The n-Category Café

Simon J. Norton was a math prodigy as a child. He went to Cambridge for grad school. Together with his advisor John Conway, he did some amazing work on group theory. In 1985, based on an idea from John McKay, they conjectured an astounding relation between the Monster group and the modular $j$-function. Conway dubbed this “Monstrous Moonshine”. The proof turned out to involve ideas from string theory, but the full implications are yet to be understood.

But in 1985 — when some mathematicians claim he suffered a “catastrophic intellectual collapse” — Simon took to collecting thousands of bus and train timetables. What happened to him? What is he doing now?

Two months ago, I told you about Hadwiger’s theorem. It’s a theorem about Euclidean space, classifying all the ways of measuring the size of convex subsets. For example, here are three ways of measuring the size of convex subsets of the plane:

• Area. This is a $2$-dimensional measure, in the sense that if you scale up a set by a factor of $t$ then its area increases by a factor of $t^2$.
• Perimeter. This is a $1$-dimensional measure, in the same obvious sense.
• Euler characteristic. This takes value $1$ on nonempty convex sets and $0$ on the empty set. It’s a $0$-dimensional measure, since if you scale up a set by a factor of $t$ then its Euler characteristic increases by a factor of $t^0 = 1$: it doesn’t change at all.

Hadwiger’s theorem in two dimensions says that these are essentially the only ways of measuring the size of convex subsets of the plane.

But Hadwiger’s theorem is all about measurement in Euclidean space. There are many other interesting metric spaces in the world! Today I’ll tell you about the quest to imitate Hadwiger in an arbitrary metric space.

Take $2$ convex bodies in $\mathbb{R}^2$, or $3$ convex bodies in $\mathbb{R}^3$, or, more generally, $n$ convex bodies in $\mathbb{R}^n$. Mixed volume assigns to each such family a single real number.

The mixed volume of convex bodies $A_1, \ldots, A_n$ is written as $V(A_1, \ldots, A_n) \in \mathbb{R}$, and it’s uniquely characterized by the following three properties:

1. Volume:  $V(A, \ldots, A) = Vol(A)$, for any convex body $A$
2. Symmetry:  $V$ is symmetric in its arguments
3. Multiadditivity:  $V(A_1 + \tilde{A}_1, A_2, \ldots, A_n)$ equals $V(A_1, A_2, \ldots, A_n) + V(\tilde{A}_1, A_2, \ldots, A_n)$ for any convex bodies $A_i$ and $\tilde{A}_1$, where $+$ denotes Minkowski sum.

This looks rather like the characterization of determinant: $det$ is unique satisfying $det(I) = 1$, antisymmetry, and multilinearity. One difference is that we have symmetry rather than antisymmetry. But a much bigger difference is that where determinant assigns a number to $n$ vectors in $\mathbb{R}^n$, mixed volume assigns a number to $n$ convex bodies in $\mathbb{R}^n$.

But maybe you don’t find the unique characterization satisfying. What is mixed volume?

There’s an interesting paper out today on the ArXiv – Joel Hamkins’ The set-theoretic multiverse.

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

So set theorists’ experience of dealing with various models of set theory, and the small modifications needed to generate models either satisfying or not satisfying the continuum hypothesis, tells them that settling its truth or falsity by devising an evident axiom from which either it or its negation follows is not a live option. Gödel’s platonism was wrong then, yet Hamkins retains a form of realism:

The multiverse view is one of higher-order realism – Platonism about universes – and I defend it as a realist position asserting actual existence of the alternative set theoretic universes into which our mathematical tools have allowed us to glimpse. The multiverse view, therefore, does not reduce via proof to a brand of formalism. In particular, we may prefer some of the universes in the multiverse to others, and there is no obligation to consider them all as somehow equal.

I wouldn’t normally post a job ad here, but if you read this one carefully, you’ll see why I made an exception. It’s from Neil Ghani at the University of Strathclyde, which is in Glasgow city centre.

6 month postdoc position

Mathematically Structured Programming Group, University of Strathclyde

We have the potential to apply for funds for a 6 month post doctoral position. The idea is that the successful candidate would spend those 6 months writing a full scale grant to fund themselves for the next 3 years. The postdoctoral position would be within the Mathematically Structured Programming group at the University of Strathclyde whose research focusses on category theory, type theory and functional programming. Current staff include Neil Ghani, Patricia Johann, Conor McBride, Peter Hancock, Robert Atkey and 6 PhD students. The candidate we are looking for should be highly self motivated and appreciate that without beauty, we are lost.

Unfortunately, the deadline is extremely short and so any interested candidates should contact me immediately. I can then tell you more about what we would need to do.

Professor Neil Ghani (ng#cis.strath.ac.uk, with obvious change)

The fixed point index is a rare example of a concept in homotopy theory that is much easier to motivate, as far as I can tell, when you think of $\infty$-groupoids as presented by topological spaces. It is possible, however, to define it in purely categorical language (and quite simply, too, without referring to any complicated technology). I want to pose this as a puzzle: can you define the fixed-point index this way (just for 1-groupoids, to make it easy) — and better, can you motivate it?

We’re having a workshop this fall:

• Fields Institute Workshop on Category Theoretic Methods in Representation Theory, October 14-16, 2011, University of Ottawa, Ottawa, Canada.

Speakers

Sabin Cautis (Columbia)

James Dolan (UC Riverside)

Ben Elias (M.I.T.)

Joel Kamnitzer (Toronto)

Aaron Lauda (Southern California)

Anthony Licata (Institute for Advanced Study)

Marco Mackaay (Universidade do Algarve)

Volodymyr Mazorchuk (Uppsala)

Kevin McGerty (Imperial College London)

Raphaël Rouquier (Oxford)

Catharina Stroppel (Bonn)

Pedro Vaz (University of Zurich and IST Lisbon)

Ben Webster (Northeastern)

Geordie Williamson (Oxford)

Oded Yacobi (Toronto)

Organizers

Alexander Hoffnung (Ottawa)

Alistair Savage (Ottawa)

Financial Support

There is some funding available for graduate students and postdoctoral fellows. Those interested should complete the funding application available on the workshop website (found below). The deadline for applying for support is September 1, 2011.

Back in May 2006, David Corfield wrote a blog entry called Klein 2-geometry, saying:

As a small experiment in collective, public thinking, I’m going to devote a post to the attempt to categorify Kleinian geometry, and update the date so it doesn’t slip off the radar of ‘Previous Posts’.

His question was:

What prevents an Erlangen program for 2-groups?

The Erlangen program, is, of course, Felix Klein’s plan to study highly symmetrical spaces by thinking of them as quotient spaces $G/H$ where $G$ is a group and $H$ a subgroup. If you’ve heard of this program but never really read about it, you might like his recent review article:

• Felix Klein, A comparative review of recent researches in geometry, arXiv:0807.3161.

Generalizing this idea to 2-groups (or beyond) seemed like a great idea, and David’s original post helped trigger the formation of the n-Category Café. The discussion went on and on, all the way to Klein Geometry XI. However, it never developed to the height of magnificence that I’d hoped, mainly because of the lack of a clear goal.

But then this summer I went to Erlangen, and talked to Derek Wise…

Here’s a really easy introduction to normed division algebras, particularly the octonions, and their role in string theory. You basically just need to have gone to high school:

• John Baez and John Huerta, The strangest numbers in string theory, Scientific American, May 1, 2011, pp. 61-65.

What follows is a guest post by Greg Weeks. If your memory extends back before the formation of this blog to the glory days of sci.physics.research, you should remember Greg.

We would like to ask for comments on an early version of an article that we are writing:

Domenico Fiorenza, Chris Rogers, U.S., A higher Chern-Weil derivation of AKSZ $\sigma$-models (pdf)

but before I say what this is about (below the fold) here some background meant to put our theorem into perspective.

In the previous entry I gave a rough indication of the original definition of the class of topological sigma-model quantum field theories called AKSZ models .

This class coincides in dimension 2 with the class of Poisson sigma-models – which in turn contains the A-model and the B-model – and in dimension 3 with the class of Courant sigma-models – which in turn contains the class of ordinary Chern-Simons theory as the special case where the base of target space is the point.

Therefore it is clear that the AKSZ models are some noteworthy type of generalization of Chern-Simons theory. Here I want to discuss a precise sense in which this is true systematically and give an alternative definition of the AKSZ models that identifies them as a canonical construction in abstract higher Chern-Weil theory. In fact, the claim is that the action functional that defines the AKSZ models is precisely the value of the higher Chern-Weil homomorphism with values in”secondary characteristic classes” and applied to a binary and non-degenerate invariant polynomial on any L-infinity algebroid.

This in turn shows that the class of AKSZ models itself is only a special case of something more general which exists on very general abstract grounds, and which we call infinity-Chern-Simons theory : this is defined for every invariant polynomial on every $L_\infty$-algebroid. Aspects of this I had mentioned before: this larger class contains of course higher dimensional abelian Chern-Simons theories (these come from the canonical invariant polynomial on line Lie n-algebras) but for instance also the class of infinity-Dijkgraaf-Witten theories with sub-classes such as ordinary Dijkgraaf-Witten theory and the Yetter models, and also for instace higher Chern-Simons supergravity.

Therefore all these topological $\sigma$-models (and many more that haven’t been given names yet) are incarnations of one single phenomenon: the higher Chern-Weil homomorphism. This exists on entirely abstract grounds in every cohesive ∞-topos. Therefore, in a sense, all these types of $\sigma$-models have an existence from “first principles”.

This is maybe noteworthy, since many of these topological QFTs (maybe all of them?) play a role in the description of genuine physics via the holographic principle: for instance the 2d Poisson $\sigma$-model as well as the A-model holographically encode ordinary quantum mechanics of particles (= 1-dimensional non-topological QFT), then 3-dimensional Chern-Simons theory holographically encodes the quantum mechanics of non-topological strings and generally higher dimensional Chern-Simons theory in dimension $D = 4k+3$ (for $k \in \mathbb{N}$) holographically encodes self-dual higher gauge theory in dimension $d = 4k+2$ (at least in the abelian case), such as the effective type II-superstring QFT in $d = 10$ – which in turn is famously thought to have vacua that look like the standard model of observed particle physics.

Due to all these relations it should be interesting to see that and how AKSZ $\sigma$-models are a special class of $\infty$-Chern-Simons theories, too. This I have tried to work out with Domenico Fiorenza and Chris Rogers. We now have an early writeup and would enjoy to hear whatever comments you might have:

A higher Chern-Weil derivation of AKSZ $\sigma$-models (pdf)

The essence of our main theorem is easily stated. See below.

This is a continuation of the series of posts on sigma-model quantum field theories. It had started as a series of comments in

∞-Dijkgraaf-Witten theory

and continued in

String Topology Operations as a Sigma-Model.

Here I indicate the original definition of the class of models called AKSZ sigma-models (see there for a hyperlinked version of the following text).

In a previous post on exposition of higher gauge theories as sigma-models I had discussed how ordinary Chern-Simons theory is a $\sigma$-model. Indeed this is also a special case of the class of AKSZ $\sigma$-models.

In a followup post I will explain that AKSZ sigma-models are characterized as precisely those ∞-Chern-Simons theories that are induced from invariant polynomials which are both binary and non-degenerate. (Which is incidentally precisely the case in which all diffeomorphisms of the worldvolume can be absorbed into gauge transformations.)