### Update

This blog has been on hiatus for a while. I’ve spent the past few years in several short-term jobs which were more teaching-heavy than the research postdocs I was working in when I started it, so a lot of my time went to a combination of teaching and applying for jobs. I know: it’s a common story.

However, as of a year ago, I’m now in a tenure-track position at SUNY Buffalo State College, in the Mathematics Department. Given the academic job market is these days, I feel pretty lucky to find such a job, and especially since it’s a supportive department with colleagues I get along with well. It’s a relatively teaching-oriented position, but they’re supportive of research too, so I’m hoping I’ll get back to updating the blog semi-regularly.

In particular, since I’ve been here I’ve been able to get out to a couple of conferences, and I’d like to take a little time to make a post about the most recent. The first I went to was the Union College Mathematics Conference, in Schenectady, here in New York state. The second was Higher Structures in Lisbon. I was able to spend some time there talking with Roger Picken, about our ongoing series of papers, and John Huerta about a variety of stuff, before the conference, which was really enjoyable.

Here’s the group picture of the participants:

The talks from the conference that had slides are all linked to from their abstracts page, but there are a few talks I’d like to comment on further. Mine was similar to talks I’ve described here in the past, about transformation structures and higher gauge theory. Hopefully there will be an arXiv paper reasonably soon, so I’ll pass over that for now. I’ll summarize what I can, though, focusing on the ones that are particularly interesting (or comprehensible) to me. I’ve linked to the slides, where available (some were whiteboard talks). I’ve grouped them into different topics. This post summarizes talks that fall under the general category of “field theory”, while the others will be in a follow-up post.

### Field Theory

One popular motivation for the use of “higher structures” is field theory, in its various forms. This makes sense: most modern physical theories are of this kind, one way or another, and physics is a major motivation for math. Specifically, one of the driving ideas is that when increasing the dimension of the theory, concepts which are best expressed with categories in low dimensions need higher $n$-categories to express them in higher categories – we see this in fully-extended TQFT’s, for instance, but also the idea that to express the homotopy $n$-type of a space (what you want, generally, for an $n$-dimensional space), you need an $n$-groupoid as a model. There are some other situations where they become useful, but this is an important one.

Ana Ros Camacho was a doctoral student with Ingo Runkel in Hamburg while I was a postdoc there, so I’ve seen her talk about her research several times (thesis). This talk, “Toward a Higher-Categorical Statement for the Landau-Ginzburg/CFT Correspondence”, was maybe the clearest overview I’ve seen so far, so this was a highlight for me. Essentially, it’s a fact of long standing that there’s a correspondence between 2D rational conformal field theory and the Landau-Ginzburg model – a certain Sigma-model (field theory where the fields are maps into some classifying space) characterized by a potential. The idea was that there’s some evidence for a conjecture (but not yet a proof that turns it into a theorem) which says that this correspondence comes from some sort of relationship – yet to be defined precisely – between two monoidal categories.

One is a category of matrix factorizations, and the other is a category which comes from representations of a vertex operator algebra $\mathcal{V}$ associated with the CFT’s. Matrix factorizations work like this: start with the polynomial ring $S = k[x_1,x_2,\dots,x_n]$, and pick a polynomial $W \in S$. If the dimension of the quotient ring of $S$ by all the derivatives of $W$ is finite-dimensional, it’s a “potential”.

This last condition is what makes it possible to talk about a “matrix factorization” of $(S,W)$, which consists of $(M,d)$, where $M = M_0 \oplus M_1$ is a free $\mathbb{Z}_2$-graded $S$-module, and $d : M \rightarrow M$ is a “twisted differential” – an $S$-linear map in degree 1 (meaning it takes $M_0$ to $M_1$ and vice versa) such that $d^2 = W \cdot Id_M$. (That is, the differential is a kind of “square root” of the potential, in this special degree-1 sense.) There is a whole bicategory of such matrix factorizations, called $LG$ (for “Landau-Ginzburg”). Its objects are algebras with a potential, $(S,W)$. The morphisms from $(S_1,W_1)$ to $(S_2,W_2)$ are matrix factorizations for $(S_1 \otimes S_2, W_1 - W_2)$ (which can be defined in a natural way), which can be composed by a kind of tensor product of modules, and the 2-morphisms are just bimodule maps.

The notion, then, is that this 2-category $LG$ is supposed to be related in some fashion to a category $Rep(\mathcal{V})$ of representations of some vertex algebra associated to a CFT. There are some partial results to the effect that there are monoidal equivalences between certain subcategories of these in particular cases (namely, for special potentials $W$). The hope is that this relationship can be expanded to explain the known relationship between the two sorts of field theory.

Tim Porter talked about “HQFT’s and Beyond” – which I’ll skimp on here mainly because I’ve written about a similar talk in a previous post. It did get into some newer ideas, such as generalizing defect-TQFT’s to HQFT and more.

Nils Carqueville gave a couple of talks – one for himself, and one for Catherine Meusburger, who had to cancel – on some joint work of theirs. One was “3D Defect TQFT’s and their Tricategories“, and the other “Orbifolds of Defect TQFT’s“. This is a use of “orbifold” that I don’t entirely understand, but I think roughly the idea is that an “orbifold completion” of a category is an extension in the same way that the category of orbifolds extends that of manifolds, and it’s connected to the idea of equivariantization – addressing symmetry.

In any case, what it’s applied to here is the notion of TQFT’s which are defined, not on just categories of manifolds with cobordisms as the morphisms, but something more general, where all of these spaces are allowed to have “defects”: embedded submanifolds of lower dimension, which can meet at still lower-dimensional junctions, and so on. The term suggests, say, a crystal in solid-state physics, where two different crystal structures meet at a “defect” plane. In defect TQFT, one has, essentially, one TQFT living on one side of the defect, and another on the other side. Then the “tricategories” in question have objects assigned to regions, morphisms to defects where regions meet, and so on (thus, this is a 3D theory). A typical case will have monoidal categories as objects, bimodule categories as morphisms, and then functors and natural transformations. The monoidal categories might be, say, representation categories for some groupoid, which is what you’ll see if the theory on each region is a gauge theory. But the formalism makes sense in a much broader situation. A later talk by Daniel Scherl addressed just such a case (the tricategory of bimodule categories) and the orbifold completion construction.

Dmitri Pavlov’s “Extended QFT’s are Local” was structured around explaining and one main theorem (and the point of view that gives it a context): that field theories $FT^G_V(T) : Man^{op} \rightarrow sSet$, which is to say covariant functors which take manifolds into simplicial sets (or, more generally, some other model of $\infty$-groupoids) have a particular kind of structure. This amounts to showing that being a field theory requires that it should have some properties. First, it should be a local theory: this amounts to the functor being a sheaf, or stack (that is, there are the usual gluing conditions which relate the $\infty$-groupoids\$ assigned to overlapping neighborhoods, and their union). Next, that there should be a classifying object $\mathcal{E}FT^G_V$ in simplicial sets so that, up to homotopy, there’s an equivalence between concordance classes of fields (which might be, say, connections on bundles, or geometric structures, or various other things) and maps into the classifying space. Then, that this classifying space can be built as a homotopy colimit in a particular way. This theorem seems like a snazzier version of the Brown Representability Theorem, which roughly says that functor satisfying some nice axioms making it somewhat like a cohomology theory (now extended to specify a “field theory” in a more physics-compatible sense) has a classifying object. The talk finished by giving examples of what the classifying object looks like for, say, the theory of vector bundles with connection, for the Stolz-Teichner theory, etc.

In a similar spirit, Alexander Schenkel’s “Towards Homotopical Algebraic QFT” is an efford to extend the formalism of Algebraic QFT (developed by people such as Roberts, and Haag) to an $\infty$-categorical – or homotopical – situation. The idea behind AQFT was that such a field theory would be a functor $F : Loc \rightarrow Alg$, which takes some category of spacetimes to a category of algebras, which are supposed to be the algebra of operators on the fields on that bit of spacetime. Then breaking down spacetime into regions, you get a net of algebras that fit together in a particular way. The axioms for AQFT say things like: the algebras for two spacelike-separated regions of space should commute with each other (as subalgebras inside the one associated to a larger region containing both). This gets at the idea that the theory is causal – acting on one region doesn’t affect the other, if there’s no timelike path from one to the other. The other conditions say that when one region is embedded in another, the algebra is also embedded; and that if a small region contains a Cauchy surface for a larger region, the two algebras are actually isomorphic (i.e. having a Cauchy surface determines the whole region). These regions get patched together by local-to-global gluing condition which makes the functor into a cosheaf (not a sheaf: it’s covariant because in general bigger regions have bigger algebras of observables). The problem was that this framework is not enough to account for things like gauge theories, essentially because the gluing has some flexibility up to gauge equivalence. So the talk describes how to extend the framework of AQFT to homotopical algebra so that the local-to-global gluing condition is a homotopy sheaf condition, and went on to talk about what such a theory looks like in some detail, including the extension to categories of structured spacetimes (in somewhat the same vein as HQFT mentioned above).

Stanislaw Szawiel spoke about “Categories of Physical Processes“, which was motivated by describing this as a “non-topological TQFT”. That is, like the Atiyah approach to TQFT, it uses a formalism of categories and functors into some category of algebras to describe various physical systems. Rather than specifically the category of bordisms used in TQFT, the precise category $Phys$ being used depends on what system one wants to model. But functors into $*Mod$, of $C^*$-algebras and bimodules, are seen as assigning algebraic data to physical content. There are a lot of details out of the theory of $C^*$-algebras, such as the GNS theorem, unitarity, and more which come into play here, which I won’t attempt to summarize. It’s interesting, though, that a bunch of different physical systems can be described with this formalism: classical Markov processes, particle scattering, and so forth. One of the main motivations seemed to be to give a language for dealing with the “Penrose Problem”, where evolution of spacetime is speculated to be dynamically related to “state vector collapse” in quantum gravity.

Theo Johnstone-Freyd’s talk on “The Moonshine Anomaly” succeeded in getting me interested in the Monster group and its relation to CFT. He did mention a couple of recent papers that calculate some elements of the fourth cohomology of the super-sized sporadic groups $C_0$ and $M$ (the Monster) which have interesting properties, and then proceeded to explain what this means. That explanation pulls in how these groups relate to the Leech Lattice – a 24-dimensional lattice with nice properties, of which they’re symmetry groups. This relates to CFT, since these are theories where the algebra of observables is a certain chiral algebra (typically described as a vertex algebra). The idea, as I understand it, is that the groups act as symmetries on some such operator, and a “gauged” or “orbifolded” theory (a longstanding idea, which is described here) ends up being related to the category of twisted representations of the group $G$. The “twisting” requires a cohomology class (which is the – nontrivial – associated of that category), and this class is what’s called the “anomaly” of the theory, which gets used in the Lagrangian action for this CFT. So the calculation of that anomaly in the papers above – an element of the Monster group’s fourth cohomology – also helps get a handle on the action of the corresponding CFT.

(More talks to come in part II)