The n-Category Café

A good place to start is Physics, Topology, Logic and Computation: A Rosetta Stone, an overview of how category theory unifies our description of “systems and processes” in four subjects. This was written by Mike Stay and me, and it appears in Bob Coecke’s volume New Structures for Physics.

After categories come n-categories — this is where things get really fun. For how n-categories show up in physics, try A Prehistory of n-Categorical Physics by Aaron Lauda and me. This will appear in Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, a book edited by Hans Halvorson. Both this and Bob Coecke’s book should be nice introductions to categories as used in physics.

We can repeatedly ‘categorify’ familiar mathematical concepts and get new ones by replacing sets with categories, 2-categories, and so on. In physics this tends to go along with boosting dimensions, for example going from theories of particles to theories of strings, 2-branes, etc. Since symmetries are so important in physics, and we use the concept of ‘Lie algebra’ to describe symmetries mathematically, it’s especially fun to categorify this concept. This gives the concept of ‘Lie 2-algebra’. We can also categorify the concept of “symplectic manifold”, which is the kind of space whose points describe states of particles. It turns out that categorified symplectic manifolds, or ‘2-plectic manifolds’, can be used to describe the states of strings. And just as any symplectic manifold gives a Lie algebra of observables, a 2-plectic manifold gives a Lie 2-algebra of observables! Alex Hoffnung, Chris Rogers and I wrote a paper developing these ideas: Categorified Symplectic Geometry and the Classical String.

The most famous Lie algebras in physics — the so-called ‘simple’ ones — can all be extended to Lie 2-algebras, and the latter show up when we describe the symmetries of strings. Chris and I wrote a paper showing how to get these Lie 2-algebras from 2-plectic geometry: Categorified Symplectic Geometry and the String Lie 2-Algebra.

Later Chris categorified these ideas further, showing quite generally that n-plectic manifolds give Lie n-algebras, in his paper L_∞-algebras from Multisymplectic geometry. In 2-Plectic Geometry, Courant Algebroids, and Categorified Prequantization, he then began describing how to quantize classical systems described by 2-plectic manifolds. In his thesis, he’ll continue to study quantization for 2-plectic manifolds.

And then there’s John Huerta, who really likes elementary particle physics. We started by writing an intro to particle physics for mathematicians, The Algebra of Grand Unified Theories. But our real goal was to understand how the normed division algebras — the real numbers, complex numbers, quaternions and octonions — are important in supersymmetric versions of categorified physics. In Division Algebras and Supersymmetry I, we reviewed how the normed division algebras give rise to an equation involving spinors and vectors that’s crucial for superstrings in spacetimes of dimensions 3, 4, 6, and 10. In Division Algebras and Supersymmetry II we developed an analogous story for super-2-branes in dimensions 4, 5, 7 and 11. As you’ve probably heard, dimensions 10 and 11 are especially interesting in string theory and M-theory — these are the cases where the octonions show up!

What does this have to do with categorification? Well, these equations involving spinors and vectors are ‘cocycle conditions’, and that means they give rise to ‘Lie 2-superalgebras’ extending the usual spacetime supersymmetries in dimensions 3,4,6 and 10 — and ‘Lie 3-superalgebras’ doing the same in dimensions 4,5,7 and 11. In his thesis, John is studying the corresponding ‘Lie 2-supergroups’ and ‘Lie 3-supergroups’.

John and I also wrote some papers to help explain these ideas: an Invitation to Higher Gauge Theory, and a gentle expository article on octonions which will appear in Scientific American.

Most of the papers so far take the continuum for granted. But it’s also tempting to use categorification to look for a ‘purely discrete’ way to do physics. One way is to use groupoids (categories where all the morphisms are isomorphisms) as a substitute for numbers. Alex Hoffnung, Christopher Walker and I wrote a paper that develops linear algebra based on this idea: Higher-Dimensional Algebra VII: Groupoidification. In this paper we sketch how to use this idea to categorify certain algebraic gadgets called ‘Hecke algebras’ and ‘Hall algebras’. These gadgets are important in the study of simple Lie algebras. We’ll study them in more detail in the next two HDA papers. For a preview of HDA8, see Alex Hoffnung’s paper The Hecke Bicategory. Christopher Walker is doing his thesis on categorified Hall algebras, and some of that work will become HDA9.

Apart from HDA8, HDA9 and a few other leftovers, I’ve moved on to other projects. So I want to point out a big hole in the above work, which I will never fill, in hopes that someone else will.

Namely: there’s a gap between the strand of work that takes the continuum for granted and the strand that explores doing math in a purely discrete way!

Luckily, there’s an obvious place to start bridging this gap.

On the one hand, we can categorify any simple Lie algebra and get a Lie 2-algebra. This work uses the real numbers, or at least the rational numbers, all over the place. On the other, we can take the quantum group corresponding to this Lie algebra. Inside the quantum group there’s a hefty piece called the Hall algebra, which we know how to categorify without ever mentioning the rational numbers: we can use groupoids instead. How are these related? They are indeed closely related: after all, both the Lie 2-algebra and the quantum group are close relatives of yet another player in this game, the centrally extended loop group! But it would be nice to clarify this relationship, and simplify it. I don’t think we’ve gotten to the bottom of the math yet, much less its possible implications for physics.

I should add that other people have other approaches to categorifying quantum groups, some of which apply to the whole quantum group. Some relevant names here include Kazhdan, Lusztig, Frenkel, Soergel, Stroppel, Khovanov, Lauda, Rouquier, and Webster — and I can think of many more, so I apologize to the rest of you. This line of work is very important, and it must hold many of the keys to the question I’m asking. But to me, alas, it still seems complicated and mysterious. Most of this is due to my ignorance, I’m sure. But I still think this work would benefit from being looked at by simple-minded folks who can’t look at a complicated formula or definition without asking “why?”

Eventually, you see, it will all turn out to be blitheringly obvious…