### conferences

To continue from the previous post

Twisted Differential Cohomology

Ulrich Bunke gave a talk introducing differential cohomology theories, and Thomas Nikolaus gave one about a twisted version of such theories (unfortunately, perhaps in the wrong order). The idea here is that cohomology can give a classification of field theories, and if we don’t want the theories to be purely topological, we would need to refine this. A cohomology theory is a (contravariant) functorial way of assigning to any space $X$, which we take to be a manifold, a $\mathbb{Z}$-graded group: that is, a tower of groups of “cocycles”, one group for each $n$, with some coboundary maps linking them. (In some cases, the groups are also rings) For example, the group of differential forms, graded by degree.

Cohomology theories satisfy some axioms – for example, the Mayer-Vietoris sequence has to apply whenever you cut a manifold into parts. Differential cohomology relaxes one axiom, the requirement that cohomology be a homotopy invariant of $X$. Given a differential cohomology theory, one can impose equivalence relations on the differential cocycles to get a theory that does satisfy this axiom – so we say the finer theory is a “differential refinement” of the coarser. So, in particular, ordinary cohomology theories are classified by spectra (this is related to the Brown representability theorem), whereas the differential ones are represented by sheaves of spectra – where the constant sheaves represent the cohomology theories which happen to be homotopy invariants.

The “twisting” part of this story can be applied to either an ordinary cohomology theory, or a differential refinement of one (though this needs similarly refined “twisting” data). The idea is that, if $R$ is a cohomology theory, it can be “twisted” over $X$ by a map $\tau: X \rightarrow Pic_R$ into the “Picard group” of $R$. This is the group of invertible $R$-modules (where an $R$-module means a module for the cohomology ring assigned to $X$) – essentially, tensoring with these modules is what defines the “twisting” of a cohomology element.

An example of all this is twisted differential K-theory. Here the groups are of isomorphism classes of certain vector bundles over $X$, and the twisting is particularly simple (the Picard group in the topological case is just $\mathbb{Z}_2$). The main result is that, while topological twists are classified by appropriate gerbes on $X$ (for K-theory, $U(1)$-gerbes), the differential ones are classified by gerbes with connection.

Fusion Categories

Scott Morrison gave a talk about Classifying Fusion Categories, the point of which was just to collect together a bunch of results constructing particular examples. The talk opens with a quote by Rutherford: “All science is either physics or stamp collecting” – that is, either about systematizing data and finding simple principles which explain it, or about collecting lots of data. This talk was unabashed stamp-collecting, on the grounds that we just don’t have a lot of data to systematically understand yet – and for that very reason I won’t try to summarize all the results, but the slides are well worth a look-over. The point is that fusion categories are very useful in constructing TQFT’s, and there are several different constructions that begin “given a fusion category $\mathcal{C}$“… and yet there aren’t all that many examples, and very few large ones, known.

Scott also makes the analogy that fusion categories are “noncommutative finite groups” – which is a little confusing, since not all finite groups are commutative anyway – but the idea is that the symmetric fusion categories are exactly the representation categories of finite groups. So general fusion categories are a non-symmetric generalization of such groups. Since classifying finite groups turned out to be difficult, and involve a laundry-list of sporadic groups, it shouldn’t be too surprising that understanding fusion categories (which, for the symmetric case, include the representation categories of all these examples) should be correspondingly tricky. Since, as he points out, we don’t have very many non-symmetric examples beyond rank 12 (analogous to knowing only finite groups with at most 12 elements), it’s likely that we don’t have a very good understanding of these categories in general yet.

There were a couple of talks – one during the workshop by Sonia Natale, and one the previous week by Sebastian Burciu, whom I also had the chance to talk with that week – about “Equivariantization” of fusion categories, and some fairly detailed descriptions of what results. The two of them have a paper on this which gives more details, which I won’t summarize – but I will say a bit about the construction.

An “equivariantization” of a category $C$ acted on by a group $G$ is supposed to be a generalization of the notion of the set of fixed points for a group acting on a set.  The category $C^G$ has objects which consist of an object $x \in C$ which is fixed by the action of $G$, together with an isomorphism $\mu_g : x \rightarrow x$ for each $g \in G$, satisfying a bunch of unsurprising conditions like being compatible with the group operation. The morphisms are maps in $C$ between the objects, which form commuting squares for each $g \in G$. Their paper, and the talks, described how this works when $C$ is a fusion category – namely, $C^G$ is also a fusion category, and one can work out its fusion rules (i.e. monoidal structure). In some cases, it’s a “group theoretical” fusion category (it looks like $Rep(H)$ for some group $H$) – or a weakened version of such a thing (it’s Morita equivalent to ).

A nice special case of this is if the group action happens to be trivial, so that every object of $C$ is a fixed point. In this case, $C^G$ is just the category of objects of $C$ equipped with a $G$-action, and the intertwining maps between these. For example, if $C = Vect$, then $C^G = Rep(G)$ (in particular, a “group-theoretical fusion category”). What’s more, this construction is functorial in $G$ itself: given a subgroup $H \subset G$, we get an adjoint pair of functors between $C^G$ and $C^H$, which in our special case are just the induced-representation and restricted-representation functors for that subgroup inclusion. That is, we have a Mackey functor here. These generalize, however, to any fusion category $C$, and to nontrivial actions of $G$ on $C$. The point of their paper, then, is to give a good characterization of the categories that come out of these constructions.

Quantizing with Higher Categories

The last talk I’d like to describe was by Urs Schreiber, called Linear Homotopy Type Theory for Quantization. Urs has been giving evolving talks on this topic for some time, and it’s quite a big subject (see the long version of the notes above if there’s any doubt). However, I always try to get a handle on these talks, because it seems to be describing the most general framework that fits the general approach I use in my own work. This particular one borrows a lot from the language of logic (the “linear” in the title alludes to linear logic).

Basically, Urs’ motivation is to describe a good mathematical setting in which to construct field theories using ingredients familiar to the physics approach to “field theory”, namely… fields. (See the description of Kevin Walker’s talk.) Also, Lagrangian functionals – that is, the notion of a physical action. Constructing TQFT from modular tensor categories, for instance, is great, but the fields and the action seem to be hiding in this picture. There are many conceptual problems with field theories – like the mathematical meaning of path integrals, for instance. Part of the approach here is to find a good setting in which to locate the moduli spaces of fields (and the spaces in which path integrals are done). Then, one has to come up with a notion of quantization that makes sense in that context.

The first claim is that the category of such spaces should form a differentially cohesive infinity-topos which we’ll call $\mathbb{H}$. The “infinity” part means we allow morphisms between field configurations of all orders (2-morphisms, 3-morphisms, etc.). The “topos” part means that all sorts of reasonable constructions can be done – for example, pullbacks. The “differentially cohesive” part captures the sort of structure that ensures we can really treat these as spaces of the suitable kind: “cohesive” means that we have a notion of connected components around (it’s implemented by having a bunch of adjoint functors between spaces and points). The “differential” part is meant to allow for the sort of structures discussed above under “differential cohomology” – really, that we can capture geometric structure, as in gauge theories, and not just topological structure.

In this case, we take $\mathbb{H}$ to have objects which are spectral-valued infinity-stacks on manifolds. This may be unfamiliar, but the main point is that it’s a kind of generalization of a space. Now, the sort of situation where quantization makes sense is: we have a space (i.e. $\mathbb{H}$-object) of field configurations to start, then a space of paths (this is WHERE “path-integrals” are defined), and a space of field configurations in the final system where we observe the result. There are maps from the space of paths to identify starting and ending points. That is, we have a span:

$A \leftarrow X \rightarrow B$

Now, in fact, these may all lie over some manifold, such as $B^n(U(1))$, the classifying space for $U(1)$ $(n-1)$-gerbes. That is, we don’t just have these “spaces”, but these spaces equipped with one of those pieces of cohomological twisting data discussed up above. That enters the quantization like an action (it’s WHAT you integrate in a path integral).

Aside: To continue the parallel, quantization is playing the role of a cohomology theory, and the action is the twist. I really need to come back and complete an old post about motives, because there’s a close analogy here. If quantization is a cohomology theory, it should come by factoring through a universal one. In the world of motives, where “space” now means something like “scheme”, the target of this universal cohomology theory is a mild variation on just the category of spans I just alluded to. Then all others come from some functor out of it.

Then the issue is what quantization looks like on this sort of scenario. The Atiyah-Singer viewpoint on TQFT isn’t completely lost here: quantization should be a functor into some monoidal category. This target needs properties which allow it to capture the basic “quantum” phenomena of superposition (i.e. some additivity property), and interference (some actual linearity over $\mathbb{C}$). The target category Urs talked about was the category of $E_{\infty}$-rings. The point is that these are just algebras that live in the world of spectra, which is where our spaces already lived. The appropriate target will depend on exactly what $\mathbb{H}$ is.

But what Urs did do was give a characterization of what the target category should be LIKE for a certain construction to work. It’s a “pull-push” construction: see the link way above on Mackey functors – restriction and induction of representations are an example . It’s what he calls a “(2-monoidal, Beck-Chevalley) Linear Homotopy-Type Theory”. Essentially, this is a list of conditions which ensure that, for the two morphisms in the span above, we have a “pull” operation for some and left and right adjoints to it (which need to be related in a nice way – the jargon here is that we must be in a Wirthmuller context), satisfying some nice relations, and that everything is functorial.

The intuition is that if we have some way of getting a “linear gadget” out of one of our configuration spaces of fields (analogous to constructing a space of functions when we do canonical quantization over, let’s say, a symplectic manifold), then we should be able to lift it (the “pull” operation) to the space of paths. Then the “push” part of the operation is where the “path integral” part comes in: many paths might contribute to the value of a function (or functor, or whatever it may be) at the end-point of those paths, because there are many ways to get from A to B, and all of them contribute in a linear way.

So, if this all seems rather abstract, that’s because the point of it is to characterize very generally what has to be available for the ideas that appear in physics notions of path-integral quantization to make sense. Many of the particulars – spectra, $E_{\infty}$-rings, infinity-stacks, and so on – which showed up in the example are in a sense just placeholders for anything with the right formal properties. So at the same time as it moves into seemingly very abstract terrain, this approach is also supposed to get out of the toy-model realm of TQFT, and really address the trouble in rigorously defining what’s meant by some of the standard practice of physics in field theory by analyzing the logical structure of what this practice is really saying. If it turns out to involve some unexpected math – well, given the underlying issues, it would have been more surprising if it didn’t.

It’s not clear to me how far along this road this program gets us, as far as dealing with questions an actual physicist would like to ask (for the most part, if the standard practice works as an algorithm to produce results, physicists seldom need to ask what it means in rigorous math language), but it does seem like an interesting question.

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!

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

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

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

Categories with Structures

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

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

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

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

Tensor Categories and Field Theories

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

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

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

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

Now, that was all conformal field theory.  Christopher Douglas described some work with Chris Schommer-Pries and Noah Snyder about fusion categories and structured topological field theories.  These are functors out of cobordism categories, the most important of which are $n$-categories, where the objects are points, morphisms are 1D cobordisms, and so on up to $n$-morphisms which are $n$-dimensional cobordisms.  To keep things under control, Chris Douglas talked about the case $Bord_0^3$, which is where $n=3$, and a “local” field theory is a 3-functor $Bord_0^3 \rightarrow \mathcal{C}$ for some 3-category $\mathcal{C}$.  Now, the (Baez-Dolan) Cobordism Hypothesis, which was proved by Jacob Lurie, says that $Bord_0^3$ is, in a suitable sense, the free symmetric monoidal 3-category with duals.  What this amounts to is that a local field theory whose target 3-category is $\mathcal{C}$ is “just” a dualizable object of $\mathcal{C}$.

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

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

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

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

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

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

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

State Sum Models

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

Higher Gauge Theory

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

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

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

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…

So apparently the “Integral” gamma-ray observatory has put some pretty strong limits on predictions of a “grain size” for spacetime, like in Loop Quantum Gravity, or other theories predicting similar violations of Lorentz invariants which would be detectable in higher- and lower-energy photons coming from distant sources.  (Original paper.)  I didn’t actually hear much about such predictions when I was the conference “Quantum Theory and Gravitation” last month in Zurich, though partly that was because it was focused on bringing together people from a variety of different approaches , so the Loop QG and String Theory camps were smaller than at some other conferences on the same subject.  It was a pretty interesting conference, however (many of the slides and such material can be found here).  As one of the organizers, Jürg Fröhlich, observed in his concluding remarks, it gave grounds for optimism about physics, in that it was clear that we’re nowhere near understanding everything about the universe.  Which seems like a good attitude to have to the situation – and it informs good questions: he asked questions in many of the talks that went right to the heart of the most problematic things about each approach.

Often after attending a conference like that, I’d take the time to do a blog about all the talks – which I was tempted to do, but I’ve been busy with things I missed while I was away, and now it’s been quite a while.  I will probably come back at some point and think about the subject of conformal nets, because there were some interesting talks by Andre Henriques at one workshop I was at, and another by Roberto Longo at this one, which together got me interested in this subject.  But that’s not what I’m going to write about this time.

This time, I want to talk about a different kind of topic.  Talking  in Zurich with various people – John Barrett, John Baez, Laurent Freidel, Derek Wise, and some others, on and off – we kept coming back to kept coming back to various seemingly strange algebraic structures.  One such structure is a “loop“, also known (maybe less confusingly) as a “quasigroup” (in fact, a loop is a quasigroup with a unit).  This was especially confusing, because we were talking about these gadgets in the context of gauge theory, where you might want to think about assigning an element of one as the holonomy around a LOOP in spacetime.  Limitations of the written medium being what they are, I’ll just avoid the problem and say “quasigroup” henceforth, although actually I tend to use “loop” when I’m speaking.

The axioms for a quasigroup look just like the axioms for a group, except that the axiom of associativity is missing.  That is, it’s a set with a “multiplication” operation, and each element $x$ has a left and a right inverse, called $x^{\lambda}$ and $x^{\rho}$.  (I’m also assuming the quasigroup is unital from here on in).  Of course, in a group (which is a special kind of quasigroup where associativity holds), you can use associativity to prove that $x^{\lambda} = x^{\rho}$, but we don’t assume it’s true in a quasigroup.  Of course, you can consider the special case where it IS true: this is a “quasigroup with two-sided inverse”, which is a weaker assumption than associativity.

In fact, this is an example of a kind of question one often asks about quasigroups: what are some extra properties we can suppose which, if they hold for a quasigroup $Q$, make life easier?  Associativity is a strong condition to ask, and gives the special case of a group, which is a pretty well-understood area.  So mostly one looks for something weaker than associativity.  Probably the most well-known, among people who know about such things, is the Moufang axiom, named after Ruth Moufang, who did a lot of the pioneering work studying quasigroups.

There are several equivalent ways to state the Moufang axiom, but a nice one is:

$y(x(yz)) = ((yx)y)z$

Which you could derive from the associative law if you had it, but which doesn’t imply associativity.   With associators, one can go from a fully-right-bracketed to a fully-left-bracketed product of four things: $w(x(yz)) \rightarrow (wx)(yz) \rightarrow ((wx)y)z$.  There’s no associator here (a quasigroup is a set, not a category – though categorifying this stuff may be a nice thing to try), but the Moufang axiom says this is an equation when $w=y$.  One might think of the stronger condition that says it’s true for all $(w,x,y,z)$, but the Moufang axiom turns out to be the more handy one.

One way this is so is found in the division algebras.  A division algebra is a (say, real) vector space with a multiplication for which there’s an identity and a notion of division – that is, an inverse for nonzero elements.  We can generalize this enough that we allow different left and right inverses, but in any case, even if we relax this (and the assumption of associativity), it’s a well-known theorem that there are still only four finite dimensional ones.  Namely, they are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, and $\mathbb{O}$: the real numbers, complex numbers, quaternions, and octonions, with real dimensions 1, 2, 4, and 8 respectively.

So the pattern goes like this.  The first two, $\mathbb{R}$ and $\mathbb{C}$, are commutative and associative.  The quaternions $\mathbb{H}$ are noncommutative, but still associative.  The octonions $\mathbb{O}$ are neither commutative nor associative.  They also don’t satisfy that stronger axiom $w(x(yz)) = ((wx)y)z$.  However, the octonions do satisfy the Moufang axiom.  In each case, you can get a quasigroup by taking the nonzero elements – or, using the fact that there’s a norm around in the usual way of presenting these algebras, the elements of unit norm.  The unit quaternions, in fact, form a group – specifically, the group $SU(2)$.  The unit reals and complexes form abelian groups (respectively, $\mathbb{Z}_2$, and $U(1)$).  These groups all have familiar names.  The quasigroup of unit octonions doesn’t have any other more familiar name.  If you believe in the fundamental importance of this sequence of four division algebras, though, it does suggest that a natural sequence in which to weaken axioms for “multiplication” goes: commutative-and-associative, associative, Moufang.

The Moufang axiom does imply some other commonly suggested weakenings of associativity, as well.  For instance, a quasigroup that satisfies the Moufang axiom must also be alternative (a restricted form of associativity when two copies of one element are next to each other: i.e. the left alternative law $x(xy) = (xx)y$, and right alternative law $x(yy) = (xy)y$).

Now, there are various ways one could go with this; the one I’ll pick is toward physics.  The first three entries in that sequence of four division algebras – and the corresponding groups – all show up all over the place in physics.  $\mathbb{Z}_2$ is the simplest nontrivial group, so this could hardly fail to be true, but at any rate, it appears as, for instance, the symmetry group of the set of orientations on a manifold, or the grading in supersymmetry (hence plays a role distinguishing bosons and fermions), and so on.  $U(1)$ is, among any number of other things, the group in which action functionals take their values in Lagrangian quantum mechanics; in the Hamiltonian setup, it’s the group of phases that characterizes how wave functions evolve in time.  Then there’s $SU(2)$, which is the (double cover of the) group of rotations of 3-space; as a consequence, its representation theory classifies the “spins”, or angular momenta, that a quantum particle can have.

What about the octonions – or indeed the quasigroup of unit octonions?  This is a little less clear, but I will mention this: John Baez has been interested in octonions for a long time, and in Zurich, gave a talk about what kind of role they might play in physics.  This is supposed to partially explain what’s going on with the “special dimensions” that appear in string theory – these occur where the dimension of a division algebra (and a Clifford algebra that’s associated to it) is the same as the codimension of a string worldsheet.  J.B.’s student, John Huerta, has also been interested in this stuff, and spoke about it here in Lisbon in February – it’s the subject of his thesis, and a couple of papers they’ve written.  The role of the octonions here is not nearly so well understood as elsewhere, and of course whether this stuff is actually physics, or just some interesting math that resembles it, is open to experiment – unlike those other examples, which are definitely physics if anything is!

So at this point, the foregoing sets us up to wonder about two questions.  First: are there any quasigroups that are actually of some intrinsic interest which don’t satisfy the Moufang axiom?  (This might be the next step in that sequence of successively weaker axioms).  Second: are there quasigroups that appear in genuine, experimentally tested physics?  (Supposing you don’t happen to like the example from string theory).

Well, the answer is yes on both counts, with one common example – a non-Moufang quasigroup which is of interest precisely because it has a direct physical interpretation.  This example is the composition of velocities in Special Relativity, and was pointed out to me by Derek Wise as a nice physics-based example of nonassociativity.  That it’s also non-Moufang is also true, and not too surprising once you start trying to check it by a direct calculation: in each case, the reason is that the interpretation of composition is very non-symmetric.  So how does this work?

Well, if we take units where the speed of light is 1, then Special Relativity tells us that relative velocities of two observers are vectors in the interior of $B_1(0) \subset \mathbb{R}^3$.  That is, they’re 3-vectors with length less than 1, since the magnitude of the relative velocity must be less than the speed of light.  In any elementary course on Relativity, you’d learn how to compose these velocities, using the “gamma factor” that describes such things as time-dilation.  This can be derived from first principles, nor is it too complicated, but in any case the end result is a new “addition” for vectors:

$\mathbf{v} \oplus_E \mathbf{u} = \frac{ \mathbf{v} + \mathbf{u}_{\parallel} + \alpha_{\mathbf{v}} \mathbf{u}_{\perp}}{1 + \mathbf{v} \cdot \mathbf{u}}$

where $\alpha_{\mathbf{v}} = \sqrt{1 - \mathbf{v} \cdot \mathbf{v}}$  is the reciprocal of the aforementioned “gamma” factor.  The vectors $\mathbf{u}_{\parallel}$ and $\mathbf{u}_{\perp}$ are the components of the vector $\mathbf{u}$ which are parallel to, and perpendicular to, $\mathbf{v}$, respectively.

The way this is interpreted is: if $\mathbf{v}$ is the velocity of observer B as measured by observer A, and $\mathbb{u}$ is the velocity of observer C as measured by observer B, then $\mathbf{v} \oplus_E \mathbf{u}$ is the velocity of observer C as measured by observer A.

Clearly, there’s an asymmetry in how $\mathbf{v}$ and $\mathbf{u}$ are treated: the first vector, $\mathbf{v}$, is a velocity as seen by the same observer who sees the velocity in the final answer.  The second, $\mathbf{u}$, is a velocity as seen by an observer who’s vanished by the time we have $\mathbf{v} \oplus_e \mathbf{u}$ in hand.  Just looking at the formular, you can see this is an asymmetric operation that distinguishes the left and right inputs.  So the fact (slightly onerous, but not conceptually hard, to check) that it’s noncommutative, and indeed nonassociative, and even non-Moufang, shouldn’t come as a big shock.

The fact that it makes $B_1(0)$ into a quasigroup is a little less obvious, unless you’ve actually worked through the derivation – but from physical principles, $B_1(0)$ is closed under this operation because the final relative velocity will again be less than the speed of light.  The fact that this has “division” (i.e. cancellation), is again obvious enough from physical principles: if we have $\mathbf{v} \oplus _E \mathbf{u}$, the relative velocity of A and C, and we have one of $\mathbf{v}$ or $\mathbf{u}$ – the relative velocity of B to either A or C – then the relative velocity of B to the other one of these two must exist, and be findable using this formula.  That’s the “division” here.

So in fact this non-Moufang quasigroup, exotic-sounding algebraic terminology aside, is one that any undergraduate physics student will have learned about and calculated with.

One point that Derek was making in pointing this example out to me was as a comment on a surprising claim someone (I don’t know who) had made, that mathematical abstractions like “nonassociativity” don’t really appear in physics.  I find the above a pretty convincing case that this isn’t true.

In fact, physics is full of Lie algebras, and the Lie bracket is a nonassociative multiplication (except in trivial cases).  But I guess there is an argument against this: namely, people often think of a Lie algebra as living inside its universal enveloping algebra.  Then the Lie bracket is defined as $[x,y] = xy - yx$, using the underlying (associative!) multiplication.  So maybe one can claim that nonassociativity doesn’t “really” appear in physics because you can treat it as a derived concept.

An even simpler example of this sort of phenomenon: the integers with subtraction (rather than addition) are nonassociative, in that $x-(y-z) \neq (x-y)-z$.  But this only suggests that subtraction is the wrong operation to use: it was derived from addition, which of course is commutative and associative.

In which case, the addition of velocities in relativity is also a derived concept.  Because, of course, really in SR there are no 3-space “velocities”: there are tangent vectors in Minkowski space, which is a 4-dimensional space.  Adding these vectors in $\mathbb{R}^4$ is again, of course, commutative and associative.  The concept of “relative velocity” of two observers travelling along given vectors is a derived concept which gets its strange properties by treating the two arguments asymmetrically, just like like “commutator” and “subtraction” do: you first use one vector to decide on a way of slicing Minkowski spacetime into space and time, and then use this to measure the velocity of the other.

Even the octonions, seemingly the obvious “true” example of nonassociativity, could be brushed aside by someone who really didn’t want to accept any example: they’re constructed from the quaternions by the Cayley-Dickson construction, so you can think of them as pairs of quaternions (or 4-tuples of complex numbers).  Then the nonassociative operation is built from associative ones, via that construction.

So are there any “real” examples of “true” nonassociativity (let alone non-Moufangness) that can’t simply be dismissed as not a fundamental operation by someone sufficiently determined?  Maybe, but none I know of right now.  It may be quite possible to consistently hold that anything nonassociative can’t possibly be fundamental (for that matter, elements of noncommutative groups can be represented by matrices of commuting real numbers).  Maybe it’s just my attitude to fundamentals, but somehow this doesn’t move me much.  Even if there are no “fundamentals” examples, I think those given above suggest a different point: these derived operations have undeniable and genuine meaning – in some cases more immediate than the operations they’re derived from.  Whether or not subtraction, or the relative velocity measured by observers, or the bracket of (say) infinitesimal rotations, are “fundamental” ideas is less important than that they’re practical ones that come up all the time.

As usual, this write-up process has been taking a while since life does intrude into blogging for some reason.  In this case, because for a little less than a week, my wife and I have been on our honeymoon, which was delayed by our moving to Lisbon.  We went to the Azores, or rather to São Miguel, the largest of the nine islands.  We had a good time, roughly like so:

Now that we’re back, I’ll attempt to wrap up with the summaries of things discussed at the workshop on Higher Gauge Theory, TQFT, and Quantum Gravity.  In the previous post I described talks which I roughly gathered under TQFT and Higher Gauge Theory, but the latter really ramifies out in a few different ways.  As began to be clear before, higher bundles are classified by higher cohomology of manifolds, and so are gerbes – so in fact these are two slightly different ways of talking about the same thing.  I also remarked, in the summary of Konrad Waldorf’s talk, the idea that the theory of gerbes on a manifold is equivalent to ordinary gauge theory on its loop space – which is one way to make explicit the idea that categorification “raises dimension”, in this case from parallel transport of points to that of 1-dimensional loops.  Next we’ll expand on that theme, and then finally reach the “Quantum Gravity” part, and draw the connection between this and higher gauge theory toward the end.

## Gerbes and Cohomology

The very first workshop speaker, in fact, was Paolo Aschieri, who has done a lot of work relating noncommutative geometry and gravity.  In this case, though, he was talking about noncommutative gerbes, and specifically referred to this work with some of the other speakers.  To be clear, this isn’t about gerbes with noncommutative group $G$, but about gerbes on noncommutative spaces.  To begin with, it’s useful to express gerbes in the usual sense in the right language.  In particular, he explain what a gerbe on a manifold $X$ is in concrete terms, giving Hitchin’s definition (viz).  A $U(1)$ gerbe can be described as “a cohomology class” but it’s more concrete to present it as:

• a collection of line bundles $L_{\alpha \beta}$ associated with double overlaps $U_{\alpha \beta} = U_{\alpha} \cap U_{\beta}$.  Note this gets an algebraic structure (multiplication $\star$ of bundles is pointwise $\otimes$, with an inverse given by the dual, $L^{-1} = L^*$, so we can require…
• $L_{\alpha \beta}^{-1} \cong L_{\beta \alpha}$, which helps define…
• transition functions $\lambda _{\alpha \beta \gamma}$ on triple overlaps $U_{\alpha \beta \gamma}$, which are sections of $L_{\alpha \beta \gamma} = L_{\alpha \beta} \star L_{\beta \gamma} \star L_{\gamma \alpha}$.  If this product is trivial, there’d be a 1-cocycle condition here, but we only insist on the 2-cocycle condition…
• $\lambda_{\beta \gamma \delta} \lambda_{\alpha \gamma \delta}^{-1} \lambda_{\alpha \beta \delta} \lambda_{\alpha \beta \gamma}^{-1} = 1$

This is a $U(1)$-gerbe on a commutative space.  The point is that one can make a similar definition for a noncommutative space.  If the space $X$ is associated with the algebra $A=C^{\infty}(X)$ of smooth functions, then a line bundle is a module for $A$, so if $A$ is noncommutative (thought of as a “space” $X$), a “bundle over $X$ is just defined to be an $A$-module.  One also has to define an appropriate “covariant derivative” operator $D$ on this module, and the $\star$-product must be defined as well, and will be noncommutative (we can think of it as a deformation of the $\star$ above).  The transition functions are sections: that is, elements of the modules in question.  his means we can describe a gerbe in terms of a big stack of modules, with a chosen algebraic structure, together with some elements.  The idea then is that gerbes can give an interpretation of cohomology of noncommutative spaces as well as commutative ones.

Mauro Spera spoke about a point of view of gerbes based on “transgressions”.  The essential point is that an $n$-gerbe on a space $X$ can be seen as the obstruction to patching together a family of  $(n-1)$-gerbes.  Thus, for instance, a $U(1)$ 0-gerbe is a $U(1)$-bundle, which is to say a complex line bundle.  As described above, a 1-gerbe can be understood as describing the obstacle to patching together a bunch of line bundles, and the obstacle is the ability to find a cocycle $\lambda$ satisfying the requisite conditions.  This obstacle is measured by the cohomology of the space.  Saying we want to patch together $(n-1)$-gerbes on the fibre.  He went on to discuss how this manifests in terms of obstructions to string structures on manifolds (already discussed at some length in the post on Hisham Sati’s school talk, so I won’t duplicate here).

A talk by Igor Bakovic, “Stacks, Gerbes and Etale Groupoids”, gave a way of looking at gerbes via stacks (see this for instance).  The organizing principle is the classification of bundles by the space maps into a classifying space – or, to get the category of principal $G$-bundles on, the category $Top(Sh(X),BG)$, where $Sh(X)$ is the category of sheaves on $X$ and $BG$ is the classifying topos of $G$-sets.  (So we have geometric morphisms between the toposes as the objects.)  Now, to get further into this, we use that $Sh(X)$ is equivalent to the category of Étale spaces over $X$ – this is a refinement of the equivalence between bundles and presheaves.  Taking stalks of a presheaf gives a bundle, and taking sections of a bundle gives a presheaf – and these operations are adjoint.

The issue at hand is how to categorify this framework to talk about 2-bundles, and the answer is there’s a 2-adjunction between the 2-category $2-Bun(X)$ of such things, and $Fib(X) = [\mathcal{O}(X)^{op},Cat]$, the 2-category of fibred categories over $X$.  (That is, instead of looking at “sheaves of sets”, we look at “sheaves of categories” here.)  The adjunction, again, involves talking stalks one way, and taking sections the other way.  One hard part of this is getting a nice definition of “stalk” for stacks (i.e. for the “sheaves of categories”), and a good part of the talk focused on explaining how to get a nice tractable definition which is (fibre-wise) equivalent to the more natural one.

Bakovic did a bunch of this work with Branislav Jurco, who was also there, and spoke about “Nonabelian Bundle 2-Gerbes“.  The paper behind that link has more details, which I’ve yet to entirely absorb, but the essential point appears to be to extend the description of “bundle gerbes” associated to crossed modules up to 2-crossed modules.  Bundles, with a structure-group $G$, are classified by the cohomology $H^1(X,G)$ with coefficients in $G$; and whereas “bundle-gerbes” with a structure-crossed-module $H \rightarrow G$ can likewise be described by cohomology $H^1(X,H \rightarrow G)$.  Notice this is a bit different from the description in terms of higher cohomology $H^2(X,G)$ for a $G$-gerbe, which can be understood as a bundle-gerbe using the shifted crossed module $G \rightarrow 1$ (when $G$ is abelian.  The goal here is to generalize this part to nonabelian groups, and also pass up to “bundle 2-gerbes” based on a 2-crossed module, or crossed complex of length 2, $L \rightarrow H \rightarrow G$ as I described previously for Joao Martins’ talk.  This would be classified in terms of cohomology valued in the 2-crossed module.  The point is that one can describe such a thing as a bundle over a fibre product, which (I think – I’m not so clear on this part) deals with the same structure of overlaps as the higher cohomology in the other way of describing things.

Finally,  a talk that’s a little harder to classify than most, but which I’ve put here with things somewhat related to string theory, was Alexander Kahle‘s on “T-Duality and Differential K-Theory”, based on work with Alessandro Valentino.  This uses the idea of the differential refinement of cohomology theories – in this case, K-theory, which is a generalized cohomology theory, which is to say that K-theory satisfies the Eilenberg-Steenrod axioms (with the dimension axiom relaxed, hence “generalized”).  Cohomology theories, including generalized ones, can have differential refinements, which pass from giving topological to geometrical information about a space.  So, while K-theory assigns to a space the Grothendieck ring of the category of vector bundles over it, the differential refinement of K-theory does the same with the category of vector bundles with connection.  This captures both local and global structures, which turns out to be necessary to describe fields in string theory – specifically, Ramond-Ramond fields.  The point of this talk was to describe what happens to these fields under T-duality.  This is a kind of duality in string theory between a theory with large strings and small strings.  The talk describes how this works, where we have a manifold with fibres at each point $M\times S^1_r$ with fibres strings of radius $r$ and $M \times S^1_{1/r}$ with radius $1/r$.  There’s a correspondence space $M \times S^1_r \times S^1_{1/r}$, which has projection maps down into the two situations.  Fields, being forms on such a fibration, can be “transferred” through this correspondence space by a “pull-back and push-forward” (with, in the middle, a wedge with a form that mixes the two directions, $exp( d \theta_r + d \theta_{1/r})$).  But to be physically the right kind of field, these “forms” actually need to be representing cohomology classes in the differential refinement of K-theory.

## Quantum Gravity etc.

Now, part of the point of this workshop was to try to build, or anyway maintain, some bridges between the kind of work in geometry and topology which I’ve been describing and the world of physics.  There are some particular versions of physical theories where these ideas have come up.  I’ve already touched on string theory along the way (there weren’t many talks about it from a physicist’s point of view), so this will mostly be about a different sort of approach.

Benjamin Bahr gave a talk outlining this approach for our mathematician-heavy audience, with his talk on “Spin Foam Operators” (see also for instance this paper).  The point is that one approach to quantum gravity has a theory whose “kinematics” (the description of the state of a system at a given time) is described by “spin networks” (based on $SU(2)$ gauge theory), as described back in the pre-school post.  These span a Hilbert space, so the “dynamical” issue of such models is how to get operators between Hilbert spaces from “foams” that interpolate between such networks – that is, what kind of extra data they might need, and how to assign amplitudes to faces and edges etc. to define an operator, which (assuming a “local” theory where distant parts of the foam affect the result independently) will be of the form:

$Z(K,\rho,P) = (\prod_f A_f) \prod_v Tr_v(\otimes P_e)$

where $K$ is a particular complex (foam), $\rho$ is a way of assigning irreps to faces of the foam, and $P$ is the assignment of intertwiners to edges.  Later on, one can take a discrete version of a path integral by summing over all these $(K, \rho, P)$.  Here we have a product over faces and one over vertices, with an amplitude $A_f$ assigned (somehow – this is the issue) to faces.  The trace is over all the representation spaces assigned to the edges that are incident to a vertex (this is essentially the only consistent way to assign an amplitude to a vertex).  If we also consider spacetimes with boundary, we need some amplitudes $B_e$ at the boundary edges, as well.  A big part of the work with such models is finding such amplitudes that meet some nice conditions.

Some of these conditions are inherently necessary – to ensure the theory is invariant under gauge transformations, or (formally) changing orientations of faces.  Others are considered optional, though to me “functoriality” (that the way of deriving operators respects the gluing-together of foams) seems unavoidable – it imposes that the boundary amplitudes have to be found from the $A_f$ in one specific way.  Some other nice conditions might be: that $Z(K, \rho, P)$ depends only on the topology of $K$ (which demands that the $P$ operators be projections); that $Z$ is invariant under subdivision of the foam (which implies the amplitudes have to be $A_f = dim(\rho_f)$).

Assuming all these means the only choice is exactly which sub-projection $P_e$ is of the projection onto the gauge-invariant part of the representation space for the faces attached to edge $e$.  The rest of the talk discussed this, including some examples (models for BF-theory, the Barrett-Crane model and the more recent EPRL/FK model), and finished up by discussing issues about getting a nice continuum limit by way of “coarse graining”.

On a related subject, Bianca Dittrich spoke about “Dynamics and Diffeomorphism Symmetry in Discrete Quantum Gravity”, which explained the nature of some of the hard problems with this sort of discrete model of quantum gravity.  She began by asking what sort of models (i.e. which choices of amplitudes) in such discrete models would actually produce a nice continuum theory – since gravity, classically, is described in terms of spacetimes which are continua, and the quantum theory must look like this in some approximation.  The point is to think of these as “coarse-graining” of a very fine (perfect, in the limit) approximation to the continuum by a triangulation with a very short length-scale for the edges.  Coarse graining means discarding some of the edges to get a coarser approximation (perhaps repeatedly).  If the $Z$ happens to be triangulation-independent, then coarse graining makes no difference to the result, nor does the converse process of refining the triangulation.  So one question is:  if we expect the continuum limit to be diffeomorphism invariant (as is General Relativity), what does this say at the discrete level?  The relation between diffeomorphism invariance and triangulation invariance has been described by Hendryk Pfeiffer, and in the reverse direction by Dittrich et al.

Actually constructing the dynamics for a system like this in a nice way (“canonical dynamics with anomaly-free constraints”) is still a big problem, which Bianca suggested might be approached by this coarse-graining idea.  Now, if a theory is topological (here we get the link to TQFT), such as electromagnetism in 2D, or (linearized) gravity in 3D, coarse graining doesn’t change much.  But otherwise, changing the length scale means changing the action for the continuum limit of the theory.  This is related to renormalization: one starts with a “naive” guess at a theory, then refines it (in this case, by the coarse-graining process), which changes the action for the theory, until arriving at (or approximating to) a fixed point.  Bianca showed an example, which produces a really huge, horrible action full of very complicated terms, which seems rather dissatisfying.  What’s more, she pointed out that, unless the theory is topological, this always produces an action which is non-local – unlike the “naive” discrete theory.  That is, the action can’t be described in terms of a bunch of non-interacting contributions from the field at individual points – instead, it’s some function which couples the field values at distant points (albeit in a way that falls off exponentially as the points get further apart).

In a more specific talk, Aleksandr Mikovic discussed “Finiteness and Semiclassical Limit of EPRL-FK Spin Foam Models”, looking at a particular example of such models which is the (relatively) new-and-improved candidate for quantum gravity mentioned above.  This was a somewhat technical talk, which I didn’t entirely follow, but  roughly, the way he went at this was through the techniques of perturbative QFT.  That is, by looking at the theory in terms of an “effective action”, instead of some path integral over histories $\phi$ with action $S(\phi)$ – which looks like $\int d\phi e^{iS(\phi)}$.  Starting with some classical history $\bar{\phi}$ – a stationary point of the action $S$ – the effective action $\Gamma(\bar{\phi})$ is an integral over small fluctuations $\phi$ around it of $e^{iS(\bar{\phi} + \phi)}$.

He commented more on the distinction between the question of triangulation independence (which is crucial for using spin foams to give invariants of manifolds) and the question of whether the theory gives a good quantum theory of gravity – that’s the “semiclassical limit” part.  (In light of the above, this seems to amount to asking if “diffeomorphism invariance” really extends through to the full theory, or is only approximately true, in the limiting case).  Then the “finiteness” part has to do with the question of getting decent asymptotic behaviour for some of those weights mentioned above so as to give a nice effective action (if not necessarily triangulation independence).  So, for instance, in the Ponzano-Regge model (which gives a nice invariant for manifolds), the vertex amplitudes $A_v$ are found by the 6j-symbols of representations.  The asymptotics of the 6j symbols then becomes an issue – Alekandr noted that to get a theory with a nice effective action, those 6j-symbols need to be scaled by a certain factor.  This breaks triangulation independence (hence means we don’t have a good manifold invariant), but gives a physically nicer theory.  In the case of 3D gravity, this is not what we want, but as he said, there isn’t a good a-priori reason to think it can’t give a good theory of 4D gravity.

Now, making a connection between these sorts of models and higher gauge theory, Aristide Baratin spoke about “2-Group Representations for State Sum Models”.  This is a project Baez, Freidel, and Wise, building on work by Crane and Sheppard (see my previous post, where Derek described the geometry of the representation theory for some 2-groups).  The idea is to construct state-sum models where, at the kinematical level, edges are labelled by 2-group representations, faces by intertwiners, and tetrahedra by 2-intertwiners.  (This assumes the foam is a triangulation – there’s a certain amount of back-and-forth in this area between this, and the Poincaré dual picture where we have 4-valent vertices).  He discussed this in a couple of related cases – the Euclidean and Poincaré 2-groups, which are described by crossed modules with base groups $SO(4)$ or $SO(3,1)$ respectively, acting on the abelian group (of automorphisms of the identity) $R^4$ in the obvious way.  Then the analogy of the 6j symbols above, which are assigned to tetrahedra (or dually, vertices in a foam interpolating two kinematical states), are now 10j symbols assigned to 4-simplexes (or dually, vertices in the foam).

One nice thing about this setup is that there’s a good geometric interpretation of the kinematics – irreducible representations of these 2-groups pick out orbits of the action of the relevant $SO$ on $R^4$.  These are “mass shells” – radii of spheres in the Euclidean case, or proper length/time values that pick out hyperboloids in the Lorentzian case of $SO(3,1)$.  Assigning these to edges has an obvious geometric meaning (as a proper length of the edge), which thus has a continuous spectrum.  The areas and volumes interpreting the intertwiners and 2-intertwiners start to exhibit more of the discreteness you see in the usual formulation with representations of the $SO$ groups themselves.  Finally, Aristide pointed out that this model originally arose not from an attempt to make a quantum gravity model, but from looking at Feynman diagrams in flat space (a sort of “quantum flat space” model), which is suggestively interesting, if not really conclusively proving anything.

Finally, Laurent Freidel gave a talk, “Classical Geometry of Spin Network States” which was a way of challenging the idea that these states are exclusively about “quantum geometries”, and tried to give an account of how to interpret them as discrete, but classical.  That is, the quantization of the classical phase space $T^*(A/G)$ (the cotangent bundle of connections-mod-gauge) involves first a discretization to a spin-network phase space $\mathcal{P}_{\Gamma}$, and then a quantization to get a Hilbert space $H_{\Gamma}$, and the hard part is the first step.  The point is to see what the classical phase space is, and he describes it as a (symplectic) quotient $T^*(SU(2)^E)//SU(2)^V$, which starts by assigning $T^*(SU(2))$ to each edge, then reduced by gauge transformations.  The puzzle is to interpret the states as geometries with some discrete aspect.

The answer is that one thinks of edges as describing (dual) faces, and vertices as describing some polytopes.  For each $p$, there’s a $2(p-3)$-dimensional “shape space” of convex polytopes with $p$-faces and a given fixed area $j$.  This has a canonical symplectic structure, where lengths and interior angles at an edge are the canonically conjugate variables.  Then the whole phase space describes ways of building geometries by gluing these things (associated to vertices) together at the corresponding faces whenever the two vertices are joined by an edge.  Notice this is a bit strange, since there’s no particular reason the faces being glued will have the same shape: just the same area.  An area-1 pentagon and an area-1 square associated to the same edge could be glued just fine.  Then the classical geometry for one of these configurations is build of a bunch of flat polyhedra (i.e. with a flat metric and connection on them).  Measuring distance across a face in this geometry is a little strange.  Given two points inside adjacent cells, you measure orthogonal distance to the matched faces, and add in the distance between the points you arrive at (orthogonally) – assuming you glued the faces at the centre.  This is a rather ugly-seeming geometry, but it’s symplectically isomorphic to the phase space of spin network states – so it’s these classical geometries that spin-foam QG is a quantization of.  Maybe the ugliness should count against this model of quantum gravity – or maybe my aesthetic sense just needs work.

(Laurent also gave another talk, which was originally scheduled as one of the school talks, but ended up being a very interesting exposition of the principle of “Relativity of Localization”, which is hard to shoehorn into the themes I’ve used here, and was anyway interesting enough that I’ll devote a separate post to it.)

Now for a more sketchy bunch of summaries of some talks presented at the HGTQGR workshop.  I’ll organize this into a few themes which appeared repeatedly and which roughly line up with the topics in the title: in this post, variations on TQFT, plus 2-group and higher forms of gauge theory; in the next post, gerbes and cohomology, plus talks on discrete models of quantum gravity and suchlike physics.

## TQFT and Variations

I start here for no better reason than the personal one that it lets me put my talk first, so I’m on familiar ground to start with, for which reason also I’ll probably give more details here than later on.  So: a TQFT is a linear representation of the category of cobordisms – that is, a (symmetric monoidal) functor $nCob \rightarrow Vect$, in the notation I mentioned in the first school post.  An Extended TQFT is a higher functor $nCob_k \rightarrow k-Vect$, representing a category of cobordisms with corners into a higher category of k-Vector spaces (for some definition of same).  The essential point of my talk is that there’s a universal construction that can be used to build one of these at $k=2$, which relies on some way of representing $nCob_2$ into $Span(Gpd)$, whose objects are groupoids, and whose morphisms in $Hom(A,B)$ are pairs of groupoid homomorphisms $A \leftarrow X \rightarrow B$.  The 2-morphisms have an analogous structure.  The point is that there’s a 2-functor $\Lambda : Span(Gpd) \rightarrow 2Vect$ which is takes representations of groupoids, at the level of objects; for morphisms, there is a “pull-push” operation that just uses the restricted and induced representation functors to move a representation across a span; the non-trivial (but still universal) bit is the 2-morphism map, which uses the fact that the restriction and induction functors are bi-ajdoint, so there are units and counits to use.  A construction using gauge theory gives groupoids of connections and gauge transformations for each manifold or cobordism.  This recovers a form of the Dijkgraaf-Witten model.  In principle, though, any way of getting a groupoid (really, a stack) associated to a space functorially will give an ETQFT this way.  I finished up by suggesting what would need to be done to extend this up to higher codimension.  To go to codimension 3, one would assign an object (codimension-3 manifold) a 3-vector space which is a representation 2-category of 2-groupoids of connections valued in 2-groups, and so on.  There are some theorems about representations of n-groupoids which would need to be proved to make this work.

The fact that different constructions can give groupoids for spaces was used by the next speaker, Thomas Nicklaus, whose talk described another construction that uses the $\Lambda$ I mentioned above.  This one produces “Equivariant Dijkgraaf-Witten Theory”.  The point is that one gets groupoids for spaces in a new way.  Before, we had, for a space $M$ a groupoid $\mathcal{A}_G(M)$ whose objects are $G$-connections (or, put another way, bundles-with-connection) and whose morphisms are gauge transformations.  Now we suppose that there’s some group $J$ which acts weakly (i.e. an action defined up to isomorphism) on $\mathcal{A}_G(M)$.  We think of this as describing “twisted bundles” over $M$.  This is described by a quotient stack $\mathcal{A}_G // J$ (which, as a groupoid, gets some extra isomorphisms showing where two objects are related by the $J$-action).  So this gives a new map $nCob \rightarrow Span(Gpd)$, and applying $\Lambda$ gives a TQFT.  The generating objects for the resulting 2-vector space are “twisted sectors” of the equivariant DW model.  There was some more to the talk, including a description of how the DW model can be further mutated using a cocycle in the group cohomology of $G$, but I’ll let you look at the slides for that.

Next up was Jamie Vicary, who was talking about “(1,2,3)-TQFT”, which is another term for what I called “Extended” TQFT above, but specifying that the objects are 1-manifolds, the morphisms 2-manifolds, and the 2-morphisms are 3-manifolds.  He was talking about a theorem that identifies oriented TQFT’s of this sort with “anomaly-free modular tensor categories” – which is widely believed, but in fact harder than commonly thought.  It’s easy enough that such a TQFT $Z$ corresponds to a MTC – it’s the category $Z(S^1)$ assigned to the circle.  What’s harder is showing that the TQFT’s are equivalent functors iff the categories are equivalent.  This boils down, historically, to the difficulty of showing the category is rigid.  Jamie was talking about a project with Bruce Bartlett and Chris Schommer-Pries, whose presentation of the cobordism category (described in the school post) was the basis of their proof.

Part of it amounts to giving a description of the TQFT in terms of certain string diagrams.  Jamie kindly credited me with describing this point of view to him: that the codimension-2 manifolds in a TQFT can be thought of as “boundaries in space” – codimension-1 manifolds are either time-evolving boundaries, or else slices of space in which the boundaries live; top-dimension cobordisms are then time-evolving slices of space-with-boundary.  (This should be only a heuristic way of thinking – certainly a generic TQFT has no literal notion of “time-evolution”, though in that (2+1) quantum gravity can be seen as a TQFT, there’s at least one case where this picture could be taken literally.)  Then part of their proof involves showing that the cobordisms can be characterized by taking vector spaces on the source and target manifolds spanned by the generating objects, and finding the functors assigned to cobordisms in terms of sums over all “string diagrams” (particle worldlines, if you like) bounded by the evolving boundaries.  Jamie described this as a “topological path integral”.  Then one has to describe the string diagram calculus – ridigidy follows from the “yanking” rule, for instance, and this follows from Morse theory as in Chris’ presentation of the cobordism category.

There was a little more discussion about what the various properties (proved in a similar way) imply.  One is “cloaking” – the fact that a 2-morphism which “creates a handle” is invisible to the string diagrams in the sense that it introduces a sum over all diagrams with a string “looped” around the new handle, but this sum gives a result that’s equal to the original map (in any “pivotal” tensor category, as here).

Chronologically before all these, one of the first talks on such a topic was by Rafael Diaz, on Homological Quantum Field Theory, or HLQFT for short, which is a rather different sort of construction.  Remember that Homotopy QFT, as described in my summary of Tim Porter’s school sessions, is about linear representations of what I’ll for now call $Cob(d,B)$, whose morphisms are $d$-dimensional cobordisms equipped with maps into a space $B$ up to homotopy.  HLQFT instead considers cobordisms equipped with maps taken up to homology.

Specifically, there’s some space $M$, say a manifold, with some distinguished submanifolds (possibly boundary components; possibly just embedded submanifolds; possibly even all of $M$ for a degenerate case).  Then we define $Cob_d^M$ to have objects which are $(d-1)$-manifolds equipped with maps into $M$ which land on the distinguished submanifolds (to make composition work nicely, we in fact assume they map to a single point).  Morphisms in $Cob_d^M$ are trickier, and look like $(N,\alpha, \xi)$: a cobordism $N$ in this category is likewise equipped with a map $\alpha$ from its boundary into $M$ which recovers the maps on its objects.  That $\xi$ is a homology class of maps from $N$ to $M$, which agrees with $\alpha$.  This forms a monoidal category as with standard cobordisms.  Then HLQFT is about representations of this category.  One simple case Rafael described is the dimension-1 case, where objects are (ordered sets of) points equipped with maps that pick out chosen submanifolds of $M$, and morphisms are just braids equipped with homology classes of “paths” joining up the source and target submanifolds.  Then a representation might, e.g., describe how to evolve a homology class on the starting manifold to one on the target by transporting along such a path-up-to-homology.  In higher dimensions, the evolution is naturally more complicated.

A slightly looser fit to this section is the talk by Thomas Krajewski, “Quasi-Quantum Groups from Strings” (see this) – he was talking about how certain algebraic structures arise from “string worldsheets”, which are another way to describe cobordisms.  This does somewhat resemble the way an algebraic structure (Frobenius algebra) is related to a 2D TQFT, but here the string worldsheets are interacting with 3-form field, $H$ (the curvature of that 2-form field $B$ of string theory) and things needn’t be topological, so the result is somewhat different.

Part of the point is that quantizing such a thing gives a higher version of what happens for quantizing a moving particle in a gauge field.  In the particle case, one comes up with a line bundle (of which sections form the Hilbert space) and in the string case one comes up with a gerbe; for the particle, this involves associated 2-cocycle, and for the string a 3-cocycle; for the particle, one ends up producing a twisted group algebra, and for the string, this is where one gets a “quasi-quantum group”.  The algebraic structures, as in the TQFT situation, come from, for instance, the “pants” cobordism which gives a multiplication and a comultiplication (by giving maps $H \otimes H \rightarrow H$ or the reverse, where $H$ is the object assigned to a circle).

There is some machinery along the way which I won’t describe in detail, except that it involves a tricomplex of forms – the gradings being form degree, the degree of a cocycle for group cohomology, and the number of overlaps.  As observed before, gerbes and their higher versions have transition functions on higher numbers of overlapping local neighborhoods than mere bundles.  (See the paper above for more)

## Higher Gauge Theory

The talks I’ll summarize here touch on various aspects of higher-categorical connections or 2-groups (though at least one I’ll put off until later).  The division between this and the section on gerbes is a little arbitrary, since of course they’re deeply connected, but I’m making some judgements about emphasis or P.O.V. here.

Apart from giving lectures in the school sessions, John Huerta also spoke on “Higher Supergroups for String Theory”, which brings “super” (i.e. $\mathbb{Z}_2$-graded) objects into higher gauge theory.  There are “super” versions of vector spaces and manifolds, which decompose into “even” and “odd” graded parts (a.k.a. “bosonic” and “fermionic” parts).  Thus there are “super” variants of Lie algebras and Lie groups, which are like the usual versions, except commutation properties have to take signs into account (e.g. a Lie superalgebra’s bracket is commutative if the product of the grades of two vectors is odd, anticommutative if it’s even).  Then there are Lie 2-algebras and 2-groups as well – categories internal to this setting.  The initial question has to do with whether one can integrate some Lie 2-algebra structures to Lie 2-group structures on a spacetime, which depends on the existence of some globally smooth cocycles.  The point is that when spacetime is of certain special dimensions, this can work, namely dimensions 3, 4, 6, and 10.  These are all 2 more than the real dimensions of the four real division algebras, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$.  It’s in these dimensions that Lie 2-superalgebras can be integrated to Lie 2-supergroups.  The essential reason is that a certain cocycle condition will hold because of the properties of a form on the Clifford algebras that are associated to the division algebras.  (John has some related material here and here, though not about the 2-group case.)

Since we’re talking about higher versions of Lie groups/algebras, an important bunch of concepts to categorify are those in representation theory.  Derek Wise spoke on “2-Group Representations and Geometry”, based on work with Baez, Baratin and Freidel, most fully developed here, but summarized here.  The point is to describe the representation theory of Lie 2-groups, in particular geometrically.  They’re to be represented on (in general, infinite-dimensional) 2-vector spaces of some sort, which is chosen to be a category of measurable fields of Hilbert spaces on some measure space, which is called $H^X$ (intended to resemble, but not exactly be the same as, $Hilb^X$, the space of “functors into $Hilb$ from the space $X$, the way Kapranov-Voevodsky 2-vector spaces can be described as $Vect^k$).  The first work on this was by Crane and Sheppeard, and also Yetter.  One point is that for 2-groups, we have not only representations and intertwiners between them, but 2-intertwiners between these.  One can describe these geometrically – part of which is a choice of that measure space $(X,\mu)$.

This done, we can say that a representation of a 2-group is a 2-functor $\mathcal{G} \rightarrow H^X$, where $\mathcal{G}$ is seen as a one-object 2-category.  Thinking about this geometrically, if we concretely describe $\mathcal{G}$ by the crossed module $(G,H,\rhd,\partial)$, defines an action of $G$ on $X$, and a map $X \rightarrow H^*$ into the character group, which thereby becomes a $G$-equivariant bundle.  One consequence of this description is that it becomes possible to distinguish not only irreducible representations (bundles over a single orbit) and indecomposible ones (where the fibres are particularly simple homogeneous spaces), but an intermediate notion called “irretractible” (though it’s not clear how much this provides).  An intertwining operator between reps over $X$ and $Y$ can be described in terms of a bundle of Hilbert spaces – which is itself defined over the pullback of $X$ and $Y$ seen as $G$-bundles over $H^*$.  A 2-intertwiner is a fibre-wise map between two such things.  This geometric picture specializes in various ways for particular examples of 2-groups.  A physically interesting one, which Crane and Sheppeard, and expanded on in that paper of [BBFW] up above, deals with the Poincaré 2-group, and where irreducible representations live over mass-shells in Minkowski space (or rather, the dual of $H \cong \mathbb{R}^{3,1}$).

Moving on from 2-group stuff, there were a few talks related to 3-groups and 3-groupoids.  There are some new complexities that enter here, because while (weak) 2-categories are all (bi)equivalent to strict 2-categories (where things like associativity and the interchange law for composing 2-cells hold exactly), this isn’t true for 3-categories.  The best strictification result is that any 3-category is (tri)equivalent to a Gray category – where all those properties hold exactly, except for the interchange law $(\alpha \circ \beta) \cdot (\alpha ' \circ \beta ') = (\alpha \cdot \alpha ') \circ (\beta \circ \beta ')$ for horizontal and vertical compositions of 2-cells, which is replaced by an “interchanger” isomorphism with some coherence properties.  John Barrett gave an introduction to this idea and spoke about “Diagrams for Gray Categories”, describing how to represent morphisms, 2-morphisms, and 3-morphisms in terms of higher versions of “string” diagrams involving (piecewise linear) surfaces satisfying some properties.  He also carefully explained how to reduce the dimensions in order to make them both clearer and easier to draw.  Bjorn Gohla spoke on “Mapping Spaces for Gray Categories”, but since it was essentially a shorter version of a talk I’ve already posted about, I’ll leave that for now, except to point out that it linked to the talk by Joao Faria Martins, “3D Holonomy” (though see also this paper with Roger Picken).

The point in Joao’s talk starts with the fact that we can describe holonomies for 3-connections on 3-bundles valued in Gray-groups (i.e. the maximally strict form of a general 3-group) in terms of Gray-functors $hol: \Pi_3(M) \rightarrow \mathcal{G}$.  Here, $\Pi_3(M)$ is the fundamental 3-groupoid of $M$, which turns points, paths, homotopies of paths, and homotopies of homotopies into a Gray groupoid (modulo some technicalities about “thin” or “laminated”  homotopies) and $\mathcal{G}$ is a gauge Gray-group.  Just as a 2-group can be represented by a crossed module, a Gray (3-)group can be represented by a “2-crossed module” (yes, the level shift in the terminology is occasionally confusing).  This is a chain of groups $L \stackrel{\delta}{\rightarrow} E \stackrel{\partial}{\rightarrow} G$, where $G$ acts on the other groups, together with some structure maps (for instance, the Peiffer commutator for a crossed module becomes a lifting $\{ ,\} : E \times E \rightarrow L$) which all fit together nicely.  Then a tri-connection can be given locally by forms valued in the Lie algebras of these groups: $(\omega , m ,\theta)$ in  $\Omega^1 (M,\mathfrak{g} ) \times \Omega^2 (M,\mathfrak{e}) \times \Omega^3(M,\mathfrak{l})$.  Relating the global description in terms of $hol$ and local description in terms of $(\omega, m, \theta)$ is a matter of integrating forms over paths, surfaces, or 3-volumes that give the various $j$-morphisms of $\Pi_3(M)$.  This sort of construction of parallel transport as functor has been developed in detail by Waldorf and Schreiber (viz. these slides, or the full paper), some time ago, which is why, thematically, they’re the next two speakers I’ll summarize.

Konrad Waldorf spoke about “Abelian Gauge Theories on Loop Spaces and their Regression”.  (For more, see two papers by Konrad on this)  The point here is that there is a relation between two kinds of theories – string theory (with $B$-field) on a manifold $M$, and ordinary $U(1)$ gauge theory on its loop space $LM$.  The relation between them goes by the name “regression” (passing from gauge theory on $LM$ to string theory on $M$), or “transgression”, going the other way.  This amounts to showing an equivalence of categories between [principal $U(1)$-bundles with connection on $LM$] and [$U(1)$-gerbes with connection on $M$].  This nicely gives a way of seeing how gerbes “categorify” bundles, since passing to the loop space – whose points are maps $S^1 \rightarrow M$ means a holonomy functor is now looking at objects (points in $LM$) which would be morphisms in the fundamental groupoid of $M$, and morphisms which are paths of loops (surfaces in $M$ which trace out homotopies).  So things are shifted by one level.  Anyway, Konrad explained how this works in more detail, and how it should be interpreted as relating connections on loop space to the $B$-field in string theory.

Urs Schreiber kicked the whole categorification program up a notch by talking about $\infty$-Connections and their Chern-Simons Functionals .  So now we’re getting up into $\infty$-categories, and particularly $\infty$-toposes (see Jacob Lurie’s paper, or even book if so inclined to find out what these are), and in particular a “cohesive topos”, where derived geometry can be developed (Urs suggested people look here, where a bunch of background is collected). The point is that $\infty$-topoi are good for talking about homotopy theory.  We want a setting which allows all that structure, but also allows us to do differential geometry and derived geometry.  So there’s a “cohesive” $\infty$-topos called $Smooth\infty Gpds$, of “sheaves” (in the $\infty$-topos sense) of $\infty$-groupoids on smooth manifolds.  This setting is the minimal common generalization of homotopy theory and differential geometry.

This is about a higher analog of this setup: since there’s a smooth classifying space (in fact, a Lie groupoid) for $G$-bundles, $BG$, there’s also an equivalence between categories $G-Bund$ of $G$-principal bundles, and $SmoothGpd(X,BG)$ (of functors into $BG$).  Moreover, there’s a similar setup with $BG_{conn}$ for bundles with connection.  This can be described topologically, or there’s also a “differential refinement” to talk about the smooth situation.  This equivalence lives within a category of (smooth) sheaves of groupoids.  For higher gauge theory, we want a higher version as in $Smooth \infty Gpds$ described above.  Then we should get an equivalence – in this cohesive topos – of $hom(X,B^n U(1))$ and a category of $U(1)$-$(n-1)$-gerbes.

Then the part about the  “Chern-Simons functionals” refers to the fact that CS theory for a manifold (which is a kind of TQFT) is built using an action functional that is found as an integral of the forms that describe some $U(1)$-connection over the manifold.  (Then one does a path-integral of this functional over all connections to find partition functions etc.)  So the idea is that for these higher $U(1)$-gerbes, whose classifying spaces we’ve just described, there should be corresponding functionals.  This is why, as Urs remarked in wrapping up, this whole picture has an explicit presentation in terms of forms.  Actually, in terms of Cech-cocycles (due to the fact we’re talking about gerbes), whose coefficients are taken in sheaves of complexes (this is the derived geometry part) of differential forms whose coefficients are in $L_\infty$-algebroids (the $\infty$-groupoid version of Lie algebras, since in general we’re talking about a theory with gauge $\infty$-groupoids now).

Whew!  Okay, that’s enough for this post.  Next time, wrapping up blogging the workshop, finally.

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

## Higher Gauge Theory

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

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

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

Actually making this precise is a lot of work when $M$ is a general 2-space – even defining open covers and setting up all the machinery properly is quite hard.  This has been done, by Toby Bartels in his thesis, but to keep things simple, John restricted his talk to the case where $M$ is just an ordinary manifold (thought of as a 2-space which has only identity morphisms).   Then a key point is that there’s an analog to how (principal) $G$-bundles (where $F \cong G$ as a $G$-set) are classified up to isomorphism by the first Cech cohomology of the manifold, $\check{H}^1(M,G)$.  This works because one can define transition functions on double overlaps $U_{ij} := U_i \cap U_j$, by $g_{ij} = f_i f_j^{-1}$.  Then these $g_{ij}$ will automatically satisfy the 1-cocycle condidion ($g_{ij} g_{jk} = g_{ik}$ on the triple overlap $U_{ijk}$) which means they represent a cohomology class $[g] = \in \check{H}^1(M,G)$.

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

Finally, having defined bundles, one usually defines connections, and so we do the same with 2-bundles.  A connection on a bundle gives a parallel transport operation for paths $\gamma$ in $M$, telling how to identify the fibres at points along $\gamma$ by means of a functor $hol : P_1(M) \rightarrow G$, thinking of $G$ as a category with one object, and where $P_1(M)$ is the path groupoid whose objects are points in $M$ and whose morphisms are paths (up to “thin” homotopy). At least, it does so once we trivialize the bundle around $\gamma$, anyway, to think of it as $M \times G$ locally – in general we need to get the transition functions involved when we pass into some other local neighborhood.  A connection on a 2-bundle is similar, but tells how to parallel transport fibres not only along paths, but along homotopies of paths, by means of $hol : P_2(M) \rightarrow \mathcal{G}$, where $\mathcal{G}$ is seen as a 2-category with one object, and $P_2(M)$ now has 2-morphisms which are (essentially) homotopies of paths.

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

## Infinite Dimensional Lie Theory and Higher Gauge Theory

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

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

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

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

Now, finding central extensions of Lie algebras is comparatively easy, and given a central extension of Lie groups, one always falls out of the induced maps.  There will be an exact sequence of Lie algebras, and now the special condition is that there must exist a continuous section of the second map.  The question is to go the other way: given one of these, get back to an extension of Lie groups.  The problem of finding extensions of $G$ by $Z$, in particular as a problem of finding a bundle with connection having specified curvature, which brings us back to gauge theory.  One type of extension is the universal cover of $G$, which appears as $\pi_1(G) \hookrightarrow \hat{G} \twoheadrightarrow G$, so that the fibre is $\pi_1(G)$.

In general, whether an extension can exist comes down to a question about a cocycle: that is, if there’s a function $f : G \times G \rightarrow Z$ which is locally smooth (i.e. in some neighborhood in $G$), and is a cocyle (so that $f(g,h) + f(gh,k) = f(g,hk) + f(h,k)$), by the same sorts of arguments we’ve already seen a bit of.  For this reason, central extensions are classified by the cohomology group $H^2(G,Z)$.  The cocycle enables a “twisting” of the multiplication associated to a nontrivial loop in $G$, and is used to construct $\hat{G}$ (by specifying how multiplication on $G$ lifts to $\hat{G}$).  Given a  2-cocycle $\omega$ at the Lie algebra level (easier to do), one would like to lift that up the Lie group.  It turns out this is possible if the period homomorphism $per_{\omega} : \Pi_2(G) \rightarrow Z$ – which takes a chain $[\sigma]$ (with $\sigma : S^2 \rightarrow G$) to the integral of the original cocycle on it, $\int_{\sigma} \omega$ – lands in a discrete subgroup of $Z$. A popular example of this is when $Z$ is just $\mathbb{R}$, and the discrete subgroup is $\mathbb{Z}$ (or, similarly, $U(1)$ and $1$ respectively).  This business of requiring a cocycle to be integral in this way is sometimes called a “prequantization” problem.

So suppose we wanted to make the “2-connected cover” $\pi_2(G) \hookrightarrow \pi_2(G) \times_{\gamma} G \twoheadrightarrow G$ as a central extension: since $\pi_2(G)$ will be abelian, this is conceivable.  If the dimension of $G$ is finite, this is trivial (since $\pi_2(G) = 0$ in finite dimensions), which is why we need some theory  of infinite-dimensional manifolds.  Moreover, though, this may not work in the context of groups: the $\gamma$ in the extension $\pi_2(G) \times_{\gamma} G$ above needs to be a “twisting” of associativity, not multiplication, being lifted from $G$.  Such twistings come from the THIRD cohomology of $G$ (see here, e.g.), and describe the structure of 2-groups (or crossed modules, whichever you like).  In fact, the solution (go read the paper for more if you like) to define a notion of central extension for 2-groups (essentially the same as the usual definition, but with maps of 2-groups, or crossed modules, everywhere).  Since a group is a trivial kind of 2-group (with only trivial automorphisms of any element), the usual notion of central extension turns out to be a special case.  Then by thinking of $\pi_2(G)$ and $G$ as crossed modules, one can find a central extension which is like the 2-connected cover we wanted – though it doesn’t work as an extension of groups because we think of $G$ as the base group of the crossed module, and $\pi_2(G)$ as the second group in the tower.

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

## Higher Spin Structures in String Theory

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

The basic point here is that, for physical reasons, we’re often interested in putting some sort of structure on a manifold, which is really best described in terms of a bundle.  For instance, a connection or spin connection on spacetime lets us transport vectors or spinors, respectively, along paths, which in turn lets us define derivatives.  These two structures really belong on vector bundles or spinor bundles.  Now, if these bundles are trivial, then one can make the connections on them trivial as well by gauge transformation.  So having nontrivial bundles really makes this all more interesting.  However, this isn’t always possible, and so one wants to the obstruction to being able to do it.  This is typically a class in one of the cohomology groups of the manifold – a characteristic class.  There are various examples: Chern classes, Pontrjagin classes, Steifel-Whitney classes, and so on, each of which comes in various degrees $i$.  Each one corresponds to a different coefficient group for the cohomology groups – in these examples, the groups $U$ and $O$ which are the limits of the unitary and orthogonal groups (such as $O := O(\infty) \supset \dots \supset O(2) \supset O(1)$)

The point is that these classes are obstructions to building certain structures on the manifold $X$ – which amounts to finding sections of a bundle.  So for instance, the first Steifel-Whitney classes, $w_1(E)$ of a bundle $E$ are related to orientations, coming from cohomology with coefficients in $O(n)$.  Orientations for the manifold $X$ can be described in terms of its tangent bundle, which is an $O(n)$-bundle (tangent spaces carry an action of the rotation group).  Consider $X = S^1$, where we have actually $O(1) \simeq \mathbb{Z}_2$.  The group $H^1(S^1, \mathbb{Z}_2)$ has two elements, and there are two types of line bundle on the circle $S^1$: ones with a nowhere-zero section, like the trivial bundle; and ones without, like the Moebius strip.  The circle is orientable, because its tangent bundle is of the first sort.

Generally, an orientation can be put on $X$ if the tangent bundle, as a map $f : X \rightarrow B(O(n))$, can be lifted to a map $\tilde{f} : X \rightarrow B(SO(n))$ – that is, it’s “secretly” an $SO(n)$-bundle – the special orthogonal group respects orientation, which is what the determinant measures.  Its two values, $\pm 1$, are what’s behind the two classes of bundles.  (In short, this story relates to the exact sequence $1 \rightarrow SO(n) \rightarrow O(n) \stackrel{det}{\rightarrow} O(1) = \mathbb{Z}_2 \rightarrow 1$; in just the same way we have big groups $SO$, $Spin$, and so forth.)

So spin structures have a story much like the above, but where the exact sequence $1 \rightarrow \mathbb{Z}_2 \rightarrow Spin(n) \rightarrow SO(n) \rightarrow 1$ plays a role – the spin groups are the universal covers (which are all double-sheeted covers) of the special rotation groups.  A spin structure on some $SO(n)$ bundle $E$, let’s say represented by $f : X \rightarrow B(SO(n))$ is thus, again, a lifting to $\tilde{f} : X \rightarrow B(Spin(n))$.  The obstruction to doing this (the thing which must be zero for the lifting to exist) is the second Stiefel-Whitney class, $w_2(E)$.  Hisham Sati also explained the example of “generalized” spin structures in these terms.  But the main theme is an analogous, but much more general, story for other cohomology groups as obstructions to liftings of some sort of structures on manifolds.  These may be bundles, for the lower-degree cohomology, or they may be gerbes or n-bundles, for higher-degree, but the setup is roughly the same.

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

This turns out to bring up another theme, which is that of special dimensions – it’s just not true that the same phenomena happen in every dimension.  In this case, this has to do with the homotopy groups  – of $O$ and its cousins.  So it turns out that the homotopy group $\pi_k(O)$ (which is the same as $\pi_k(O_n)$ as long as $n$ is bigger than $k$) follows a pattern, where $\pi_k(O) = \mathbb{Z}_2$ if $k = 0,1 (mod 8)$, and $\pi_k(O) = \mathbb{Z}$ if $k = 3,7 (mod 8)$.  The fact that this pattern repeats mod-8 is one form of the (real) Bott Periodicity theorem.  These homotopy groups reflect that, wherever there’s nontrivial homotopy in some dimension, there’s an obstruction to contracting maps into $O$ from such a sphere.

All of this plays into the question of what kinds of nontrivial structures can be put on orthogonal bundles on manifolds of various dimensions.  In the dimensions where these homotopy groups are non-trivial, there’s an obstruction to the lifting, and therefore some interesting structure one can put on $X$ which may or may not exist.  Hisham Sati spoke of “killing” various homotopy groups – meaning, as far as I can tell, imposing conditions which get past these obstructions.  In string theory, his application of interest, one talks of “anomaly cancellation” – an anomaly being the obstruction to making these structures.  The first part of the punchline is that, since these are related to nontrivial cohomology groups, we can think of them in terms of defining structures on n-bundles or gerbes.  These structures are, essentially, connections – they tell us how to parallel-transport objects of various dimensions.  It turns out that the $\pi_k$ homotopy group is related to parallel transport along $(k-1)$-dimensional surfaces in $X$, which can be thought of as the world-sheets of $(k-2)$-dimensional “particles” (or rather, “branes”).

So, for instance, the fact that $\pi_1(O)$ is nontrivial means there’s an obstruction to a lifting in the form of a class in $H^2(X,\mathbb{Z})$, which has to do with spin structure – as above.  “Cancelling” this “anomaly” means that for a theory involving such a spin structure to be well-defined, then this characteristic class for $X$ must be zero.  The fact that $\pi_3(O) = \mathbb{Z}$ is nontrivial means there’s an obstruction to a lifting in the form of a class in $H^4(X, \mathbb{Z})$.  This has to do with “string bundles”, where the string group is a higher analog of $Spin$ in exactly the sense we’ve just described.  If such a lifting exists, then there’s a “string-structure” on $X$ which is compatible with the spin structure we lifted (and with the orientation a level below that).  Similarly, $\pi_7(O) = \mathbb{Z}$ being nontrivial, by way of an obstruction in $H^8$, means there’s an interesting notion of “five-brane” structure, and a $Fivebrane$ group, and so on.  Personally, I think of these as giving a geometric interpretation for what the higher cohomology groups actually mean.

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

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

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

I’d like to continue describing the talks that made up the HGTQGR workshop, in particular the ones that took place during the school portion of the event.  I’ll save one “school” session, by Laurent Freidel, to discuss with the talks because it seems to more nearly belong there. This leaves five people who gave between two and four lectures each over a period of a few days, all intermingled. Here’s a very rough summary in the order of first appearance:

## 2D Extended TQFT

Chris Schommer-Pries gave the longest series of talks, about the classification of 2D extended TQFT’s.  A TQFT is a kind of topological invariant for manifolds, which has a sort of “locality” property, in that you can decompose the manifold, compute the invariant on the parts, and find the whole by gluing the pieces back together.  This is expressed by saying it’s a monoidal functor $Z : (Cob_d, \sqcup) \rightarrow (Vect, \otimes)$, where the “locality” property is now functoriality property that composition is preserved.  The key thing here is the cobordism category $Cob_d$, which has objects (d-1)-dimensional manifolds, and morphisms d-dimensional cobordisms (manifolds with boundary, where the objects are components of the boundary).  Then a closed d-manifold is just a cobordism from $latex\emptyset$ to itself.

Making this into a category is actually a bit nontrivial: gluing bits of smooth manifolds, for instance, won’t necessarily give something smooth.  There are various ways of handling this, such as giving the boundaries “collars”, but Chris’ preferred method is to give boundaries (and, ultimately, corners, etc.) a”halation”.  This word originally means the halo of light around bright things you sometimes see in photos, but in this context, a halation for $X$ is an equivalence class of embeddings into neighborhoods $U \subset \mathbb{R}^d$.  The equivalence class says two such embeddings into $U$ and $V$ are equivalent if there’s a compatible refinement into some common $W$ that embeds into both $U$ and $V$.  The idea is that a halation is a kind of d-dimensional “halo”, or the “germ of a d-manifold” around $X$.  Then gluing compatibly along (d-1)-boundaries with halations ensures that we get smooth d-manifolds.  (One can also extend this setup so that everything in sight is oriented, or has some other such structure on it.)

In any case, an extended TQFT will then mean an n-functor $Z : (Bord_d,\sqcup) \rightarrow (\mathcal{C},\otimes)$, where $(\mathcal{C},\otimes)$ is some symmetric monoidal n-category (which is supposed to be similar to $Vect$).  Its exact nature is less important than that of $Bord_d$, which has:

• 0-Morphisms (i.e. Objects): 0-manifolds (collections of points)
• 1-Morphisms: 1-dimensional cobordisms between 0-manifolds (curves)
• 2-Morphisms: 2-dim cobordisms with corners between 1-Morphisms (surfaces with boundary)
• d-Morphisms: d-dimensional cobordisms between (d-1)-Morphisms (n-manifolds with corners), up to isomorphism

(Note: the distinction between “Bord” and “Cobord” is basically a matter of when a given terminology came in.  “Cobordism” and “Bordism”, unfortunately, mean the same thing, except that “bordism” has become popular more recently, since the “co” makes it sound like it’s the opposite category of something else.  This is kind of regrettable, but that’s what happened.  Sorry.)

The crucial point, is that Chris wanted to classify all such things, and his approach to this is to give a presentation of $Bord_d$.  This is based on stuff in his thesis.  The basic idea is to use Morse theory, and its higher-dimensional generalization, Cerf theory.  The idea is that one can put a Morse function  on a cobordism (essentially, a well-behaved “time order” on points) and look at its critical points.  Classifying these tells us what the generators for the category of cobordisms must be: there need to be enough to capture all the most general sorts of critical points.

Cerf theory does something similar, but one dimension up: now we’re talking about “stratified” families of Morse functions.  Again one studies critical points, but, for instance, on a 2-dim surface, there can be 1- and 0-dimensional parts of the set of cricical points.  In general, this gets into the theory of higher-dimensional singularities, catastrophe theory, and so on.  Each extra dimension one adds means looking at how the sets of critical points in the previous dimension can change over “time” (i.e. within some stratified family of Cerf functions).  Where these changes themselves go through critical points, one needs new generators for the various j-morphisms of the cobordism category.  (See some examples of such “catastrophes”, such as folds, cusps, swallowtails, etc. linked from here, say.)  Showing what such singularities can be like in the “generic” situation, and indeed, even defining “generic” in a way that makes sense in any dimension, required some discussion of jet bundles.  These are generalizations of tangent bundles that capture higher derivatives the way tangent bundles capture first-derivatives.  The essential point is that one can find a way to decompose these into a direct sum of parts of various dimensions (capturing where various higher derivatives are zero, say), and these will eventually tell us the dimension of a set of critical points for a Cerf function.

Now, this gives a characterization of what cobordisms can be like – part of the work in the theorem is to show that this is sufficient: that is, given a diagram showing the critical points for some Morse/Cerf function, one needs to be able to find the appropriate generators and piece together the cobordism (possibly a closed manifold) that it came from.  Chris showed how this works – a slightly finicky process involving cutting a diagram of the singular points (with some extra labelling information) into parts, and using a graphical calculus to work out how pasting works – and showed an example reconstruction of a surface this way.  This amounts to a construction of an equivalence between an “abstract” cobordism category given in terms of generators (and relations) which come from Cerf theory, and the concrete one.  The theorem then says that there’s a correspondence between equivalence classes of 2D cobordisms, and certain planar diagrams, up to some local moves.  To show this properly required a digression through some theory of symmetric monoidal bicategories, and what the right notion of equivalence for them is.

This all done, the point is that $Bord_d$ has a characterization in terms of a universal property, and so any ETQFT $Z : Bord_d \rightarrow \mathcal{C}$ amounts to a certain kind of object in $\mathcal{C}$ (corresponding to the image of the point – the generating object in $Bord_d$).  For instance, in the oriented situation this object needs to be “fully dualizable”: it should have a dual (the point with opposite orientation), and a whole bunch of maps that specify the duality: a cobordism from $(+,-)$ to nothing (just the “U”-shaped curve), which has a dual – and some 2-D cobordisms which specify that duality, and so on.  Specifying all this dualizability structure amounts to giving the image of all the generators of cobordisms, and determines the functors $Z$, and vice versa.

This is a rapid summary of six hours of lectures, of course, so for more precise versions of these statements, you may want to look into Chris’ thesis as linked above.

## Homotopy QFT and the Crossed Menagerie

The next series of lectures in the school was Tim Porter’s, about relations between Homotopy Quantum Field Theory (HQFT) and various sort of crossed gizmos.  HQFT is an idea introduced by Vladimir Turaev, (see his paper with Tim here, for an intro, though Turaev also now has a book on the subject).  It’s intended to deal with similar sorts of structures to TQFT, but with various sorts of extra structure.  This structure is related to the “Crossed Menagerie”, on which Tim has written an almost unbelievably extensive bunch of lecture notes, of which a special short version was made for this lecture series that’s a mere 350 pages long.

Anyway, the cobordism category $Bord_d$ described above is replaced by one Tim called $HCobord(d,B)$ (see above comment about “bord” and “cobord”, which mean the same thing).  Again, this has d-dimensional cobordisms as its morphisms and (d-1)-dimensional manifolds as its objects, but now everything in sight is equipped with a map into a space $B$ – almost.  So an object is $X \rightarrow B$, and a morphism is a cobordism with a homotopy class of maps $M \rightarrow B$ which are compatible with the ones at the boundaries.  Then just as a d-TQFT is a representation (i.e. a functor) of $Cob_d$ into $Vect$, a $(d,B)$-HQFT is a representation of $HCobord(d,B)$.

The motivating example here is when $B = B(G)$, the classifying space of a group.  These spaces are fairly complicated when you describe them as built from gluing cells (in homotopy theory, one typically things of spaces as something like CW-complexes: a bunch of cells in various dimensions glued together with face maps etc.), but $B(G)$ has the property that its fundamental group is $G$, and all other homotopy groups are trivial (ensuring this part is what makes the cellular decomposition description tricky).

The upshot is that there’s a correspondence between (homotopy classes of) maps $Map(X ,B(G)) \simeq Hom(\pi(X),G)$ (this makes a good alternative definition of the classifying space, though one needs to ).  Since a map from the fundamental group into $G$ amounts to a flat principal $G$-bundle, we can say that $HCobord(d,B(G))$ is a category of manifolds and cobordisms carrying such a bundle.  This gets us into gauge theory.

But we can go beyond and into higher gauge theory (and other sorts of structures) by picking other sorts of $B$.  To begin with, notice that the correspondence above implies that mapping into $B(G)$ means that when we take maps up to homotopy, we can only detect the fundamental group of $X$, and not any higher homotopy groups.  We say we can only detect the “homotopy 1-type” of the space.  The “homotopy n-type” of a given space $X$ is just the first $n$ homotopy groups $(\pi_1(X), \dots, \pi_n(X))$.  Alternatively, an “n-type” is an equivalence class of spaces which all have the same such groups.  Or, again, an “n-type” is a particular representative of one of these classes where these are the only nonzero homotopy groups.

The point being that if we’re considering maps $X \rightarrow B$ up to homotopy, we may only be detecting the n-type of $X$ (and therefore may as well assume $X$ is an n-type in the last sense when it’s convenient).  More precisely, there are “Postnikov functors” $P_n(-)$ which take a space $X$ and return the corresponding n-type.  This can be done by gluing in “patches” of higher dimensions to “fill in the holes” which are measured by the higher homotopy groups (in general, the result is infinite dimensional as a cell complex).  Thus, there are embeddings $X \hookrightarrow P_n(X)$, which get along with the obvious chain

$\dots \rightarrow P_{n+1}(X) \rightarrow P_n(X) \rightarrow P_{n-1}(X) \rightarrow \dots$

There was a fairly nifty digression here explaining how this is a “coskeleton” of $X$, in that $P_n$ is a right adjoint to the “n-skeleton” functor (which throws away cells above dimension n, not homotopy groups), so that $S(Sk_n(M),X) \cong S(M,P_n(X))$.  To really explain it properly, though I would have to really explain what that $S$ is (it refers to maps in the category of simplicial sets, which are another nice model of spaces up to homotopy).  This digression would carry us away from higher gauge theory, which is where I’m going.

One thing to say is that if $X$ is d-dimensional, then any HQFT is determined entirely by the d-type of $B$.  Any extra jazz going on in $B$‘s higher homotopy groups won’t be detected when we’re only mapping a d-dimensional space $X$ into it.  So one might as well assume that $B$ is just a d-type.

We want to say we can detect a homotopy n-type of a space if, for example, $B = B(\mathcal{G})$ where $\mathcal{G}$ is an “n-group”.  A handy way to account for this is in terms of a “crossed complex”.  The first nontrivial example of this would be a crossed module, which consists of

• Two groups, $G$ and $H$ with
• A map $\partial : H \rightarrow G$ and
• An action of $G$ on $H$ by automorphisms, $G \rhd H$
• all such that action looks as much like conjugation as possible:
• $\partial(g \rhd h) = g (\partial h) g^{-1}$ (so that $\partial$ is $G$-equivariant)
• $\partial h \rhd h' = h h' h^{-1}$ (the “Peiffer identity”)

This definition looks a little funny, but it does characterize “2-groups” in the sense of categories internal to $\mathbf{Groups}$ (the definition used elsewhere), by taking $G$ to be the group of objects, and $H$ the group of automorphisms of the identity of $G$.  In the description of John Huerta’s lectures, I’ll get back to how that works.

The immediate point is that there are a bunch of natural examples of crossed modules.  For instance: from normal subgroups, where $\partial: H \subset G$ is inclusion and the action really is conjugation; from fibrations, using fundamental groups of base and fibre; from a canonical case where $H = Aut(G)$  and $\partial = 1$ takes everything to the identity; from modules, taking $H$ to be a $G$-module as an abelian group and $\partial = 1$ again.  The first and last give the classical intuition of these guys: crossed modules are simultaneous generalizations of (a) normal subgroups of $G$, and (b) $G$-modules.

There are various other examples, but the relevant thing here is a theorem of MacLane and Whitehead, that crossed modules model all connected homotopy 2-types.  That is, there’s a correspondence between crossed modules up to isomorphism and 2-types.  Of course, groups model 1-types: any group is the fundmental group for a 1-type, and any 1-type is the classifying space for some group.  Likewise, any crossed module determines a 2-type, and vice versa.  So this theorem suggests why crossed modules might deserve to be called “2-groups” even if they didn’t naturally line up with the alternative definition.

To go up to 3-types and 4-types, the intuitive idea is: “do for crossed modules what we did for groups”.  That is, instead of a map of groups $\partial : H \rightarrow G$, we consider a map of crossed modules (which is given by a pair of maps between the groups in each) and so forth.  The resulting structure is a square diagram in $\mathbf{Groups}$ with a bunch of actions.  Each of these maps is the $\partial$ map for a crossed module.  (We can think of the normal subgroup situation: there are two normal subgroups $H,K$ of $G$, and in each of them, the intersection $H \cap K$ is normal, so it determines a crossed module).  This is a “crossed square”, and things like this correspond exactly to homotopy 3-types.  This works roughly as before, since there is a notion of a classifying space $B(\mathcal{G})$ where $\mathcal{G} = (G,H,\partial,\rhd)$, and similarly on for crossed n-cubes.   We can carry on in this way to define a “crossed n-cube”, which correspond to homotopy (n+1)-types.  The correspondence is a little bit more fiddly than it was for groups, but it still exists: any (n+1)-type is the classifying space for a crossed n-cube, and any such crossed n-cube has an (n+1)-type for its classifying space.

This correspondence is the point here.  As we said, when looking at HQFT’s from $HCobord(d,B)$, we may as well assume that $B$ is a d-type.  But then, it’s a classifying space for some crossed (d-1)-cube.  This is a sensible sort of $B$ to use in an HQFT, and it ends up giving us a theory which is related to higher gauge theory: a map $X \rightarrow B(\mathcal{G})$ up to homotopy, where $\mathcal{G}$ is a crossed n-cube will correspond to the structure of a flat $(n+1)$-bundle on $X$, and similarly for cobordisms.  HQFT’s let us look at the structure of this structured cobordism category by means of its linear representations.  Now, it may be that this crossed-cube point of view isn’t the best way to look at $B$, but it is there, and available.

To say more about this, I’ll have to talk more directly about higher gauge theory in its own terms – which I’ll do in part IIb, since this is already pretty long.

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

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

### Pre-School

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

Quantum Gravity

Aleksandr’s first talk set out the basic problem of quantizing the gravitational field (this isn’t the only attitude to what the problem of quantum gravity is, but it’s a good starting point), starting with the basic ingredients.  He summarized how general relativity describes gravity in terms of a metric $g_{\mu \nu}$ which is supposed to satisfy the Einstein equation, relating the curvature of the metric to a source field $T_{\mu \nu}$ which comes from matter.  Quantization then, starting from a classical picture involving trajectories of particles (or sections of fibre bundles to describe fields), one gets a picture where states are vectors in a Hilbert space, and there’s an algebra of operators including observables (self-adjoint operators) and time-evolution (hermitian ones).   An initial try at quantum gravity was to do this using the metric as the field, using the methods of perturbative QFT: treating the metric in terms of “small” fluctuations from some background metric like the flat Minkowski metric.  This uses the Einstein-Hilbert action $S=\frac{1}{G} \int \sqrt{det(g)}R$, where $G$ is the gravitational constant and $R$ is the Ricci scalar that summarizes the curvature of $g$.  This runs into problems: things diverge in various calculations, and since the coupling constant $G$ has units, one can’t “renormalize” the divergences away.  So one needs a non-perturbative approach,  one of which is “canonical quantization“.

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

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

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

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

String Theory

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

To begin with, he introduced the basic tools of “classical” string theory – the analog of classical mechanics of point particles.  This is the string analog of the following: one can describe a moving particle by its worldline – a path $x : \mathbb{R} \rightarrow M^{(D)}$ from a “generic” worldline into a ($D$-dimensional) manifold $M^{(D)}$.  This $M^{(D)}$ is generally taken to be like physical spacetime, which in this context means that it has a metric $g$ with signature $(-1,1,\dots,1)$ (that is, locally there’s a basis for tangent spaces with one timelike vector and $D-1$ spacelike ones).  Then one can define an action for a moving particle which is just determined by the length of the line’s image.  The nicest way to say this is $S[x] = m \int d\tau \sqrt{x*g}$, where $x*g$ means the pullback of the metric along the map $x$, $\tau$ is some parameter along the generic worldline, and $m$, the particle’s mass, is a coupling constant which doesn’t happen to affect the result in this simple case, but eventually becomes important.  One can do the usual variational-calculus of the Lagrangian approach here, finding a critical point of the action occurs when the particle is travelling in a geodesic – a straight line, in flat space, or the closest available approximation.  In paritcular, the Euler-Lagrange equations say that the covariant derivative of the path should be zero.

There’s an analogous action for a string, the Nambu-Goto action.  Instead of a single-parameter $x$, we now have an embedding of a “generic string worldsheet” – let’s say $\Sigma^{(2)} \cong S^1 \times \mathbb{R}$ into spacetime: $x : \Sigma^{(2)} \rightarrow M^{(D)}$.  Then then the analogous action is just $S[x] = \int_{\Sigma^{(2)}} \star_{x*g} 1$.  This is pretty much the same as before: we pull back the metric to get $x*g$, and integrate over the generic worldsheet.  A slight subtlety comes because we’re taking the Hodge dual $\star$.  This is conceptually clean, but expands out to a fairly big integral when you express it in coordinates, where the leading term  involves $\sqrt{det(\partial_{\mu} x^m \partial_{\nu} x^n g_{mn}}$ (the determinant is taken over $(\mu,\nu)$.  Varying this to get the equations of motion produces:

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

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

A souped up version of the Nambu-Goto action is the Polyakov action, which is a natural variation that comes up when $\Sigma^{(2)}$ has a metric of its own, $h$.  You can check out the details behind that link, but part of what makes this action nice is that the corresponding Euler-Lagrange equation from varying $h$ says that $x*g \sim h$.  That is, the worldsheet $\Sigma^{(2)}$ will have an image with a shape such that its own metric agrees with the one induced from the spacetime $M^{(D)}$.   This action is called the Polyakov action (even though it was introduced by Deser and Zumino, among others) because Polyakov used it for quantizing the string.

Other variations on this action add additional terms which represent fields which the string might be affected by: a scalar $\phi(x)$, and a 2-form field $B_{mn}(x)$ (here we’re using the physics convention where $x$ represents both the function, and its values at particular points, in this case, values of parameters $(\sigma_0,\sigma_1)$ on $\Sigma^{(2)}$).

That 2-form, the “B-field”, is an important field in string theory, and eventually links up with higher gauge theory, which we’ll get to as we go on: one can interpret the B-field as part of a higher connection, to which the string is coupled (as in Baez and Perez, say).  The scalar field $\phi$ essentially determines how strongly the shape of the string itself affects the action – it’s a “string coupling” term, or string coupling “constant” if it’s chosen to be just a number $\phi_0$.  (In such a case, the action includes a term that looks like $\phi_0$ times the Euler characteristic of the surface $\Sigma^{(2)}$.)

Sebastian briefly explained some of the physical intuition for why these are the kinds of couplings which it makes sense to introduce.  Essentially, any coupling one writes in coordinates has to get along with gauge symmetries, changes of coordinates, etc.  That is, there should be no physical difference between the class of solutions one finds in a given set of coordinates, and the coordinates one gets by doing some diffeomorphism on the spacetime $M^{(D)}$, or by changing the metric on $\Sigma^{(2)}$ by some conformal transformation $h_{\mu \nu} \mapsto exp(2 \omega(\sigma^0,\sigma^1)) h_{\mu \nu}$ (that is, scaling by some function of position on the worldsheet – underlying string theory is Conformal Field Theory in that the scale of the generic worldsheet is irrelevant – only the light-cones).  Anything a string couples to should be a field that transforms in a way that respects this.  One important upshot for the quantum theory is that when one quantizes a string coupled to such a field, this makes sure that time evolution is unitary.

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

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

Remember: if one is finding amplitudes for various observables, the partition function is a normalizing factor, and finding the value of any observables means squeezing them into a similar-looking integral (and normalizing by this factor).  So this says that they’re found by summing over all the string topologies which go from the input to the output, and integrating over all embeddings $x : \Sigma^{(2)} \rightarrow M^{(D)}$ and metrics on $\Sigma^{(2)}$.  (The denominator in that fraction is dividing out by the volumes of the symmetry groups, as usual is quantum field theory since these symmetries mean one is “overcounting” physically identical situations.)

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

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

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