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.