mathcal{G}$ (given by an element in $G \rtimes H$), and one from $\mathbf{C}$.

The horizontal arrow on the bottom of this square is:

$(\partial \eta) \gamma \rhd f \circ \Phi(\gamma,\eta)_x = \Phi(\gamma,\eta)_y \circ \gamma \rhd f$

The fact that these are equal is exactly the fact that $\Phi(\gamma,\eta)$ is a natural transformation.

The double category $\mathbf{C} /\!\!/ \mathcal{G}$ turns out to have a very natural example which occurs in higher gauge theory.

### Higher Symmetry of the Moduli Space

The point of the talk is to show how the 2-groupoid of connections, previously described as $Hom(\Pi_2(M),\mathcal{G})$, can be seen as coming from a 2-group action on a category – the objects of this category being exactly the connections. In the slides above, for various reasons, we did this in a discretized setting – a manifold with a decomposition into cells. This is useful for writing things down explicitly, but not essential to the idea behind the 2-symmetry of the moduli space.

The point is that there is a category we call $\mathbf{Conn}$, whose objects are the connections: these assign $G$-holonomies to edges of our discretization (in general, to paths), and $H$-holonomies to 2D faces. (Without discretization, one would describe these in terms of $Lie(G)$-valued 1-forms and $Lie(H)$-valued 2-forms.)

The morphisms of $\mathbf{Conn}$ are one type of “gauge transformation”: namely, those which assign $H$-holonomies to edges. (Or: $Lie(H)$-valued 1-forms). They affect the edge holonomies of a connection just like a 2-morphism in $\mathcal{G}$.  Face holonomies are affected by the $H$-value that comes from the boundary of the face.

What’s physically significant here is that both objects and morphisms of $\mathbf{Conn}$ describe nonlocal geometric information.  They describe holonomies over edges and surfaces: not what happens at a point.  The “2-group of gauge transformations”, which we call $\mathbf{Gauge}$, on the other hand, is purely about local transformations.  If $V$ is the vertex set of the discretized manifold, then $\mathbf{Gauge} = \mathcal{G}^V$: one copy of the gauge 2-group at each vertex.  (Keeping this finite dimensional and avoiding technical details was one main reason we chose to use a discretization.  In principle, one could also talk about the 2-group of $\mathcal{G}$-valued functions, whose objects and morphisms, thinking of it as a group object in $\mathbf{Cat}$, are functions valued in morphisms of $\mathcal{G}$.)

Now, the way $\mathbf{Gauge}$ acts on $\mathbf{Conn}$ is essentially by conjugation: edge holonomies are affected by pre- and post-multiplication by the values at the two vertices on the edge – whether objects or morphisms of $\mathbf{Gauge}$.  (Face holonomies are unaffected).  There are details about this in the slides, but the important thing is that this is a 2-group of purely local changes.  The objects of $\mathbf{Gauge}$ are gauge transformations of this other type.  In a continuous setting, they would be described by $G$-valued functions.  The morphisms are gauge modifications, and could be described by $H$-valued functions.

The main conceptual point here is that we have really distinguished between two kinds of gauge transformation, which are the horizontal and vertical arrows of the double category $\mathbf{Conn} /\!\!/ \mathbf{Gauge}$.  This expresses the 2-symmetry by moving some gauge transformations into the category of connections, and others into the 2-group which acts on it.  But physically, we would like to say that both are “gauge transformations”.  So one way to do this is to “collapse” the double category to a bicategory: just formally allow horizontal and vertical arrows to compose, so that there is only one kind of arrow.  Squares become 2-cells.

So then if we collapse the double category expressing our 2-symmetry relation this way, the result is exactly equivalent to the functor category way of describing connections.  (The morphisms will all be invertible because $\mathbf{Conn}$ is a groupoid and $\mathbf{Gauge}$ is a 2-group).

I’m interested in this kind of geometrical example partly because it gives a good way to visualize something new happening here.  There appears to be some natural 2-symmetry on this space of fields, which is fairly easy to see geometrically, and distinguishes in a fundamental way between two types of gauge transformation.  This sort of phenomenon doesn’t occur in the world of $\mathbf{Sets}$ – a set $S$ has no morphisms, after all, so the transformation groupoid for a group action on it is much simpler.

In broad terms, this means that 2-symmetry has qualitatively new features that familiar old 1-symmetry doesn’t have.  Higher categorical versions – $n$-groups acting on $n$-groupoids, as might show up in more complicated HQFT – will certainly be even more complicated.  The 2-categorical version is just the first non-trivial situation where this happens, so it gives a nice starting point to understand what’s new in higher symmetry that we didn’t already know.

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 $U_q(sl_2)$, 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 $sl_2$ on categories.  This starting point is nice, because it can work by just mimicking the construction of $sl_2$ and $U_q(sl_2)$ representations in terms of weight spaces: one gets categories $V_{-N}, \dots, V_N$ which correspond to the “weight spaces” (usually just vector spaces), and the $E$ and $F$ 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 $n$-dimensional space.  A simple example is the Grassmannian $Gr(1,V)$, which is the space of all 1-dimensional subspaces of $V$ (i.e. the projective space $P(V)$), which is of course an algebraic variety.  Likewise, $Gr(k,V)$, the space of all $k$-dimensional subspaces of $V$ is a variety.  The flag variety $Fl(k,k+1,V)$ consists of all pairs $W_k \subset W_{k+1}$, of a $k$-dimensional subspace of $V$, inside a $(k+1)$-dimensional subspace (the case $k=2$ 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”, $Fl(1,2,\dots,n,V)$ (where $V$ is $n$-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 $Fl(k,k+1,V)$ as relating the Grassmanians: there are projections $Fl(k,k+1,V) \rightarrow Gr(k,V)$ and $Fl(k,k+1,V) \rightarrow Gr(k+1,V)$, 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 $H_0, H_1, \dots, H_n$, 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 $U_q(sl_2)$ (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 $n$ commuting variables, $\mathbb{C}[x_1,\dots,x_n]$, generated by the multiplication operators $x_i$, 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 $q$-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 $x_i$ 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 $sl_3$ web algebra, describing the results of this paper with Weiwei Pan and Daniel Tubbenhauer.  This is the analog of the above, for $U_q(sl_3)$, 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 $h$ (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 $G$.

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”  $H \bowtie K$ of two groups $H$ and $K$ (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:

$Poinc = (SO(3) \bowtie (SO(1; 1) \bowtie N)) \ltimes \mathbb{R}_4$

($N$ is certain group of matrices).  So there’s a unique double group which corresponds to it – it has squares labelled by $\mathbb{R}_4$, and the horizontial and vertical morphisms by elements of $SO(3)$ and $N$ respectively.  Dany finished by explaining that there are higher-dimensional analogs of all this – $n$-tuple categories can be defined recursively by internalization (“internal categories in $(n-1)$-tuple-Cat”).  There are somewhat more sophisticated versions of the same kind of structure, and finally leading up to a special class of $n$-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 $n$-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 $X$ is a quotient of the space of closed forms (so the $k^{th}$ cohomology, $H^{k}(X) = \Omega^{k}$, is a quotient of the space of closed $k$-forms – the quotient being that forms differing by a coboundary are considered the same).  There’s a similar construction for the $K$-theory $K(X)$, which can be modelled as a quotient of the space of vector bundles over $X$.  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 $(n-1)$-dimensional manifolds and $n$-dimensional cobordisms

$Z : d-Bord \rightarrow Vect$

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 $(M,\mathcal{O})$, 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 $d|\delta$-dimensional space has the sheaf of rings where one introduces some new antisymmetric coordinate functions $\theta_i$, the “fermionic dimensions”:

$\mathcal{O}(U) = C^{\infty}(U) \otimes \bigwedge^{\ast}[\theta_1,\dots,\theta_{\delta}]$

Then a supersymmetric TFT is a functor:

$E : (d|\delta)-Bord \rightarrow STV$

(where $STV$ 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 $0|1$-dimensional extended TFT (that is, with 0 bosonic and 1 fermionic dimension), and $K$-theory to a $1|1$-dimensional extended TFT.  The Stoltz-Teichner Conjecture is that the third example (topological modular forms) is related in the same way to a $2_1$-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 $L(X)$ shifts things by one categorical level: its points are loops in $X$, and its paths are therefore certain 2-morphisms of $X$.  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 $X$.   The general notion of n-plectic geometry, where the usual symplectic geometry is the case $n=1$, involves a $(n+1)$-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 $G$ 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!

So I’ve been travelling a lot in the last month, spending more than half of it outside Portugal. I was in Ottawa, Canada for a Fields Institute workshop, “Categorical Methods in Representation Theory“. Then a little later I was in Erlangen, Germany for one called “Categorical and Representation-Theoretic Methods in Quantum Geometry and CFT“. Despite the similar-sounding titles, these were on fairly different themes, though Marco Mackaay was at both, talking about categorifying the $q$-Schur algebra by diagrams.  I’ll describe the meetings, but for now I’ll start with the first.  Next post will be a summary of the second.

The Ottawa meeting was organized by Alistair Savage, and Alex Hoffnung (like me, a former student of John Baez). Alistair gave a talk here at IST over the summer about a $q$-deformation of Khovanov’s categorification of the Heisenberg Algebra I discussed in an earlier entry. A lot of the discussion at the workshop was based on the Khovanov-Lauda program, which began with categorifying quantum version of the classical Lie groups, and is now making lots of progress in the categorification of algebras, representation theory, and so on.

The point of this program is to describe “categorifications” of particular algebras. This means finding monoidal categories with the property that when you take the Grothendieck ring (the ring of isomorphism classes, with a multiplication given by the monoidal structure), you get back the integral form of some algebra. (And then recover the original by taking the tensor over $\mathbb{Z}$ with $\mathbb{C}$). The key thing is how to represent the algebra by generators and relations. Since free monoidal categories with various sorts of structures can be presented as categories of string diagrams, it shouldn’t be surprising that the categories used tend to have objects that are sequences (i.e. monoidal products) of dots with various sorts of labelling data, and morphisms which are string diagrams that carry those labels on strands (actually, usually they’re linear combinations of such diagrams, so everything is enriched in vector spaces). Then one imposes relations on the “free” data given this way, by saying that the diagrams are considered the same morphism if they agree up to some local moves. The whole problem then is to find the right generators (labelling data) and relations (local moves). The result will be a categorification of a given presentation of the algebra you want.

So for instance, I was interested in Sabin Cautis and Anthony Licata‘s talks connected with this paper, “Heisenberg Categorification And Hilbert Schemes”. This is connected with a generalization of Khovanov’s categorification linked above, to include a variety of other algebras which are given a similar name. The point is that there’s such a “Heisenberg algebra” associated to different subgroups $\Gamma \subset SL(2,\mathbf{k})$, which in turn are classified by Dynkin diagrams. The vertices of these Dynkin diagrams correspond to some generators of the Heisenberg algebra, and one can modify Khovanov’s categorification by having strands in the diagram calculus be labelled by these vertices. Rules for local moves involving strands with different labels will be governed by the edges of the Dynkin diagram. Their paper goes on to describe how to represent these categorifications on certain categories of Hilbert schemes.

Along the same lines, Aaron Lauda gave a talk on the categorification of the NilHecke algebra. This is defined as a subalgebra of endomorphisms of $P_a = \mathbb{Z}[x_1,\dots,x_a]$, generated by multiplications (by the $x_i$) and the divided difference operators $\partial_i$. There are different from the usual derivative operators: in place of the differences between values of a single variable, they measure how a function behaves under the operation $s_i$ which switches variables $x_i$ and $x_{i+1}$ (that is, the reflection in the hyperplane where $x_i = x_{i+1}$). The point is that just like differentiation, this operator – together with multiplication – generates an algebra in $End(\mathbb{Z}[x_1,\dots,x_a]$. Aaron described how to categorify this presentation of the NilHecke algebra with a string-diagram calculus.

So anyway, there were a number of talks about the explosion of work within this general program – for instance, Marco Mackaay’s which I mentioned, as well as that of Pedro Vaz about the same project. One aspect of this program is that the relatively free “string diagram categories” are sometimes replaced with categories where the objects are bimodules and morphisms are bimodule homomorphisms. Making the relationship precise is then a matter of proving these satisfy exactly the relations on a “free” category which one wants, but sometimes they’re a good setting to prove one has a nice categorification. Thus, Ben Elias and Geordie Williamson gave two parts of one talk about “Soergel Bimodules and Kazhdan-Lusztig Theory” (see a blog post by Ben Webster which gives a brief intro to this notion, including pointing out that Soergel bimodules give a categorification of the Hecke algebra).

One of the reasons for doing this sort of thing is that one gets invariants for manifolds from algebras – in particular, things like the Jones polynomial, which is related to the Temperley-Lieb algebra. A categorification of it is Khovanov homology (which gives, for a manifold, a complex, with the property that the graded Euler characteristic of the complex is the Jones polynomial). The point here is that categorifying the algebra lets you raise the dimension of the kind of manifold your invariants are defined on.

So, for instance, Scott Morrison described “Invariants of 4-Manifolds from Khonanov Homology“.  This was based on a generalization of the relationship between TQFT’s and planar algebras.  The point is, planar algebras are described by the composition of diagrams of the following form: a big circle, containing some number of small circles.  The boundaries of each circle are labelled by some number of marked points, and the space between carries curves which connect these marked points in some way.  One composes these diagrams by gluing big circles into smaller circles (there’s some further discussion here including a picture, and much more in this book here).  Scott Morrison described these diagrams as “spaghetti and meatball” diagrams.  Planar algebras show up by associating a vector spaces to “the” circle with $n$ marked points, and linear maps to each way (up to isotopy) of filling in edges between such circles.  One can think of the circles and marked-disks as a marked-cobordism category, and so a functorial way of making these assignments is something like a TQFT.  It also gives lots of vector spaces and lots of linear maps that fit together in a particular way described by this category of marked cobordisms, which is what a “planar algebra” actually consists of.  Clearly, these planar algebras can be used to get some manifold invariants – namely the “TQFT” that corresponds to them.

Scott Morrison’s talk described how to get invariants of 4-dimensional manifolds in a similar way by boosting (almost) everything in this story by 2 dimensions.  You start with a 4-ball, whose boundary is a 3-sphere, and excise some number of 4-balls (with 3-sphere boundaries) from the interior.  Then let these 3D boundaries be “marked” with 1-D embedded links (think “knots” if you like).  These 3-spheres with embedded links are the objects in a category.  The morphisms are 4-balls which connect them, containing 2D knotted surfaces which happen to intersect the boundaries exactly at their embedded links.  By analogy with the image of “spaghetti and meatballs”, where the spaghetti is a collection of 1D marked curves, Morrison calls these 4-manifolds with embedded 2D surfaces “lasagna diagrams” (which generalizes to the less evocative case of “$(n,k)$ pasta diagrams”, where we’ve just mentioned the $(2,1)$ and $(4,2)$ cases, with $k$-dimensional “pasta” embedded in $n$-dimensional balls).  Then the point is that one can compose these pasta diagrams by gluing the 4-balls along these marked boundaries.  One then gets manifold invariants from these sorts of diagrams by using Khovanov homology, which assigns to

Ben Webster talked about categorification of Lie algebra representations, in a talk called “Categorification, Lie Algebras and Topology“. This is also part of categorifying manifold invariants, since the Reshitikhin-Turaev Invariants are based on some monoidal category, which in this case is the category of representations of some algebra.  Categorifying this to a 2-category gives higher-dimensional equivalents of the RT invariants.  The idea (which you can check out in those slides) is that this comes down to describing the analog of the “highest-weight” representations for some Lie algebra you’ve already categorified.

The Lie theory point here, you might remember, is that representations of Lie algebras $\mathfrak{g}$ can be analyzed by decomposing them into “weight spaces” $V_{\lambda}$, associated to weights $\lambda : \mathfrak{g} \rightarrow \mathbf{k}$ (where $\mathbf{k}$ is the base field, which we can generally assume is $\mathbb{C}$).  Weights turn Lie algebra elements into scalars, then.  So weight spaces generalize eigenspaces, in that acting by any element $g \in \mathfrak{g}$ on a “weight vector” $v \in V_{\lambda}$ amounts to multiplying by $\lambda{g}$.  (So that $v$ is an eigenvector for each $g$, but the eigenvalue depends on $g$, and is given by the weight.)  A weight can be the “highest” with respect to a natural order that can be put on weights ($\lambda \geq \mu$ if the difference is a nonnegative combination of simple weights).  Then a “highest weight representation” is one which is generated under the action of $\mathfrak{g}$ by a single weight vector $v$, the “highest weight vector”.

The point of the categorification is to describe the representation in the same terms.  First, we introduce a special strand (which Ben Webster draws as a red strand) which represents the highest weight vector.  Then we say that the category that stands in for the highest weight representation is just what we get by starting with this red strand, and putting all the various string diagrams of the categorification of $\mathfrak{g}$ next to it.  One can then go on to talk about tensor products of these representations, where objects are found by amalgamating several such diagrams (with several red strands) together.  And so on.  These categorified representations are then supposed to be usable to give higher-dimensional manifold invariants.

Now, the flip side of higher categories that reproduce ordinary representation theory would be the representation theory of higher categories in their natural habitat, so to speak. Presumably there should be a fairly uniform picture where categorifications of normal representation theory will be special cases of this. Vlodymyr Mazorchuk gave an interesting talk called 2-representations of finitary 2-categories.  He gave an example of one of the 2-categories that shows up a lot in these Khovanov-Lauda categorifications, the 2-category of Soergel Bimodules mentioned above.  This has one object, which we can think of as a category of modules over the algebra $\mathbb{C}[x_1, \dots, x_n]/I$ (where I  is some ideal of homogeneous symmetric polynomials).  The morphisms are endofunctors of this category, which all amount to tensoring with certain bimodules – the irreducible ones being the Soergel bimodules.  The point of the talk was to explain the representations of 2-categories $\mathcal{C}$ – that is, 2-functors from $\mathcal{C}$ into some “classical” 2-category.  Examples would be 2-categories like “2-vector spaces”, or variants on it.  The examples he gave: (1) [small fully additive $\mathbf{k}$-linear categories], (2) the full subcategory of it with finitely many indecomposible elements, (3) [categories equivalent to module categories of finite dimensional associative $\mathbf{k}$-algebras].  All of these have some claim to be a 2-categorical analog of [vector spaces].  In general, Mazorchuk allowed representations of “FIAT” categories: Finitary (Two-)categories with Involutions and Adjunctions.

Part of the process involved getting a “multisemigroup” from such categories: a set $S$ with an operation which takes pairs of elements, and returns a subset of $S$, satisfying some natural associativity condition.  (Semigroups are the case where the subset contains just one element – groups are the case where furthermore the operation is invertible).  The idea is that FIAT categories have some set of generators – indecomposable 1-morphisms – and that the multisemigroup describes which indecomposables show up in a composite.  (If we think of the 2-category as a monoidal category, this is like talking about a decomposition of a tensor product of objects).  So, for instance, for the 2-category that comes from the monoidal category of $\mathfrak{sl}(2)$ modules, we get the semigroup of nonnegative integers.  For the Soergel bimodule 2-category, we get the symmetric group.  This sort of thing helps characterize when two objects are equivalent, and in turn helps describe 2-representations up to some equivalence.  (You can find much more detail behind the link above.)

On the more classical representation-theoretic side of things, Joel Kamnitzer gave a talk called “Spiders and Buildings”, which was concerned with some geometric and combinatorial constructions in representation theory.  These involved certain trivalent planar graphs, called “webs”, whose edges carry labels between 1 and $(n-1)$.  They’re embedded in a disk, and the outgoing edges, with labels $(k_1, \dots, k_m)$ determine a representation space for a group $G$, say $G = SL_n$, namely the tensor product of a bunch of wedge products, $\otimes_j \wedge^{k_j} \mathbb{C}^n$, where $SL_n$ acts on $\mathbb{C}^n$ as usual.  Then a web determines an invariant vector in this space.  This comes about by having invariant vectors for each vertex (the basic case where $m =3$), and tensoring them together.  But the point is to interpret this construction geometrically.  This was a bit outside my grasp, since it involves the Langlands program and the geometric Satake correspondence, neither of which I know much of anything about, but which give geometric/topological ways of constructing representation categories.  One thing I did pick up is that it uses the “Langlands dual group” $\check{G}$ of $G$ to get a certain metric space called $Gn_{\check{G}}$.  Then there’s a correspondence between the category of representations of $G$ and the category of (perverse, constructible) sheaves on this space.  This correspondence can be used to describe the vectors that come out of these webs.

Jim Dolan gave a couple of talks while I was there, which actually fit together as two parts of a bigger picture – one was during the workshop itself, and one at the logic seminar on the following Monday. It helped a lot to see both in order to appreciate the overall point, so I’ll mix them a bit indiscriminately. The first was called “Dimensional Analysis is Algebraic Geometry”, and the second “Toposes of Quasicoherent Sheaves on Toric Varieties”. For the purposes of the logic seminar, he gave the slogan of the second talk as “Algebraic Geometry is a branch of Categorical Logic”. Jim’s basic idea was inspired by Bill Lawvere’s concept of a “theory”, which is supposed to extend both “algebraic theories” (such as the “theory of groups”) and theories in the sense of physics.  Any given theory is some structured category, and “models” of the theory are functors into some other category to represent it – it thus has a functor category called its “moduli stack of models”.  A physical theory (essentially, models which depict some contents of the universe) has some parameters.  The “theory of elastic scattering”, for instance, has the masses, and initial and final momenta, of two objects which collide and “scatter” off each other.  The moduli space for this theory amounts to assignments of values to these parameters, which must satisfy some algebraic equations – conservation of energy and momentum (for example, $\sum_i m_i v_i^{in} = \sum_i m_i v_i^{out}$, where $i \in 1, 2$).  So the moduli space is some projective algebraic variety.  Jim explained how “dimensional analysis” in physics is the study of line bundles over such varieties (“dimensions” are just such line bundles, since a “dimension” is a 1-dimensional sort of thing, and “quantities” in those dimensions are sections of the line bundles).  Then there’s a category of such bundles, which are organized into a special sort of symmetric monoidal category – in fact, it’s contrained so much it’s just a graded commutative algebra.

In his second talk, he generalized this to talk about categories of sheaves on some varieties – and, since he was talking in the categorical logic seminar, he proposed a point of view for looking at algebraic geometry in the context of logic.  This view could be summarized as: Every (generalized) space studied by algebraic geometry “is” the moduli space of models for some theory in some doctrine.  The term “doctrine” is Bill Lawvere’s, and specifies what kind of structured category the theory and the target of its models are supposed to be (and of course what kind of functors are allowed as models).  Thus, for instance, toposes (as generalized spaces) are supposed to be thought of as “geometric theories”.  He explained that his “dimensional analysis doctrine” is a special case of this.  As usual when talking to Jim, I came away with the sense that there’s a very large program of ideas lurking behind everything he said, of which only the tip of the iceberg actually made it into the talks.

Next post, when I have time, will talk about the meeting at Erlangen…

One talk at the workshop was nominally a school talk by Laurent Freidel, but it’s interesting and distinctive enough in its own right that I wanted to consider it by itself.  It was based on this paper on the “Principle of Relative Locality”. This isn’t so much a new theory, as an exposition of what ought to happen when one looks at a particular limit of any putative theory that has both quantum field theory and gravity as (different) limits of it. This leads through some ideas, such as curved momentum space, which have been kicking around for a while. The end result is a way of accounting for apparently non-local interactions of particles, by saying that while the particles themselves “see” the interactions as local, distant observers might not.

Whereas Einstein’s gravity describes a regime where Newton’s gravitational constant $G_N$ is important but Planck’s constant $\hbar$ is negligible, and (special-relativistic) quantum field theory assumes $\hbar$ significant but $G_N$ not.  Both of these assume there is a special velocity scale, given by the speed of light $c$, whereas classical mechanics assumes that all three can be neglected (i.e. $G_N$ and $\hbar$ are zero, and $c$ is infinite).   The guiding assumption is that these are all approximations to some more fundamental theory, called “quantum gravity” just because it accepts that both $G_N$ and $\hbar$ (as well as $c$) are significant in calculating physical effects.  So GR and QFT incorporate two of the three constants each, and classical mechanics incorporates neither.  The “principle of relative locality” arises when we consider a slightly different approximation to this underlying theory.

This approximation works with a regime where $G_N$ and $\hbar$ are each negligible, but the ratio is not – this being related to the Planck mass $m_p \sim \sqrt{\frac{\hbar}{G_N}}$.  The point is that this is an approximation with no special length scale (“Planck length”), but instead a special energy scale (“Planck mass”) which has to be preserved.   Since energy and momentum are different parts of a single 4-vector, this is also a momentum scale; we expect to see some kind of deformation of momentum space, at least for momenta that are bigger than this scale.  The existence of this scale turns out to mean that momenta don’t add linearly – at least, not unless they’re very small compared to the Planck scale.

So what is “Relative Locality”?  In the paper linked above, it’s stated like so:

Physics takes place in phase space and there is no invariant global projection that gives a description of processes in spacetime.  From their measurements local observers can construct descriptions of particles moving and interacting in a spacetime, but different observers construct different spacetimes, which are observer-dependent slices of phase space.

Motivation

This arises from taking the basic insight of general relativity – the requirement that physical principles should be invariant under coordinate transformations (i.e. diffeomorphisms) – and extend it so that instead of applying just to spacetime, it applies to the whole of phase space.  Phase space (which, in this limit where $\hbar = 0$, replaces the Hilbert space of a truly quantum theory) is the space of position-momentum configurations (of things small enough to treat as point-like, in a given fixed approximation).  Having no $G_N$ means we don’t need to worry about any dynamical curvature of “spacetime” (which doesn’t exist), and having no Planck length means we can blithely treat phase space as a manifold with coordinates valued in the real line (which has no special scale).  Yet, having a special mass/momentum scale says we should see some purely combined “quantum gravity” effects show up.

The physical idea is that phase space is an accurate description of what we can see and measure locally.  Observers (whom we assume small enough to be considered point-like) can measure their own proper time (they “have a clock”) and can detect momenta (by letting things collide with them and measuring the energy transferred locally and its direction).  That is, we “see colors and angles” (i.e. photon energies and differences of direction).  Beyond this, one shouldn’t impose any particular theory of what momenta do: we can observe the momenta of separate objects and see what results when they interact and deduce rules from that.  As an extension of standard physics, this model is pretty conservative.  Now, conventionally, phase space would be the cotangent bundle of spacetime $T^*M$.  This model is based on the assumption that objects can be at any point, and wherever they are, their space of possible momenta is a vector space.  Being a bundle, with a global projection onto $M$ (taking $(x,v)$ to $x$), is exactly what this principle says doesn’t necessarily obtain.  We still assume that phase space will be some symplectic manifold.   But we don’t assume a priori that momentum coordinates give a projection whose fibres happen to be vector spaces, as in a cotangent bundle.

Now, a symplectic manifold  still looks locally like a cotangent bundle (Darboux’s theorem). So even if there is no universal “spacetime”, each observer can still locally construct a version of “spacetime”  by slicing up phase space into position and momentum coordinates.  One can, by brute force, extend the spacetime coordinates quite far, to distant points in phase space.  This is roughly analogous to how, in special relativity, each observer can put their own coordinates on spacetime and arrive at different notions of simultaneity.  In general relativity, there are issues with trying to extend this concept globally, but it can be done under some conditions, giving the idea of “space-like slices” of spacetime.  In the same way, we can construct “spacetime-like slices” of phase space.

Geometrizing Algebra

Now, if phase space is a cotangent bundle, momenta can be added (the fibres of the bundle are vector spaces).  Some more recent ideas about “quasi-Hamiltonian spaces” (initially introduced by Alekseev, Malkin and Meinrenken) conceive of momenta as “group-valued” – rather than taking values in the dual of some Lie algebra (the way, classically, momenta are dual to velocities, which live in the Lie algebra of infinitesimal translations).  For small momenta, these are hard to distinguish, so even group-valued momenta might look linear, but the premise is that we ought to discover this by experiment, not assumption.  We certainly can detect “zero momentum” and for physical reasons can say that given two things with two momenta $(p,q)$, there’s a way of combining them into a combined momentum $p \oplus q$.  Think of doing this physically – transfer all momentum from one particle to another, as seen by a given observer.  Since the same momentum at the observer’s position can be either coming in or going out, this operation has a “negative” with $(\ominus p) \oplus p = 0$.

We do have a space of momenta at any given observer’s location – the total of all momenta that can be observed there, and this space now has some algebraic structure.  But we have no reason to assume up front that $\oplus$ is either commutative or associative (let alone that it makes momentum space at a given observer’s location into a vector space).  One can interpret this algebraic structure as giving some geometry.  The commutator for $\oplus$ gives a metric on momentum space.  This is a bilinear form which is implicitly defined by the “norm” that assigns a kinetic energy to a particle with a given momentum. The associator given by $p \oplus ( q \oplus r ) - (p \oplus q ) \oplus r)$, infinitesimally near $0$ where this makes sense, gives a connection.  This defines a “parallel transport” of a finite momentum $p$ in the direction of a momentum $q$ by saying infinitesimally what happens when adding $dq$ to $p$.

Various additional physical assumptions – like the momentum-space “duals” of the equivalence principle (that the combination of momenta works the same way for all kinds of matter regardless of charge), or the strong equivalence principle (that inertial mass and rest mass energy per the relation $E = mc^2$ are the same) and so forth can narrow down the geometry of this metric and connection.  Typically we’ll find that it needs to be Lorentzian.  With strong enough symmetry assumptions, it must be flat, so that momentum space is a vector space after all – but even with fairly strong assumptions, as with general relativity, there’s still room for this “empty space” to have some intrinsic curvature, in the form of a momentum-space “dual cosmological constant”, which can be positive (so momentum space is closed like a sphere), zero (the vector space case we usually assume) or negative (so momentum space is hyperbolic).

This geometrization of what had been algebraic is somewhat analogous to what happened with velocities (i.e. vectors in spacetime)) when the theory of special relativity came along.  Insisting that the “invariant” scale $c$ be the same in every reference system meant that the addition of velocities ceased to be linear.  At least, it did if you assume that adding velocities has an interpretation along the lines of: “first, from rest, add velocity v to your motion; then, from that reference frame, add velocity w”.  While adding spacetime vectors still worked the same way, one had to rephrase this rule if we think of adding velocities as observed within a given reference frame – this became $v \oplus w = (v + w) (1 + uv)$ (scaling so $c =1$ and assuming the velocities are in the same direction).  When velocities are small relative to $c$, this looks roughly like linear addition.  Geometrizing the algebra of momentum space is thought of a little differently, but similar things can be said: we think operationally in terms of combining momenta by some process.  First transfer (group-valued) momentum $p$ to a particle, then momentum $q$ – the connection on momentum space tells us how to translate these momenta into the “reference frame” of a new observer with momentum shifted relative to the starting point.  Here again, the special momentum scale $m_p$ (which is also a mass scale since a momentum has a corresponding kinetic energy) is a “deformation” parameter – for momenta that are small compared to this scale, things seem to work linearly as usual.

There’s some discussion in the paper which relates this to DSR (either “doubly” or “deformed” special relativity), which is another postulated limit of quantum gravity, a variation of SR with both a special velocity and a special mass/momentum scale, to consider “what SR looks like near the Planck scale”, which treats spacetime as a noncommutative space, and generalizes the Lorentz group to a Hopf algebra which is a deformation of it.  In DSR, the noncommutativity of “position space” is directly related to curvature of momentum space.  In the “relative locality” view, we accept a classical phase space, but not a classical spacetime within it.

Physical Implications

We should understand this scale as telling us where “quantum gravity effects” should start to become visible in particle interactions.  This is a fairly large scale for subatomic particles.  The Planck mass as usually given is about 21 micrograms: small for normal purposes, about the size of a small sand grain, but very large for subatomic particles.  Converting to momentum units with $c$, this is about 6 kg m/s: on the order of the momentum of a kicked soccer ball or so.  For a subatomic particle this is a lot.

This scale does raise a question for many people who first hear this argument, though – that quantum gravity effects should become apparent around the Planck mass/momentum scale, since macro-objects like the aforementioned soccer ball still seem to have linearly-additive momenta.  Laurent explained the problem with this intuition.  For interactions of big, extended, but composite objects like soccer balls, one has to calculate not just one interaction, but all the various interactions of their parts, so the “effective” mass scale where the deformation would be seen becomes $N m_p$ where $N$ is the number of particles in the soccer ball.  Roughly, the point is that a soccer ball is not a large “thing” for these purposes, but a large conglomeration of small “things”, whose interactions are “fundamental”.  The “effective” mass scale tells us how we would have to alter the physical constants to be able to treat it as a “thing”.  (This is somewhat related to the question of “effective actions” and renormalization, though these are a bit more complicated.)

There are a number of possible experiments suggested in the paper, which Laurent mentioned in the talk.  One involves a kind of “twin paradox” taking place in momentum space.  In “spacetime”, a spaceship travelling a large loop at high velocity will arrive where it started having experienced less time than an observer who remained there (because of the Lorentzian metric) – and a dual phenomenon in momentum space says that particles travelling through loops (also in momentum space) should arrive displaced in space because of the relativity of localization.  This could be observed in particle accelerators where particles make several transits of a loop, since the effect is cumulative.  Another effect could be seen in astronomical observations: if an observer is observing some distant object via photons of different wavelengths (hence momenta), she might “localize” the object differently – that is, the two photons travel at “the same speed” the whole way, but arrive at different times because the observer will interpret the object as being at two different distances for the two photons.

This last one is rather weird, and I had to ask how one would distinguish this effect from a variable speed of light (predicted by certain other ideas about quantum gravity).  How to distinguish such effects seems to be not quite worked out yet, but at least this is an indication that there are new, experimentally detectible, effects predicted by this “relative locality” principle.  As Laurent emphasized, once we’ve noticed that not accepting this principle means making an a priori assumption about the geometry of momentum space (even if only in some particular approximation, or limit, of a true theory of quantum gravity), we’re pretty much obliged to stop making that assumption and do the experiments.  Finding our assumptions were right would simply be revealing which momentum space geometry actually obtains in the approximation we’re studying.

A final note about the physical interpretation: this “relative locality” principle can be discovered by looking (in the relevant limit) at a Lagrangian for free particles, with interactions described in terms of momenta.  It so happens that one can describe this without referencing a “real” spacetime: the part of the action that allows particles to interact when “close” only needs coordinate functions, which can certainly exist here, but are an observer-dependent construct.  The conservation of (non-linear) momenta is specified via a Lagrange multiplier.  The whole Lagrangian formalism for the mechanics of colliding particles works without reference to spacetime.  Now, even though all the interactions (specified by the conservation of momentum terms) happen “at one location”, in that there will be an observer who sees them happening in the momentum space of her own location.  But an observer at a different point may disagree about whether the interaction was local – i.e. happened at a single point in spacetime.  Thus “relativity of localization”.

Again, this is no more bizarre (mathematically) than the fact that distant, relatively moving, observers in special relativity might disagree about simultaneity, whether two events happened at the same time.  They have their own coordinates on spacetime, and transferring between them mixes space coordinates and time coordinates, so they’ll disagree whether the time-coordinate values of two events are the same.  Similarly, in this phase-space picture, two different observers each have a coordinate system for splitting phase space into “spacetime” and “energy-momentum” coordinates, but switching between them may mix these two pieces.  Thus, the two observers will disagree about whether the spacetime-coordinate values for the different interacting particles are the same.  And so, one observer says the interaction is “local in spacetime”, and the other says it’s not.  The point is that it’s local for the particles themselves (thinking of them as observers).  All that’s going on here is the not-very-astonishing fact that in the conventional picture, we have no problem with interactions being nonlocal in momentum space (particles with very different momenta can interact as long as they collide with each other)… combined with the inability to globally and invariantly distinguish position and momentum coordinates.

What this means, philosophically, can be debated, but it does offer some plausibility to the claim that space and time are auxiliary, conceptual additions to what we actually experience, which just account for the relations between bits of matter.  These concepts can be dispensed with even where we have a classical-looking phase space rather than Hilbert space (where, presumably, this is even more true).

Edit: On a totally unrelated note, I just noticed this post by Alex Hoffnung over at the n-Category Cafe which gives a lot of detail on issues relating to spans in bicategories that I had begun to think more about recently in relation to developing a higher-gauge-theoretic version of the construction I described for ETQFT. In particular, I’d been thinking about how the 2-group analog of restriction and induction for representations realizes the various kinds of duality properties, where we have adjunctions, biadjunctions, and so forth, in which units and counits of the various adjunctions have further duality. This observation seems to be due to Jim Dolan, as far as I can see from a brief note in HDA II. In that case, it’s really talking about the star-structure of the span (tri)category, but looking at the discussion Alex gives suggests to me that this theme shows up throughout this subject. I’ll have to take a closer look at the draft paper he linked to and see if there’s more to say…

So there’s a lot of preparations going on for the workshop HGTQGR coming up next week at IST, and the program(me) is much more developed – many of the talks are now listed, though the schedule has yet to be finalized.  This week we’ll be having a “pre-school school” to introduce the local mathematicans to some of the physics viewpoints that will be discussed at the workshop – Aleksandar Mikovic will be introducing Quantum Gravity (from the point of view of the loop/spin-foam approach), and Sebastian Guttenberg will be giving a mathematician’s introduction to String theory.

These are by no means the only approaches physicists have taken to the problem of finding a theory that incorporates both General Relativity and Quantum Field Theory.  They are, however, two approaches where lots of work has been done, and which appear to be amenable to using the mathematical tools of (higher) category theory which we’re going to be talking about at the workshop.  These are “higher gauge theory”, which very roughly is the analog of gauge theory (which includes both GR and QFT) using categorical groups, and TQFT, which is a very simple type of quantum field theory that has a natural description in terms of categories, which can be generalized to higher categories.

I’ll probably take a few posts after the workshop to write up these, and the many other talks and mini-courses we’ll be having, but right now, I’d like to say a little bit about another talk we had here recently.  Actually, the talk was in Porto, but several of us at IST in Lisbon attended by a videoconference.  This was the first time I’ve seen this for a colloquium-style talk, though I did once take a course in General Relativity from Eric Poisson that was split between U of Waterloo and U of Guelph.  I thought it was a great idea then, and it worked quite well this time, too.  This is the way of the future – and unfortunately it probably will be for some time to come…

Anyway, the talk in question was by Thomasz Brzezinski, about “Synthetic Non-Commutative Geometry” (link points to the slides).  The point here is to take two different approaches to extending differential geometry (DG) and combine the two insights.  The “Synthetic” part refers to synthetic differential geometry (SDG), which is a program for doing DG in a general topos.  One aspect of this is that in a topos where the Law of the Excluded Middle doesn’t apply, it’s possible for the real-numbers object to have infinitesimals: that is, elements which are smaller than any positive element, but bigger than zero.  This lets one take things which have to be treated in a roundabout way in ordinary DG, like $dx$, and take them at face value – as an infinitesimal change in $x$.  It also means doing geometry in a completely constructive way.

However, these aspects aren’t so important here.  The important fact about it here is that it’s based on building a theory that was originally defined in terms of sets, or topological spaces – that is, in the toposes $Sets$, or $Top$  – and transplanting it to another category.  This is because Brzezinski’s goal was to do something similar for a different extension of DG, namely non-commutative geometry (NCG).  This is a generalisation of DG which is based on the equivalence $CommAlg^{op} \simeq lCptHaus$ between the categories of commutative $C^{\star}$-algebras (and algebra maps, read “backward” as morphisms in $CommAlg^{op}$), and that of locally compact Hausdorff spaces (which, for objects, equates a space $X$ with the algebra $C(X)$ of continuous functions on it, and an algebra $A$ with its spectrum $Spec(A)$, the space of maximal ideals).  The generalization of NCG is to take structures defined for $lCptHaus$ that create DG, and make similar definitions in the category $Alg^{op}$, of not-necessarily-commutative $C^{\star}$-algebras.

This category is the one which plays the role of the topos $Top$.  It isn’t a topos, though: it’s some sort of monoidal category.  And this is what “synthetic NCG” is about: taking the definitions used in NCG and reproducing them in a generic monoidal category (to be clear, a braided monoidal category).

The way he illustrated this is by explaining what a principal bundle would be in such a generic category.

To begin with, we can start by giving a slightly nonstandard definition of the concept in ordinary DG: a principal $G$-bundle $P$ is a manifold with a free action of a (compact Lie) group $G$ on it.  The point is that this always looks like a “base space” manifold $B$, with a projection $\pi : P \rightarrow B$ so that the fibre at each point of $B$ looks like $G$.  This amounts to saying that $\pi$ is an equalizer:

$P \times G \stackrel{\longrightarrow}{\rightarrow} P \stackrel{\pi}{\rightarrow} B$

where the maps from $G\times P$ to $P$ are (a) the action, and (b) the projection onto $P$.  (Being an equalizer means that $\pi$ makes this diagram commute – has the same composite with both maps – and any other map $\phi$ that does the same factors uniquely through $\pi$.)  Another equivalent way to say this is that since $P \times G$ has two maps into $P$, then it has a map into the pullback $P \times_B P$ (the pullback of two copies of $P \stackrel{\pi}{\rightarrow} B$), and the claim is that it’s actually ismorphic.

The main points here are that (a) we take this definition in terms of diagrams and abstract it out of the category $Top$, and (b) when we do so, in general the products will be tensor products.

In particular, this means we need to have a general definition of a group object $G$ in any braided monoidal category (to know what $G$ is supposed to be like).  We reproduce the usual definition of a group objects so that $G$ must come equipped with a “multiplication” map $m : G \otimes G \rightarrow G$, an “inverse” map $\iota : G \rightarrow G$ and a “unit” map $u : I \rightarrow G$, where $I$ is the monoidal unit (which takes the role of the terminal object in a topos like $Top$, the unit for $\times$).  These need to satisfy the usual properties, such as the monoid property for multiplication:

$m \circ (m \otimes id_G) = m \circ (id_G \otimes m) : G \otimes G \otimes G \rightarrow G$

(usually given as a diagram, but I’m being lazy).

The big “however” is this: in $Sets$ or $Top$, any object $X$ is always a comonoid in a canonical way, and we use this implictly in defining some of the properties we need.  In particular, there’s always the diagonal map $\Delta : X \rightarrow X \times X$ which satisfies the dual of the monoid property:

$(id_X \times \Delta) \circ \Delta = (\Delta \times id_X) \circ \Delta$

There’s also a unique counit $\epsilon \rightarrow \star$, the map into the terminal object, which makes $(X,\Delta,\epsilon)$ a counital comonoid automatically.  But in a general braided monoidal category, we have to impose as a condition that our group object also be equipped with $\Delta : G \rightarrow G \otimes G$ and $\epsilon : G \rightarrow I$ making it a counital comonoid.  We need this property to even be able to make sense of the inverse axiom (which this time I’ll do as a diagram):

This diagram uses not only $\Delta$ but also the braiding map $\sigma_{G,G} : G \otimes G \rightarrow G \otimes G$ (part of the structure of the braided monoidal category which, in $Top$ or $Sets$ is just the “switch” map).  Now, in fact, since any object in $Set$ or $Top$ is automatically a comonoid, we’ll require that this structure be given for anything we look at: the analog of spaces (like $P$ above), or our group object $G$.  For the group object, we also must, in general, require something which comes for free in the topos world and therefore generally isn’t mentioned in the definition of a group.  Namely, the comonoid and monoid aspects of $G$ must get along.  (This comes for free in a topos essentially because the comonoid structure is given canonically for all objects.)  This means:

For a group in $Sets$ or $Top$, this essentially just says that the two ways we can go from $(x,y)$ to $(xy,xy)$ (duplicate, swap, then multiply, or on the other hand multiply then duplicate) are the same.

All these considerations about how honest-to-goodness groups are secretly also comonoids does explain why corresponding structures in noncommutative geometry seem to have more elaborate definitions: they have to explicitly say things that come for free in a topos.  So, for instance, a group object in the above sense in the braided monoidal category $Vect = (Vect_{\mathbb{F}}, \otimes_{\mathbb{F}}, \mathbb{F}, flip)$ is a Hopf algebra.  This is a nice canonical choice of category.  Another is the opposite category $Vect^{op}$ – this is a standard choice in NCG, since spaces are supposed to be algebras – this would be given the comonoid structure we demanded.

So now once we know all this, we can reproduce the diagrammatic definition of a principal $G$-bundle above: just replace the product $\times$ with the monoidal operation $\otimes$, the terminal object by $I$, and so forth.  The diagrams are understood to be diagrams of comonoids in our braided monoidal category.  In particular, we have an action $\rho : P \otimes G \rightarrow P$,which is compatible with the $\Delta$ maps – so in $Vect$ we would say that a noncommutative principal $G$-bundle $P$ is a right-module coalgebra over the Hopf algebra $G$.  We can likewise take this (in a suitably abstract sense of “algebra” or “module”) to be the definition in any braided monoidal category.

To have the “freeness” of the action, there needs to be an equalizer of:

$\rho, (id_P \otimes \epsilon) : P \otimes G \stackrel{\longrightarrow}{\rightarrow} P \stackrel{\pi}{\rightarrow} B$

The “freeness” condition for the action is likewise defined using a monoidal-category version of the pullback (fibre product) $P \times_B P$.

This was as far as Brzezinski took the idea of synthetic NCG in this particular talk, but the basic idea seems quite nice.  In SDG, one can define all sorts of differential geometric structures synthetically, that is, for a general topos: for example, Gonzalo Reyes has gone and defined the Einstein field equations synthetically.  Presumably, a lot of what’s done in NCG could also be done in this synthetic framework, and transplanted to other categories than the usual choices.

Brzezinski said he was mainly interested in the “usual” choices of category, $Vect$ and $Vect^{op}$ – so for instance in $Vect^{op}$, a “principal $G$-bundle” is what’s called a Hopf-Galois extension.  Roger Picken did, however, ask an interesting question about other possible candidates for the category to work in.  Given that one wants a braided monoidal category, a natural one to look at is the category whose morphisms are braids.  This one, as a matter of fact, isn’t quite enough (there’s no braid $m : n \otimes n \rightarrow n$, because this would be a braid with $2n$ strands in and $n$ strands out – which is impossible.  But some sort of category of tangles might make an interestingly abstract setting in which to see what NCG looks like.  So far, this doesn’t seem to have been done as far as I can see.

Marco Mackaay recently pointed me at a paper by Mikhail Khovanov, which describes a categorification of the Heisenberg algebra $H$ (or anyway its integral form $H_{\mathbb{Z}}$) in terms of a diagrammatic calculus.  This is very much in the spirit of the Khovanov-Lauda program of categorifying Lie algebras, quantum groups, and the like.  (There’s also another one by Sabin Cautis and Anthony Licata, following up on it, which I fully intend to read but haven’t done so yet. I may post about it later.)

Now, as alluded to in some of the slides I’ve from recent talks, Jamie Vicary and I have been looking at a slightly different way to answer this question, so before I talk about the Khovanov paper, I’ll say a tiny bit about why I was interested.

Groupoidification

The Weyl algebra (or the Heisenberg algebra – the difference being whether the commutation relations that define it give real or imaginary values) is interesting for physics-related reasons, being the algebra of operators associated to the quantum harmonic oscillator.  The particular approach to categorifying it that I’ve worked with goes back to something that I wrote up here, and as far as I know, originally was suggested by Baez and Dolan here.  This categorification is based on “stuff types” (Jim Dolan’s term, based on “structure types”, a.k.a. Joyal’s “species”).  It’s an example of the groupoidification program, the point of which is to categorify parts of linear algebra using the category $Span(Gpd)$.  This has objects which are groupoids, and morphisms which are spans of groupoids: pairs of maps $G_1 \leftarrow X \rightarrow G_2$.  Since I’ve already discussed the backgroup here before (e.g. here and to a lesser extent here), and the papers I just mentioned give plenty more detail (as does “Groupoidification Made Easy“, by Baez, Hoffnung and Walker), I’ll just mention that this is actually more naturally a 2-category (maps between spans are maps $X \rightarrow X'$ making everything commute).  It’s got a monoidal structure, is additive in a fairly natural way, has duals for morphisms (by reversing the orientation of spans), and more.  Jamie Vicary and I are both interested in the quantum harmonic oscillator – he did this paper a while ago describing how to construct one in a general symmetric dagger-monoidal category.  We’ve been interested in how the stuff type picture fits into that framework, and also in trying to examine it in more detail using 2-linearization (which I explain here).

Anyway, stuff types provide a possible categorification of the Weyl/Heisenberg algebra in terms of spans and groupoids.  They aren’t the only way to approach the question, though – Khovanov’s paper gives a different (though, unsurprisingly, related) point of view.  There are some nice aspects to the groupoidification approach: for one thing, it gives a nice set of pictures for the morphisms in its categorified algebra (they look like groupoids whose objects are Feynman diagrams).  Two great features of this Khovanov-Lauda program: the diagrammatic calculus gives a great visual representation of the 2-morphisms; and by dealing with generators and relations directly, it describes, in some sense1, the universal answer to the question “What is a categorification of the algebra with these generators and relations”.  Here’s how it works…

Heisenberg Algebra

One way to represent the Weyl/Heisenberg algebra (the two terms refer to different presentations of isomorphic algebras) uses a polynomial algebra $P_n = \mathbb{C}[x_1,\dots,x_n]$.  In fact, there’s a version of this algebra for each natural number $n$ (the stuff-type references above only treat $n=1$, though extending it to “$n$-sorted stuff types” isn’t particularly hard).  In particular, it’s the algebra of operators on $P_n$ generated by the “raising” operators $a_k(p) = x_k \cdot p$ and the “lowering” operators $b_k(p) = \frac{\partial p}{\partial x_k}$.  The point is that this is characterized by some commutation relations.  For $j \neq k$, we have:

$[a_j,a_k] = [b_j,b_k] = [a_j,b_k] = 0$

but on the other hand

$[a_k,b_k] = 1$

So the algebra could be seen as just a free thing generated by symbols $\{a_j,b_k\}$ with these relations.  These can be understood to be the “raising and lowering” operators for an $n$-dimensional harmonic oscillator.  This isn’t the only presentation of this algebra.  There’s another one where $[p_k,q_k] = i$ (as in $i = \sqrt{-1}$) has a slightly different interpretation, where the $p$ and $q$ operators are the position and momentum operators for the same system.  Finally, a third one – which is the one that Khovanov actually categorifies – is skewed a bit, in that it replaces the $a_j$ with a different set of $\hat{a}_j$ so that the commutation relation actually looks like

$[\hat{a}_j,b_k] = b_{k-1}\hat{a}_{j-1}$

It’s not instantly obvious that this produces the same result – but the $\hat{a}_j$ can be rewritten in terms of the $a_j$, and they generate the same algebra.  (Note that for the one-dimensional version, these are in any case the same, taking $a_0 = b_0 = 1$.)

Diagrammatic Calculus

To categorify this, in Khovanov’s sense (though see note below1), means to find a category $\mathcal{H}$ whose isomorphism classes of objects correspond to (integer-) linear combinations of products of the generators of $H$.  Now, in the $Span(Gpd)$ setup, we can say that the groupoid $FinSet_0$, or equvialently $\mathcal{S} = \coprod_n \mathcal{S}_n$, represents Fock space.  Groupoidification turns this into the free vector space on the set of isomorphism classes of objects.  This has some extra structure which we don’t need right now, so it makes the most sense to describe it as $\mathbb{C}[[t]]$, the space of power series (where $t^n$ corresponds to the object $[n]$).  The algebra itself is an algebra of endomorphisms of this space.  It’s this algebra Khovanov is looking at, so the monoidal category in question could really be considered a bicategory with one object, where the monoidal product comes from composition, and the object stands in formally for the space it acts on.  But this space doesn’t enter into the description, so we’ll just think of $\mathcal{H}$ as a monoidal category.  We’ll build it in two steps: the first is to define a category $\mathcal{H}'$.

The objects of $\mathcal{H}'$ are defined by two generators, called $Q_+$ and $Q_-$, and the fact that it’s monoidal (these objects will be the categorifications of $a$ and $b$).  Thus, there are objects $Q_+ \otimes Q_- \otimes Q_+$ and so forth.  In general, if $\epsilon$ is some word on the alphabet $\{+,-\}$, there’s an object $Q_{\epsilon} = Q_{\epsilon_1} \otimes \dots \otimes Q_{\epsilon_m}$.

As in other categorifications in the Khovanov-Lauda vein, we define the morphisms of $\mathcal{H}'$ to be linear combinations of certain planar diagrams, modulo some local relations.  (This type of formalism comes out of knot theory – see e.g. this intro by Louis Kauffman).  In particular, we draw the objects as sequences of dots labelled $+$ or $-$, and connect two such sequences by a bunch of oriented strands (embeddings of the interval, or circle, in the plane).  Each $+$ dot is the endpoint of a strand oriented up, and each $-$ dot is the endpoint of a strand oriented down.  The local relations mean that we can take these diagrams up to isotopy (moving the strands around), as well as various other relations that define changes you can make to a diagram and still represent the same morphism.  These relations include things like:

which seems visually obvious (imagine tugging hard on the ends on the left hand side to straighten the strands), and the less-obvious:

and a bunch of others.  The main ingredients are cups, caps, and crossings, with various orientations.  Other diagrams can be made by pasting these together.  The point, then, is that any morphism is some $\mathbf{k}$-linear combination of these.  (I prefer to assume $\mathbf{k} = \mathbb{C}$ most of the time, since I’m interested in quantum mechanics, but this isn’t strictly necessary.)

The second diagram, by the way, are an important part of categorifying the commutation relations.  This would say that $Q_- \otimes Q_+ \cong Q_+ \otimes Q_- \oplus 1$ (the commutation relation has become a decomposition of a certain tensor product).  The point is that the left hand sides show the composition of two crossings $Q_- \otimes Q_+ \rightarrow Q_+ \otimes Q_-$ and $Q_+ \otimes Q_- \rightarrow Q_- \otimes Q_+$ in two different orders.  One can use this, plus isotopy, to show the decomposition.

That diagrams are invariant under isotopy means, among other things, that the yanking rule holds:

(and similar rules for up-oriented strands, and zig zags on the other side).  These conditions amount to saying that the functors $- \otimes Q_+$ and $- \otimes Q_-$ are two-sided adjoints.  The two cups and caps (with each possible orientation) give the units and counits for the two adjunctions.  So, for instance, in the zig-zag diagram above, there’s a cup which gives a unit map $\mathbf{k} \rightarrow Q_- \otimes Q_+$ (reading upward), all tensored on the right by $Q_-$.  This is followed by a cap giving a counit map $Q_+ \otimes Q_- \rightarrow \mathbf{k}$ (all tensored on the left by $Q_-$).  So the yanking rule essentially just gives one of the identities required for an adjunction.  There are four of them, so in fact there are two adjunctions: one where $Q_+$ is the left adjoint, and one where it’s the right adjoint.

Karoubi Envelope

Now, so far this has explained where a category $\mathcal{H}'$ comes from – the one with the objects $Q_{\epsilon}$ described above.  This isn’t quite enough to get a categorification of $H_{\mathbb{Z}}$: it would be enough to get the version with just one $a$ and one $b$ element, and their powers, but not all the $a_j$ and $b_k$.  To get all the elements of the (integral form of) the Heisenberg algebras, and in particular to get generators that satisfy the right commutation relations, we need to introduce some new objects.  There’s a convenient way to do this, though, which is to take the Karoubi envelope of $\mathcal{H}'$.

The Karoubi envelope of any category $\mathcal{C}$ is a universal way to find a category $Kar(\mathcal{C})$ that contains $\mathcal{C}$ and for which all idempotents split (i.e. have corresponding subobjects).  Think of vector spaces, for example: a map $p \in End(V)$ such that $p^2 = p$ is a projection.  That projection corresponds to a subspace $W \subset V$, and $W$ is actually another object in $Vect$, so that $p$ splits (factors) as $V \rightarrow W subset V$.  This might not happen in any general $\mathcal{C}$, but it will in $Kar(\mathcal{C})$.  This has, for objects, all the pairs $(C,p)$ where $p : C \rightarrow C$ is idempotent (so $\mathcal{C}$ is contained in $Kar(\mathcal{C})$ as the cases where $p=1$).  The morphisms $f : (C,p) \rightarrow (C',p')$ are just maps $f : C \rightarrow C'$ with the compatibility condition that $p' f = p f = f$ (essentially, maps between the new subobjects).

So which new subobjects are the relevant ones?  They’ll be subobjects of tensor powers of our $Q_{\pm}$.  First, consider $Q_{+^n} = Q_+^{\otimes n}$.  Obviously, there’s an action of the symmetric group $\mathcal{S}_n$ on this, so in fact (since we want a $\mathbf{k}$-linear category), its endomorphisms contain a copy of $\mathbf{k}[\mathcal{S}_n]$, the corresponding group algebra.  This has a number of different projections, but the relevant ones here are the symmetrizer,:

$e_n = \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} \sigma$

which wants to be a “projection onto the symmetric subspace” and the antisymmetrizer:

$e'_n = \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} sign(\sigma) \sigma$

which wants to be a “projection onto the antisymmetric subspace” (if it were in a category with the right sub-objects). The diagrammatic way to depict this is with horizontal bars: so the new object $S^n_+ = (Q_{+^n}, e)$ (the symmetrized subobject of $Q_+^{\oplus n}$) is a hollow rectangle, labelled by $n$.  The projection from $Q_+^{\otimes n}$ is drawn with $n$ arrows heading into that box:

The antisymmetrized subobject $\Lambda^n_+ = (Q_{+^n},e')$ is drawn with a black box instead.  There are also $S^n_-$ and $\Lambda^n_-$ defined in the same way (and drawn with downward-pointing arrows).

The basic fact – which can be shown by various diagram manipulations, is that $S^n_- \otimes \Lambda^m_+ \cong (\Lambda^m_+ \otimes S^n_-) \oplus (\Lambda_+^{m-1} \otimes S^{n-1}_-)$.  The key thing is that there are maps from the left hand side into each of the terms on the right, and the sum can be shown to be an isomorphism using all the previous relations.  The map into the second term involves a cap that uses up one of the strands from each term on the left.

There are other idempotents as well – for every partition $\lambda$ of $n$, there’s a notion of $\lambda$-symmetric things – but ultimately these boil down to symmetrizing the various parts of the partition.  The main point is that we now have objects in $\mathcal{H} = Kar(\mathcal{H}')$ corresponding to all the elements of $H_{\mathbb{Z}}$.  The right choice is that the $\hat{a}_j$  (the new generators in this presentation that came from the lowering operators) correspond to the $S^j_-$ (symmetrized products of “lowering” strands), and the $b_k$ correspond to the $\Lambda^k_+$ (antisymmetrized products of “raising” strands).  We also have isomorphisms (i.e. diagrams that are invertible, using the local moves we’re allowed) for all the relations.  This is a categorification of $H_{\mathbb{Z}}$.

Some Generalities

This diagrammatic calculus is universal enough to be applied to all sorts of settings where there are functors which are two-sided adjoints of one another (by labelling strands with functors, and the regions of the plane with categories they go between).  I like this a lot, since biadjointness of certain functors is essential to the 2-linearization functor $\Lambda$ (see my link above).  In particular, $\Lambda$ uses biadjointness of restriction and induction functors between representation categories of groupoids associated to a groupoid homomorphism (and uses these unit and counit maps to deal with 2-morphisms).  That example comes from the fact that a (finite-dimensional) representation of a finite group(oid) is a functor into $Vect$, and a group(oid) homomorphism is also just a functor $F : H \rightarrow G$.  Given such an $F$, there’s an easy “restriction” $F^* : Fun(G,Vect) \rightarrow Fun(H,Vect)$, that just works by composing with $F$.  Then in principle there might be two different adjoints $Fun(H,Vect) \rightarrow Fun(G,Vect)$, given by the left and right Kan extension along $F$.  But these are defined by colimits and limits, which are the same for (finite-dimensional) vector spaces.  So in fact the adjoint is two-sided.

Khovanov’s paper describes and uses exactly this example of biadjointness in a very nice way, albeit in the classical case where we’re just talking about inclusions of finite groups.  That is, given a subgroup $H < G$, we get a functors $Res_G^H : Rep(G) \rightarrow Rep(H)$, which just considers the obvious action of $H$ act on any representation space of $G$.  It has a biadjoint $Ind^G_H : Rep(H) \rightarrow Rep(G)$, which takes a representation $V$ of $H$ to $\mathbf{k}[G] \otimes_{\mathbf{k}[H]} V$, which is a special case of the formula for a Kan extension.  (This formula suggests why it’s also natural to see these as functors between module categories $\mathbf{k}[G]-mod$ and $\mathbf{k}[H]-mod$).  To talk about the Heisenberg algebra in particular, Khovanov considers these functors for all the symmetric group inclusions $\mathcal{S}_n < \mathcal{S}_{n+1}$.

Except for having to break apart the symmetric groupoid as $S = \coprod_n \mathcal{S}_n$, this is all you need to categorify the Heisenberg algebra.  In the $Span(Gpd)$ categorification, we pick out the interesting operators as those generated by the $- \sqcup \{\star\}$ map from $FinSet_0$ to itself, but “really” (i.e. up to equivalence) this is just all the inclusions $\mathcal{S}_n < \mathcal{S}_{n+1}$ taken at once.  However, Khovanov’s approach is nice, because it separates out a lot of what’s going on abstractly and uses a general diagrammatic way to depict all these 2-morphisms (this is explained in the first few pages of Aaron Lauda’s paper on ambidextrous adjoints, too).  The case of restriction and induction is just one example where this calculus applies.

There’s a fair bit more in the paper, but this is probably sufficient to say here.

1 There are two distinct but related senses of “categorification” of an algebra $A$ here, by the way.  To simplify the point, say we’re talking about a ring $R$.  The first sense of a categorification of $R$ is a (monoidal, additive) category $C$ with a “valuation” in $R$ that takes $\otimes$ to $\times$ and $\oplus$ to $+$.  This is described, with plenty of examples, in this paper by Rafael Diaz and Eddy Pariguan.  The other, typical of the Khovanov program, says it is a (monoidal, additive) category $C$ whose Grothendieck ring is $K_0(C) = R$.  Of course, the second definition implies the first, but not conversely.  The objects of the Grothendieck ring are isomorphism classes in $C$.  A valuation may identify objects which aren’t isomorphic (or, as in groupoidification, morphisms which aren’t 2-isomorphic).

So a categorification of the first sort could be factored into two steps: first take the Grothendieck ring, then take a quotient to further identify things with the same valuation.  If we’re lucky, there’s a commutative square here: we could first take the category $C$, find some surjection $C \rightarrow C'$, and then find that $K_0(C') = R$.  This seems to be the relation between Khovanov’s categorification of $H_{\mathbb{Z}}$ and the one in $Span(Gpd)$. This is the sense in which it seems to be the “universal” answer to the problem.

On a tangential note, let me point out John Baez’ most recent “This Week’s Finds”, which has an accessible but fairly in-depth discussion of climate modelling.  There have been many years of very loud public discussion of this which, for reasons of politics, seems to involve putting the “Mathematical models are inherently elitist gibberish” and “Science knows everything so shut up, moron” positions on display and letting viewer decide.  This is known in the journalism trade as “balance”.  Obviously, within the research community working on them, there’s a mountain of literature on what the models model, how detailed they are, how they work, etc., but it mostly goes over my head, so John’s post strikes a nice balance for me.

Like most computer simulation models, they’re basically discrete approximations to big systems of differential equations – but exactly which systems, how they’re developed, how accurately they model the real thing, and the relative merits of simple vs. complex models is the main point.  The use of Monte Carlo methods and Bayesian analysis to tune the various free parameters is a key part of the matter of how accurate they should be.  Anyway – check it out.

Meanwhile, the TQFT club at IST recently started up its series of seminars.  The first few speakers were Rui Carpentier, Anne-Laure Thiel, and Marco Mackaay.  Rui is faculty here at IST, and a former student of Roger Picken (his thesis was on a topic closely related to what he was talking about).  Anne-Laure is a post-doc here at IST, mainly working with Marco, who, however, is actually at the University of the Algarve in Faro, Portugal, and had to come up to Lisbon specially for the seminar.  Anne-Laure and Marco were both speaking mainly about some of the Soergel bimodule stuff which came up at the Oporto meeting on categorification, which I posted about previously, so I’ll go over that in a bit more detail here.

First, though, Rui Carpentier’s talk:

## 3-colourings of Cubic Graphs and Operators

All these talks involve algebraic representations of categories that can be represented by some graphical calculus, but in this case, one starts with a category whose morphisms are precisely graphs with loose ends.  (The objects are non-negative integers, or, if you like, finite sets of dots which act as the vertices of the loose ends).  The graphs are trivalent (except at the input and output vertices, which are 1-valent), hence “cubic graphs”.  This category is therefore called $\mathbf{CG}$, and it has a small number of generators, which happen to be quite similar to those which generate the category of 2D-cobordisms (one of the connections to TQFT), though the relations are slightly different.

Roughly, and without drawing the pictures: the generators are cup and cap (the shapes $\cup$ and $\cap$), two different trivalent vertices (a $Y$, and the same upside-down), the swap (an $X$ where the strands cross without a vertex), and the identity (just a vertical line).  There are a number of relations, including Reidemeister moves, on these generating pictures, which ensure that they’re enough to identify graphs up to isotopy of the pictures.

Then the point is to describe graphs using operators – that is, construct a representation $F :\mathbf{CG} \rightarrow \mathbf{Vect}$.   Given any such representation, these generators provide all the structure maps of a bialgebra – chiefly, unit, counit, multiplication and co-multiplication – and the relations imposed by isotopy make this work (though unlike some other situations, it’s neither commutative nor cocommutative).  The representation $F$ he constructs is based on 3-colourings of the edges of the graphs.  At the object level, it assigns to a dot the 3-dimensional vector space $V= span(e_1,e_2,e_3)$.  Being monoidal, $F$ takes the object $n$ to $V^{\otimes n}$ – the tensor product of the spaces at each vertex.

The idea is that choosing a basis vector in this space amounts to picking a colouring of the incoming and outgoing edges.  For morphisms, we should note that the rule that says when a colouring is admissible is that all the edges incident to a given vertex must have different colours.  Then, given a morphism (graph) $G : m \rightarrow n$, we can describe the linear map $F(G)$ most easily by saying that the component in the matrix, given an incoming and outgoing basis vector, just counts the number of admissible graphs that agree with the chosen colourings on the in-edges and out-edges.

There’s another functor, $\hat{F}$, which counts these graphs with a sign, which marks whether the graph contains an odd or an even number of crossings of differently-coloured edges – negative for odd, positive for even.  This  is the “Penrose evaluation” of the graph.

So these maps give the “operators” of the title, and the rest of the point is to use them to study graphs and their colourings.  One can, in this setup, rewrite some graphs as linear combinations of others – so-called “Skein relations” hold, for example, so that, after applying $F$, the composite of multiplication and comultiplication (taking two points to two points, through one cut-edge) is the same as the identity minus the swap.  This sort of thing appears in formal knot theory all the time, and is a key tool for recoupling in spin networks, and so on…

Given this “recoupling” idea, there are some important facts: first, any graph can be rewritten as a linear combination of planar graphs, and any planar graph with cycles can be reduced to a sum of planar graphs without cycles.  (Rui gave the example of decomposing a pentagonal cycle as a linear combination of four other graphs, three of which are disconnected).  So in fact any graph decomposes as a linear combination of forests (cycle-free graphs, the connected components of which are called “trees”, hence the name).  Another essential fact is that, due to the Euler characteristic of the plane, any planar graph can be split into two parts with at most five edges between them (the basis of the solution to the three utilities puzzle).  Then it so happens that the space of graphs connecting zero in-edges to five out-edges is a 6-dimensional space, $\mathcal{V}^o_5$, generated by just six forests (including one lonesome tree).

So one theorem which Rui told us about, which can be shown using the so-called Penrose relations (provable using the representations $F$ and $\hat{F}$), is that there’s just one such graph (which he described in the particular basis above) that evaluates to zero when composed with some other graph.  The proof of this uses the Four Colour Theorem (3-colouring of graph edges being related to 4-colouring of planar regions); in fact, the two theorems are equivalent so if anyone can find an alternative proof of this one, the bonus is another proof of the FCT.

Finally, he gave a conjecture that, if true, would help recognize planar graphs just by the operators produced by the representation $\hat{F}$ (at least it proposes a necessary condition).  This conjecture says that if a planar graph with five output edges (the maximum, remember) is written in the basis mentioned above, then the sum of the coefficients of the five disconnected trees is nonnegative.  (Thus, the connected tree doesn’t contribute to this measure).  This is still just a conjecture – Rui said that to date neither proof nor counterexample has been found.

## Soergel Bimodules, Singular and Virtual Braids

As I mentioned up top, I previously posted a bit about work on Soergel bimodules when describing Catharina Stroppel’s talk at the meeting in Faro in July.  To recap: they are associated with categories of modules over rings – specifically, rings of certain classes of symmetric functions.  Even more specifically, given a partition $\lambda$ of an integer $n$, there is a subgroup of the symmetric group $S_{\lambda} \subset S_n$ which fixes the partition.  All such groups act on the ring of $n$-variable polynomial functions $R =\mathbb{Q}[x_1, \dots, x_n]$, and the ones fixed by $S_{\lambda}$ form the ring $R^{\lambda}$.

Now, these groups are all related to each other in a web of containments, hence so are the rings.  So the module categories $R^{\lambda}$ are connected by various functors.  Given a containment $R^{\lambda '} \subset R^{\lambda}$, modules over $R^{\lambda}$ can be restricted to ones over $R^{\lambda '}$, and modules over $R^{\lambda '}$ can be induced up to ones over $R^{\lambda}$.  The restriction and induction functors can be represented as “tensor with a bimodule” (this is much the same classification as that for 2-linear maps which I’ve said a bunch about here, except that those must be free).  Applying induction functors repeatedly gives abitrarily large bimodules, but they are built as direct sums of simple parts.  Those simple parts, and any direct sums of them, are Soergel bimodules.  The point is that such bimodules describe morphisms.

So in the TQFT club, Marco Mackaay gave the first of a series of survey talks on this topic, and Anne-Laure Thiel gave a talk about the “Categorification of Singular Braid Monoids and Virtual Braid Groups”.  Since Marco’s talk was the first in a series of surveys, and a lot of what it surveyed was work described in my post on the Faro meeting, I’ll just mention that it dealt with the original motivation of a lot of this work in categorifying representation theory of Lie algebras (c.f. the discussion of the Khovanov-Lauda categorification of quantum groups in the previous post), and also got a bit into some of the different diagrammatic calculi created for that purpose, along the lines of the talks by Ben Webster and Geordie Williamson at that meeting.  Maybe when Marco has given more of these talks, I’ll return to this one here as well.

Now, the starting point of Anne-Laure’s talk was that the setup above lets one define a category with a presentation like that of the Hecke algebra (a quotient of the group algebra of the braid group), where exact relations become isomorphisms.  That is, we go from a category where morphisms are braids (up to isotopy and Reidemeister moves and so forth as usual) to a 2-category where the morphisms are bimodules, which happen to satisfy the same relations.  (The 2-morphisms, bimodule maps, are what allow relations to hold weakly…)

Specifically, the generators of the braid group are $\sigma_i$, the braids taking the $i^{th}$ strand over the $(i+1)^{st}$.  The parallel thing is $B_i = R \otimes_{R^{\sigma_i}} R$, where here we’re talking about the subgroup generated by the transposition of $i$ and $i+1$.  In the language of partitions, this corresponds to a $\lambda$ with one part of size two, $(i,i+1)$, and the rest of size one.  Now, since this bimodule is actually built from polynomials in $R$, it naturally has a grading – this corresponds to the degree of $q$, since the Hecke algebra involves a quotient giving q-deformed relations – so there is a degree-shift operation that categorifies multiplication by $q$.  This much is due to Soergel.

Anne-Laure’s talk was about extending this to talk about a categorification, first of the braid group in terms of complexes of these bimodules (due actually to Rouquier), then virtual and singular braids.  These, again, are basically creatures of formal knot theory (see link above).  They can be described by a presentation similar to that for braids – just as the braid group has a generators-and-relations presentation in terms of over-crossings of adjacent strands, these incorporate other kinds of crossings.  Singular braids allow a sort of “through” crossing, where the $i^{th}$ strand goes neither over nor under the $(i+1)^{st}$.  Virtual braids (the braid variant on virtual knots) have a special type of marked crossing called the “virtual crossing”, drawn with a little circle around it.  These are included as new generators in describing the virtual braid group, and of course some new relations are added to show how they relate to the original generators – variations on the Reidemeister moves, for example.

To categorify this, Anne-Laure explained that these new generators can also be represented by bimodules, but these ones need to be twisted.  In particular, twisting the bimodule $R$ by the action of a permutation $\omega \in S_n$ gives $R_{\omega}$, which is the same as $R$ as a left $R$-module, but is acted on by an element $a \in R$ on the right through multiplication by $\omega(a)$, so that $b \cdot p \cdot a = bp(\omega(a))$.  Then the new generators, beyond the $B_i = R \otimes_{R^{\sigma_i}} R$, are of the form $R_{\omega} \otimes_{R^{\omega '}} R$.  These then satsify the right relations for this to categorify the virtual braid group.

So this is a couple of weeks backdated.  I’ve had a pretty serious cold for a while – either it was bad in its own right, or this was just a case of the difference in native viruses between two different continents that my immune system wasn’t prepared for.  Then, too, last week was Republic Day – the 100th anniversary of the middle of three revolutions (the Liberal, the Republican, and the Carnation revolution that ousted the dictatorship regime in 1974 – and let me say that it’s refreshing for a North American to be reminded that Republicanism is a refinement of Liberalism, though how the flowers fit into it is less straightforward).  So my family and I went to attend some of the celebrations downtown, which were impressive.

Anyway, with the TQFT club seminars starting up very shortly, I wanted to finish this post on the first talks I got to see here at IST, which were on pretty widely different topics.  The first was by Ivan Smith, entitled “Quadrics, 3-Manifolds and Floer Cohomology”.  The second was a recorded video talk arranged by the string theory group.  This was a recording of a talk given by Kostas Skenderis a couple of years ago, entitled “The Fuzzball Proposal for Black Holes”.

## Ivan Smith – Quadrics, 3-Manfolds and Floer Cohomology

Ivan Smith’s talk began with some motivating questions from topology, symplectic geometry, and from the study of moduli spaces.  The topological question talks about 3-manifolds $Y$ and the space of representations $Hom(\pi_1(Y),G)$ of its fundamental group into a compact Lie group $G$, which was generally $SO(3)$ or $SU(2)$.  Specifically, the question is how this space is affected by operations on $Y$ such as surgery, taking covering spaces, etc.  The symplectic geometry question asks, for a symplectic manifold $(X,\omega)$, what the “mapping class group” of symplectic transformations – that is, the group $\pi_0(Symp(X))$ of connected components of symplectomorphisms from $X$ to itself – in a sense, this is asking how much of the geometry is seen by the symplectic situation.  The question about moduli spaces asks to characterize the action of the (again, mapping class group of) diffeomorphisms of a Riemann surface on the moduli space of bundles on it.  (This space, for  $\Sigma$ with genus $g \geq 2$, look like $M_g \simeq Hom(\pi_1(\Sigma),SU(2))$ modulo conjugation.  It is the complex-manifold version of the space of flat connections which I’ve been quite interested in for purposes of TQFT, though this is a coarse quotient, not a stack-like quotient.  Lots of people are interested in this space in its various hats.)

The point of the talk being to elucidate how these all fit together.  The first part of the title, “Quadrics”, referred to the fact that, when $\Sigma$ has genus 2, the moduli space we’ll be looking at can be described as an intersection of some varieties (defined by quadric equations) in the projective space $\mathbb{CP}^5$.  Knowing this, one can describe some of its properties just by looking at intersections of curves.

In general we’re talking about complex manifolds, here.  To start with, for Riemann surfaces (one-dimensional complex manifolds), he pointed out that there is an isomorphism between the mapping class groups of symplectomorphisms and diffeomorphisms: $\pi_0(Symp(\Sigma)) \simeq \pi_0(Diff(\Sigma))$.  But in general, for example, for 3-dimensional manifolds, there is structure in the symplectic maps which is forgotten by the smooth ones – there’s still a map $\pi_0(Symp(\Sigma)) \rightarrow \pi_0(Diff(\Sigma))$, but it has a kernel – there are distinct symplectic maps that all look like the identity up to smooth deformation.

Now, our original question was what the action of the diffeomorphisms of on the moduli space $M_g$ of bundles over $\Sigma$.  An element $h$ of $\pi_0(Diff(\Sigma))$ acts (by symplectic map) on it.  The discrepancy we mentioned is that the map corresponding to $h$ will always have fixed points, but be smoothly equivalent to one that doesn’t.  So the smooth mapping class group can’t detect the property of having fixed points.  What it CAN detect, however, is information about intersections.  In particular,   as mentioned above, the moduli space of bundles over a genus 2 surface is an intersection; in this situation, there is an injective map back from the smooth mapping class group into the group of classes of symplectic maps.  So looking symplectically loses nothing from the smooth case.

Now, these symplectic maps tie into the third part of the title, “Floer Homology”, as follows.  Given a symplectic map $\phi : (X,\omega) \rightarrow (X,\omega)$, one can define a complex of vector spaces $HF(\phi)$ which is the usual cohomology of a chain complex generated by fixed points of the map $\phi$, and with a differential $\partial$ which is defined by counting certain curves.  The way this is set up, if $\phi$ is the identity so that all points are fixed points, one gets the usual cohomology of the space $X$ – except that it’s defined so as to be the quantum cohomology of $X$ (for more, check out this tutorial by Givental).  This has the same complex as the usual cohomology, but with the cup product replaced by a deformed product.  It’s an older theorem (due to Donaldson) that, at least for genus 2, the quantum cohomology of the moduli space of bundles over $\Sigma$ splits into a direct sum of rings:

$QH^*(M_2) \cong \mathbb{C} \oplus QH^*(\Sigma_2) \oplus \mathbb{C}$

So one of the key facts is that this works also with Floer homology for other maps than the identity (so this becomes a special case).  So replacing $QH^*$ in the above with $HF^*(\phi)$ for any $\phi$ (acting either on the surface $\Sigma$, or the induced action on the moduli space) still gives a true statement.  Note that this actually implies the theorem that there are fixed points in the space of bundles, since the right hand side is always nontrivial.

So at this point we have some idea of how Floer cohomology is part of what ties the original three questions together.  To take a further look at these we can start to build a category combining much of the same information.  This is the (derived) Fukaya category.  The objects are Lagrangian submanifolds of a symplectic manifold $(X,\omega)$ – ones where the symplectic form vanishes.  To start building the category, consider what we can build from pairs of such objects $(L_1,L_2)$.  This is rather like the above – we define a complex of vector spaces, which is the cohomology of another complex.  Instead of being the complex freely generated by fixed points, though, it’s generated by intersection points of $L_1$ and $L_2$.  This automatically becomes a module over $QH^*(X)$, so the category we’re building is enriched over these.

Defining the structure of this category is apparently a little bit complicated – in particular, there is a composition product $HF(L_1,L_2) \otimes HF(L_2,L_3) \rightarrow HF(L_1,L_3)$ in the form of a cohomology operation.  Furthermore, which Ivan Smith didn’t have time to describe in detail, there are other “higher” products.  These are Massey type products, which is to say higher-order cohomology operations, which involve more than two inputs.  These give the whole structure (where one takes the direct sum of all those hom-modules $HF(L_i,L_j)$ to get one big module) the structure of an $A_{\infty}$algebra (so the Fukaya category is an $A_{\infty}$-category, I suppose).  This is one way of talking about weak higher categories (the higher products give the associator for composition, and its higher analogs), so in fact this is a pretty complex structure, which the talk didn’t dwell on in detail.  But in any case, the point is that the operations in the category correspond to cohomology operations.

Then one deals with the “derived” Fukaya category $\mathcal{DF}(X)$.  I understand derived categories to be (at least among other examples) a way of taking categories of complexes “up to homotopy”, perhaps as a way of getting rid of some of this complication.  Again, the talk didn’t elaborate too much on this.  However, the fundamental theorem about this category is a generalization of the theorem above above quantum cohomology:

$\mathcal{DF}(M_2) \cong \mathcal{DF}(pt) \oplus \mathcal{DF}(\Sigma_2) \oplus \mathcal{DF}(pt)$

That is, the derived Fukaya category for the moduli space of bundles over $\Sigma_2$ is the category for the Riemann surface itself, summed with two copies of the category for a single point (which is replacing the two copies of $\mathbb{C}$).  This reduces to the previous theorem when we’re looking at the map $\phi = id$, just as before.

So the last question Ivan Smith addressed about this is the fact that these sorts of categories are often hard to calculate explicitly, but they can be described in terms of some easily-described data.  He gave the analogy of periodic functions – which may be quite complicated, but by means of Fourier decompositions, can be easily described in terms of sines and cosines, which are easy to analyze.  In the same way, although the Fukaya categories for particular spaces might be complicated, they can be described in terms of the (derived) category of modules over the $A_{\infty}$-algebras.  In particular, every category $\mathcal{DF}(X)$ embeds in a generic example $\mathcal{D}(mod-A_{\infty}-alg)$.  So by understanding categories like this, one can understand a lot about the categories that come from spaces, which generalize quantum cohomology as described above.

I like this punchline of the analogy with Fourier analysis, as imprecise as it might be, because it suggests a nice way to approach complex entities by finding out the parts that can generate them, or simple but large things you might discover them inside.

## Fuzzballs

The Skenderis talk about black holes was interesting, in that it was a recorded version of a talk given somewhere else – I haven’t seen this done before, but apparently the String Theory group does it pretty regularly.  This has some obvious advantages – they can get a wider range of talks by many different speakers.  There was some technical problem – I suppose due to the way the video was encoded – that meant the slides were sometimes unreadably blurry, but that’s still better than not getting the speaker at all.  I don’t have the background in string theory to be able to really get at the meat of the talk, though it did involve the AdS/CFT correspondence.  However, I can at least say a few concrete things about the motivation.  First, the “fuzzball” proposal is a more-or-less specific proposal to deal with the problem of black hole entropy.

The problem, basically, is that it’s known that the thermodynamic entropy associated to a black hole – which can be computed in completely macroscopic terms – is proportional to the area of its horizon.  On the other hand, in essentially every other setting, entropy has an interpretation in terms of counting microstates, so that the entropy of a “macrostate” is proportional to the logarithm of the number of microstates.  (Or, in a thermal state, which is a statistical distribution, this is weighted by the probability of the microstate).  So, for example, with a gas in a box, there are many macrostates that correspond to a relatively even distribution of position and momentum among the molecules, and relatively few in which all molecules are all in one small corner of the box.

The reason this is a problem is that, classically, the state of a black hole is characterized by very few numbers: the mass, angular momentum, and electric charge.   There doesn’t seem to be room for “microstates” in a classical black hole.  So the overall point of the proposal is to describe what microstates would be.  The specific way this is done with “fuzzballs” is somewhat mysterious to me, but the overall idea makes sense.  One interesting consequence of this approach is that event horizons would be strictly a property of thermal states, in whatever underlying theory one takes to be the quantum theory behind classical gravity (here assumed to be some specific form of string theory – the example he was using is something called the B1-B5 black hole, which I know nothing about).  That’s because a pure state would have a single microstate, hence have zero entropy, hence no horizon.

Now, what little I do understand about the particular model relies on the fact that near a (classical) event horizon, the background metric has a component that looks like anti-deSitter space – a vacuum solution to the Einstein equations with a negative cosmological constant.  (This part isn’t so hard to see – AdS space has that “saddle-shaped” appearance of a hyperbolic surface, and so does the area around a horizon, even when you draw it like this.)  But then, there is the AdS/CFT correspondence that says states for a gravitational field in (asymptotically) anti-deSitter space correspond to states for a conformal field theory (CFT) at the boundary.  So the way to get microstates, in the “fuzzball” proposal, is to look at this CFT, and find geometries that correspond to them.  Some would be well-approximated by the classical, horizon-ridden geometry, but others would be different.  The fact that this CFT is defined at the boundary explains why entropy would be proportional to area, not volume, of the black hole – this being a manifestation of the so-called “holographic principle”.  The “fuzziness” that one throws away by reducing a thermal state that combines these many geometries to the classical “no-hair” black hole determined by just three numbers is exactly the information described by the entropy.

I couldn’t follow some parts of it, not having much string-theory background – I don’t feel qualified to judge whether string theory makes sense as physics, but it isn’t an approach I’ve studied much.  Still, this talk did reinforce my feeling that the AdS/CFT correspondence, at the very least, is something well-worth learning about and important in its own right.

Coming soon: descriptions of the TQFT club seminars which are starting up at IST.

Whatever ultimately becomes of some aspects of the Standard Model – the Higgs boson, for example – here is a report (based on an experiment described here) that some of the fundamentals hold up well to experimental test. Specifically, the Spin-Statistics Theorem – the relationship between quantum numbers of elementary particles and the representation theory of the Poincare group. It would have been very surprising if things had been otherwise, but as usual, the more you rely on an idea, the more important it is to be sure it fits the facts. The association between physics and representation theory is one of those things.

So the fact that it all seems to work correctly is a bit of a relief for me. See below.

Since the paperwork is now well on its way, I may as well now mention here that I’ve taken a job as a postdoctoral researcher at CAMGSD, a centre at IST in Lisbon, starting in September. In a week or so I will be heading off to visit there – there are quite a few people there doing things I find quite interesting, so it should be an interesting trip. After that, I’ll be heading down to the south of the country for the Oporto meeting on Geometry, Topology and Physics, which is held this year in Faro. This year the subject is “categorification”, so my talk will be mainly about my paper on ETQFT. There are a bunch of interesting speakers – two I happen to know personally are Aaron Lauda and Joel Kamnitzer, but many others look quite promising.

In particular, one of the main invited speakers is Mikhail Khovanov, whose name is famously (for some values of “famous”) attached to Khovanov Homology, which is a categorification of the Jones Polynomial. Instead of a polynomial, it associates a graded complex of vector spaces to a knot. (Dror Bar-Natan wrote an intro, with many pictures and computations). Khovanov’s more recent work, with Aaron Lauda, has been on categorifying quantum groups (starting with this).

Now, as for me, since my talk in Faro will only be about 20 minutes, I’m glad of the opportunity to give some more background during the visit at IST. In particular, a bunch of the background to the ETQFT paper really depends on this paper on 2-linearization. I’ve given some previous talks on the subject, but this time I’m going to try to get a little further into how this fits into a more general picture. To repeat a bit of what’s in this post, 2-linearization describes a (weak) 2-functor:

$\Lambda : Span(Gpd) \rightarrow 2Vect$

where $Span(Gpd)$ has groupoids as its objects, spans of groupoid homomorphisms as its arrows, and spans-of-span-maps as 2-morphisms. $2Vect$ is the 2-category of 2-vector spaces, which I’ve explained before. This 2-functor is supposed to be a sort of “linearization”, which is a very simple functor

$L : Span(FinSet) \rightarrow Vect$

It takes a set $X$ to the free vector space $L(X) = \mathbb{C}^X$, and a span $X \stackrel{s}{\leftarrow} S \stackrel{t}{\rightarrow} Y$ to a linear map $L(S) : L(X) \rightarrow L(Y)$. This can be described in two stages, starting with a vector in $L(S)$, namely, a function $\psi : X \rightarrow \mathbb{C}$. The two stages are:

• First, “pull” $\psi$ up along $s$ to $\mathbb{C}^S$ (note: I’m conflating the set $S$ with the span $(S,s,t)$), to get the function $s^*\psi = \psi \circ s : S \rightarrow \mathbb{C}$.
• Then “push” this along $t$ to get $t_*(s^*\psi)$. The “push” operation $f_*$ along any map $f : X \rightarrow Y$ is determined by the fact that it takes the basis vector $\delta_x \in \mathbb{C}^X$ to the basis vector $\delta_{f(x)} \in \mathbb{C}^Y$ (these are the delta functions which are 1 on the given element and 0 elsewhere)

It’s helpful to note that, for a given map $f : X \rightarrow Y$, are linear adjoints (using the standard inner product where the delta functions are orthonormal). Combining them together – it’s easy to see – gives a linear map which can be described in the basis of delta functions by a matrix. The $(x,y)$-entry of the matrix counts the elements of $S$ which map to $(x,y)$ under $(s,t) : S \rightarrow X \times Y$. We interpret this by saying the matrix “counts histories” connecting $x$ to $y$.

In groupoidification, a-la Baez and Dolan (see the various references beyond the link), one replaces $FinSet$ with $FinGpd$, the 2-category of (essentially) finite groupoids, but we still have a functor into $Vect$. In fact, into $FinHilb$: the vector space $D(G)$ is the free one on isomorphism classes in $G$, but the linear maps (and the inner product) are tweaked using the groupoid cardinality, which can be any positive rational number. Then we say the matrix does a “sum over histories” of certain weights. In this paper, I extend this to “$U(1)$-groupoids”, which are labelled by phases – which represent the exponentiated action in quantum mechanics – and end up with complex matrices. So far so good.

The 2-linearization process is really “just” a categorification of what happens for sets, where we treat “groupoid” as the right categorification of “set”, and “Kapranov-Voevodsky 2-vector space” as the right categorification of “vector space”. (To treat “category” as the right categorification of “set”, one would have to use Elgueta’s “generalized 2-vector space“, which is probably morally the right thing to do, but here I won’t.) To a groupoid $X$, we assign the category of functors into $Vect$ – that is, $Rep(X)$ (in smooth cases, we might want to restrict what kind of representations we mean – see below).

To pull such a functor along a groupoid homomorphism $f : X \rightarrow Y$ is again done by precomposition: $f^*F = F \circ f$. The push map in 2-linearization is the Kan extension of the functor $\Psi$ along $f$. This is the universal way to push a functor forward, and is the (categorical!) adjoint to the pull map. (Kan extensions are supposed to come equipped with some natural transformations: these are the ones associated to the adjunction). Then composing “pull” and “push”, one categorifies “sum over histories”.

So here’s one thing this process is related to: in the case where our groupoids have just one object (i.e. are groups), and the homomorphism $f : X \rightarrow Y$ is an inclusion (conventionally written $H < G$), this goes by a familiar name in representation theory: restriction and induction. So, given a representation $\rho$ of $G$ (that is, a functor from $Y$ into $Vect$), there is an induced representation $res_H^G \rho = f^*\rho$, which is just the same representation space, acted on only by elements of $H$ (that is, $X$). This is the easy one. The harder one is the induced representation of $G$ from a representation $\tau$ of $H$ (i.e. $\tau : X \rightarrow Vect$, which is to say $ind^G_H \tau = f_* \tau : Y \rightarrow Vect$. The fact that these operations are adjoints goes in representation theory by the name “Frobenius reciprocity”.

These two operations were studied by George Mackey (in particular, though I’ve been implicitly talking about discrete groups, Mackey’s better known for looking at the case of unitary representations of compact Lie groups). The notion of a Mackey functor is supposed to abstract the formal properties of these operations. (A Mackey functor is really a pair of functors, one covariant and one contravariant – giving restriction and “transfer”/induction maps for – which have formal properties similar to the functor from groups into their representation rings – which it’s helpful to think of as the categories of representations, decategorificatied. In nice cases, a Mackey functor from a category $C$ is the same as a functor out of $Span(C)$).

Anyway, by way of returning to groupoids: the induced representation for groups is found by $\mathbb{C}[G] \otimes_{\mathbb{C}[H]} V$, where $V$ is the representation space of $\tau$. (For compact Lie groups, replace the group algebra $\mathbb{C}[G]$ with $L^2(G)$, and likewise for $H$). A similar formula shows up in the groupoid case, but with a contribution from each object (see the paper on 2-linearization for more details). This is also the formula for the Kan extension.

“Now wait a minute”, the categorically aware may ask, “do you mean the left Kan extension, or the right Kan extension?” That’s a good question! For one thing, they have different formulas: one involving limits, and the other involving colimits. Instead of answering it, I’ll talk about something not entirely unrelated – and a little more context for 2-linearization.

The setup here is actually a rather special case of Grothendieck’s six-operation framework, in the algebro-geometric context, for sheaves on (algebraic) spaces (there’s an overview in this talk by Joseph Lipman, the best I’ve been able to find online). Now, , these operations as extended to derived categories of sheaves (see this intro by R.P. Thomas). The derived category $D(X)$ is described concretely in terms of chain complexes of sheaves in $Sh(X)$, taken “up to homotopy” – it is a sort of categorification of cohomology. But of course, this contains $Sh(X)$ as trivial complexes (i.e. concentrated at level zero). The fact that our sheaves come from functors into $Vect$, which form a 2-vector space, so that functors between these are exact, means that there’s no nontrivial homology – so in our special case, the machinery of derived categories is more than we need.

This framework has been extended to groupoids – so the sheaves are on the space of objects, and are equivariant – as described in a paper by Moerdijk called “Etale Groupoids, Derived Categories, and Operations” (the situation of sheaves that are equivariant under a group action is described in more detail by Bernstein and Lunts in the Springer lecture notes “Equivariant Sheaves and Functors”). Sheaves on groupoids are essentially just equivariant sheaves on the space of objects. Now, given a morphism $f : X \ra Y$, there are four induced operations:

• $f^* , f^! : D(Y) \rightarrow D(X)$
• $f_*, f^! : D(X) \rightarrow D(Y)$ (in general right adjoint to $f^*$ and $f^!$)

(The other operations of the “six” are $hom$ and $\otimes$). The basic point here is that we can “pull” and “push” sheaves along the map $f$ in various ways. For our purposes, it’s enough to consider $f^*$ and $f_*$. The sheaves we want come from functors into $Vect$ (we actually have a vector space at each point in the space of objects). These are equivariant “bundles”, albeit not necessarily locally trivial. The fact that we can think of these as sheaves – of sections – tends to stay in the background most of the time, but in particular, being functors automatically makes the resulting sheaves equivariant. In the discrete case, we can just think of these as sheaves of vector spaces: just take $F(U)$ to be the direct sum of all the vector spaces at each object in any subset $U$ – all subsets are open in the discrete topology… For the smooth situation, it’s better not to do this, and think of the space of sections as a module over the ring of suitable functions.

Now to return to your very good question about “left or right Kan extension”… the answer is both. since for $Vect$-valued functors (where $Vect$ is the category of finite dimensional vector spaces), we have natural isomorphisms $f^* \cong f^!$ and $f_* \cong f_!$: these functors are \textit{ambiadjoint} (ie. both left and right adjoint). We use this to define the effect of $\Lambda$ on 2-morphisms in $Span_2(Gpd)$.

This isomorphism is closely related to the fact that finite-dimensional vector spaces are canonically isomorphic to their double-dual: $V \cong V^{**}$. That’s because the functors $f^*$ and $f_*$ are 2-linear maps. These are naturally isomorphic to maps represented as matrices of vector spaces. Taking an adjoint – aside from transposing the matrix, naturally replaces the matrices with their duals. Doing this twice, we get the isomorphisms above. So the functors are both left and right adjoint to each other, and thus in particular we have what is both left and right Kan extension. (This is also connected with the fact that, in $Vect$, the direct sum is both product and coproduct – i.e. limit and colimit.)

It’s worth pointing out, then, that we wouldn’t generally expect this to happen for infinite-dimensional vector spaces, since these are generally not canonically isomorphic to their double-duals. Instead, for this case we would need to be looking at functors valued in $Hilb$, since Hilbert spaces do have that property. That’s why, in the case of smooth groupoids (say, Lie groupoids), we end up talking about “(measurable) equivariant Hilbert bundles”. (In particular, the ring of functions over which our sheaves are modules is: the measurable ones. Why this is the right choice would be a bit of a digression, but roughly it’s analogous to the fact that $L^2(X)$ is a space of measurable functions. This is the limitation on which representations we want that I alluded to above.).

Now, $\Lambda$ is supposed to be a 2-functor. In general, given a category $C$ with all pullbacks, $Span_2(C)$ is the universal 2-category faithfully containing $C$ such that every morphism has an ambiadjoint. So the fact that the “pull” and “push” operations are ambiadjoint lets this 2-functor respect that property. It’s the unit and counits of the adjunctions which produce the effect of $\Lambda$ on 2-morphisms: given a span of span-maps, we take the two maps in the middle, consider the adjoint pairs of functors that come from them, and get a natural transformation which is just the composite of the counit of one adjunction and the unit of the other.

Here’s where we understand how this fits into the groupoidification program – because the effect of $\Lambda$ on 2-morphisms exactly reproduces the “degroupoidification” functor of Baez and Dolan, from spans of groupoids into $Vect$, when we think of such a span as a 2-morphism in $Hom(1,1)$ – that is, a span of maps of spans from the terminal groupoid to itself. In other words, degroupoidification is an example something we can do between ANY pair of groupoids – but in the special case where the representation theory all becomes trivial. (This by no means makes it uninteresting: in fact, it’s a perfect setting to understand almost everything else about the subject).

Now, to actually get all the coefficients to work out to give the groupoid cardinality, one has to be a bit delicate – the exact isomorphism between the construction of the left and right adjoint has some flexibility when we’re working over the field of complex numbers. But there’s a general choice – the Nakayama isomorphism – which works even when we’re replace $Vect$ by $R$-modules for some ring $R$. To make sure, for general $R$, that we have a true isomorphism, the map needs some constants. These happen to be, in our case, exactly the groupoid cardinalities to make the above statement true!

To me, this last part is a rather magical aspect of the whole thing, since the motivation I learned for groupoid cardinalities is quite remote from this – it’s just a valuation on groupoids which gets along with products and coproducts, and also with group actions (so that $|X/G| = |X|/|G|$, even when the action isn’t free). So one thing I’d like to know, but currently don’t is: how is it that this is “secretly” the same thing as the Nakayama isomorphism?

I say this is about a “recent” talk, though of course it was last year… But to catch up: Ivan Dynov was visiting from York and gave a series of talks, mainly to the noncommutative geometry group here at UWO, about the problem of classifying von Neumann algebras. (Strictly speaking, since there is not yet a complete set of invariants for von Neumann algebras known, one could dispute the following is a “classification”, but here it is anyway).

The first point is that any von Neumann algebra $\mathcal{A}$ is a direct integral of factors, which are highly noncommutative in that the centre of a factor consists of just the multiples of the identity. The factors are the irreducible building blocks of the noncommutative features of $\mathcal{A}$.

There are two basic tools that provide what classification we have for von Neumann algebras: first, the order theory for projections; second, the Tomita-Takesaki theory. I’ve mentioned the Tomita flow previously, but as for the first part:

A projection (self-adjoint idempotent) is just what it sounds like, if you reprpsent $\mathcal{M}$ as an algebra of bounded operators on a Hilbert space. An extremal but informative case is $\mathcal{M} = \mathcal{B}(H)$, but in general not every bounded operator appears in $\mathcal{M}$.

In the case where $\mathcal{M} = \mathcal{B}(H)$, then a projection in $\mathcal{M}$ is the same thing as a subspace of $H$. There is an (orthomodular) lattice of them (in general, the lattice of projections is $\mathcal{P(M)}$). For subspaces, the dimension characterizes $H$ up to isomorphism – any any two subspaces of the same dimension are isomorphic by some operator in $\mathcal{B}(H)$ (but not necessarily in a general $\mathcal{M}$).

The idea is to generalize this to projections in a general $\mathcal{A}$, and get some characterization of $\mathcal{A}$. The kind of isomorphism that matters for subspaces is a partial isometry – a map $u$ which preserves the metric on some subspace, and otherwise acts as a projection. In fact, the corresponding projections are then conjugate by $u$. So we define, for a general $\mathcal{M}$, an equivalence relation on projections, which amounts to saying that $e \sim f$ if there’s a partial isometry $u \in \mathcal{M}$ with $e = u*u$, and $f = uu*$ (i.e. the projections are conjugate by $u$).

Then there’s an order relation on the equivalence classes of projections – which, as suggested above, we should think of as generalizing “dimension” from the case $\mathcal{M} = \mathcal{B}(H)$. The order relation says that $e \leq f$ if $e \sim e_0$ where $e_0 \leq f$ as a projection (i.e. inclusion thinking of a projection as its image subspace of $H$). But the fact that $\mathcal{M}$ may not be all of $\mathcal{B}(H)$ has some counterintuitive consequences. For example, we can define a projection $e \in \mathcal{M}$ to be finite if the only time $e \sim e_0 \leq e$ is when $e_0 = e$ (which is just the usual definition of finite, relativized to use only maps in $\mathcal{M}$). We can call $e \in \mathcal{M}$ a minimal projection if it is nonzero and $f \leq e$ imples $f = e$ or $f = 0$.

Then the first pass at a classification of factors (i.e. “irreducible” von Neumann algebras) says a factor $\mathcal{M}$ is:

• Type $I$: If $\mathcal{M}$ contains a minimal projection
• Type $II$: If $\mathcal{M}$ contains no minimal projection, but contains a (nontrivial) finite projection
• Type $III$: If $\mathcal{M}$ contains no minimal or nontrivial finite projection

We can further subdivide them by following the “dimension-function” analogy, which captures the ordering of projections for $\mathcal{M} = \mathcal{B}(H)$, since it’s a theorem that there will be a function $d : \mathcal{P(M)} \rightarrow [0,\infty]$ which has the properties of “dimension” in that it gets along with the equivalence relation $\sim$, respects finiteness, and “dimension” of direct sums. Then letting $D$ be the range of this function, we have a few types. There may be more than one function $d$, but every case has one of the types:

• Type $I_n$: When $D = \{0,1,\dots,n\}$ (That is, there is a maximal, finite projection)
• Type $I_\infty$: When $D = \{ 0, 1, \dots, \infty \}$ (If there is an infinite projection in $\mathcal{M}$
• Type $II_1$: When $D = [ 0 , 1 ]$ (The maximal projection is finite – such a case can always be rescaled so the maximum $d$ is $1$)
• Type $II_\infty$: When $D = [ 0 , \infty ]$ (The maximal projection is infinite – notice that this has the same order type as type $II_1$)
• Type $III_\infty$ \: When $D = [0,\infty]$ (An infinite maximal projection)
• Type $III$: $D = \{0,1\}$, (these are called properly infinite)

The type $I$ case are all just (equivalent to) matrix algebras on some countable or finite dimensional vector space – which we can think of as a function space like $l_2(X)$ for some set $X$. Types $II$ and $III$ are more interesting. Type $II$ algebras are related to what von Neumann called “continuous geometries” – analogs of projective geometry (i.e. geometry of subspaces), with a continuous dimension function.

(If we think of these algebras $\mathcal{M}$ as represented on a Hilbert space $H$, then in fact, thought of as subspaces of $H$, all the projections give infinite dimensional subspaces. But since the definition of “finite” is relative to $\mathcal{M}$, and any partial isometry from a subspace $H' \leq H$ to a proper subspace $H'' < H'$ of itself that may exist in $\mathcal{B}(H)$ is not in $M$.)

In any case, this doesn’t exhaust what we know about factors. In his presentation, Ivan Dynov described some examples constructed from crossed products of algebras, which is important later, but for the moment, I’ll finish describing another invariant which helps pick apart the type $III$ factors. This is related to Tomita-Takesaki theory, which I’ve mentioned in here before.

You’ll recall that the Tomita flow (associated to a given state $\phi$) is given by $\sigma^{\phi}_t(A) = e^{i \Delta t} A e^{-i \Delta t}$, where $\Delta$ is the self-adjoint part of the conjugation operator $S$ (which depends on the state $\phi$ because it refers to the GNS representation of $\mathcal{M}$ on a Hilbert space $H$). This flow is uninteresting for Type $I$ or $II$ factors, but for type $III$ factors, it’s the basis of Connes’ classification.

In particular, the we can understand the Tomita flow in terms of eigenvalues of $\Delta$, since it comes from exponentials of $\Delta$. Moreover, as I commented last time, the really interesting part of the flow is independent of which state we pick. So we are interested in the common eigenvalues of the $\Delta$ associated to different states $\phi$, and define

$S(\mathcal{M}) = \cap_{\phi \in W} Spec(\Delta_{\phi})$

(where $W$ is the set of all states on $\mathcal{M}$, or actually “weights”)

Then $S(\mathcal{M}) - \{ 0 \}$, it turns out, is always a multiplicative subgroup of the positive real line, and the possible cases refine to these:

• $S(\mathcal{M}) = \{ 1 \}$ : This is when $\mathcal{M}$ is type $I$ or $II$
• $S(\mathcal{M}) = [0, \infty )$ : Type $III_1$
• $S(\mathcal{M}) = \{ 0 \} \cup \{ \lambda^n : n \in \mathbb{Z}, 0 < \lambda < 1 \}$ : Type $III_{\lambda}$ (for each $\lambda$ in the range $(0,1)$, and
• $S(\mathcal{M}) = \{ 0 , 1 \}$ : Type $III_0$

(Taking logarithms, $S(\mathcal{M}) - \{ 0 \}$ gives an additive subgroup of $\mathbb{R}$, $\Gamma(\mathcal{M})$ which gives the same information). So roughly, the three types are: $I$ finite and countable matrix algebras, where the dimension function tells everything; $II$ where the dimension function behaves surprisingly (thought of as analogous to projective geometry); and $III$, where dimensions become infinite but a “time flow” dimension comes into play.  The spectra of $\Delta$ above tell us about how observables change in time by the Tomita flow:  high eigenvalues cause the observable’s value to change faster with time, while low ones change slower.  Thus the spectra describe the possible arrangements of these eigenvalues: apart from the two finite cases, the types are thus a continuous positive spectrum, and a discrete one with a single generator.  (I think of free and bound energy spectra, for an analogy – I’m not familiar enough with this stuff to be sure it’s the right one).

This role for time flow is interesting because of the procedures for constructing examples of type $III$, which Ivan Dynov also described to us. These are examples associated with dynamical systems. These show up as crossed products. See the link for details, but roughly this is a “product” of an algebra by a group action – a kind of von Neumann algebra equivalent of the semidirect product of groups $H \rtimes K$ incorporating an action of $K$ on $H$. Indeed, if a (locally compact) group $K$ acts on group $H$ then the crossed product of algebras is just the von Neumann algebra of the semidirect product group.

In general, a ($W*$)-dynamical system is $(\mathcal{M},G,\alpha)$, where $G$ is a locally compact group acting by automorphisms on the von Neumann algebra $\mathcal{M}$, by the map $\alpha : G \rightarrow Aut(\mathcal{M})$. Then the crossed product $\mathcal{M} \rtimes_{\alpha} G$ is the algebra for the dynamical system.

A significant part of the talks (which I won’t cover here in detail) described how to use some examples of these to construct particular type $III$ factors. In particular, a theorem of Murray and von Neumann says $\mathcal{M} = L^{\infty}(X,\mu) \rtimes_{\alpha} G$ is a factor if the action of discrete group $G$ on a finite measure space $X$ is ergodic (i.e. has no nontrivial proper invariant sets – roughly, each orbit is dense). Another says this factor is type $III$ unless there’s a measure equivalent to (i.e. absolutely continuous with) $\mu$, and which is equivariant. Some clever examples I won’t reconstruct gave some factors like this explicitly.

He concluded by talking about some efforts to improve the classification: the above is not a complete set of invariants, so a lot of work in this area is improving the completeness of the set. One set of results he told us about do this somewhat for the case of hyperfinite factors (i.e. ones which are limits of finite ones), namely that if they are type $III$, they are crossed products of with a discrete group.

At any rate, these constructions are interesting, but it would take more time than I have here to look in detail – perhaps another time.

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