Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

October 22, 2010

The Art of Math

Posted by John Baez

Here’s a gentle introduction to the work my students have been doing on categorification and physics:

• Sophie Hebden, The art of math: a pictorial branch of mathematics could help physicists draw new conclusions about quantum gravity and the nature of time.

It was put out by the Foundational Questions Institute, or FQXi. This is an organization that funds innovative research on hard questions like

what is the nature of time?

what is ultimately possible in physics?

and

how come there’s an ‘X’ in the acronym for ‘Foundational Questions Institute’?

A while back they gave me a grant to help out three of my grad students: John Huerta, Chris Rogers and Christopher Walker. It made a huge difference! Instead of working as teaching assistants all the time, they could write lots of papers, go to lots of conferences, and make progress much faster. They’re all finishing up this spring, and they’ll need jobs. You should hire them.

Unfortunately it’s a bit hard to describe their work in simple terms.

Fortunately, Sophie Hebden’s article does a great job! How do you explain categorification to people who haven’t studied math since high school? It may sound impossible, but this article does it.

But if you know some math, you’ll probably want to see more technical details: without the details, our work might sound like fluff with no substance. So: let me describe the papers we wrote with the help of this FQXi grant. For most I’ll include links, not only to the papers themselves, but to conversations about them here on 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…

Posted at October 22, 2010 4:01 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2293

10 Comments & 0 Trackbacks

Re: The Art of Math

I can’t resist adding the plug that the first Baez-Huerta paper on Division Algebras and Supersymmetry is about to appear in our CBMS conference proceedings
Superstrings, Geometry, Topology, and C*-algebras, which should be appearing in print imminently.

Posted by: Greg Friedman on October 22, 2010 8:26 AM | Permalink | Reply to this

Re: The Art of Math

I just got the volume! It looks great.

Posted by: John Huerta on October 28, 2010 2:43 AM | Permalink | Reply to this

fundamental physics from higher category theory

Recently I had the chance to – finally – exapand the nnLab entry

higher category theory and physics

a bit. It’s still far from where I imaging it to be, but now at least it is beginning to show an inkling of what it can be.

This starts with describing a path that leads from just the general abstract notion of space and process in \infty-topos theory to action functionals for those QFTs that we started calling \infty-Chern-Simons theories . Which includes quite few, not the least the QFTs of AKSZ-type.

I am saying this not to hijack John’s promotion of his student’s work, but on the contrary, to add to it: with Chris Rogers I am currently working on writing up an article on action functionals for \infty-Chern-Simons theories. See

\infty-Chern-Simons theory

for a hint of what we are doing.

I had also tried to get John Huerta involved into this: the Chern-Simons elements on an \infty-Lie algebroid that govern these action functionals are, when applied to the supergravity Lie 6-algebra of the type that he has been investigating, essentially the Lagrangians for supergravity in their form that D’Auria-Fré called by the curious name “cosmo-cocycles”, as described here. So that’s very closely related to John Huerta’s work. He should think about this.

Posted by: Urs Schreiber on October 22, 2010 8:59 AM | Permalink | Reply to this

Re: fundamental physics from higher category theory

I am very interested in this idea. Once I have some time, I’ll work on it.

Posted by: John Huerta on October 24, 2010 12:50 AM | Permalink | Reply to this

Re: The Art of Math

…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!

I remember our discussion on the reals emerging from the classification of locally compact Hausdorff abelian groups back here. I wonder if the coincidence of so very many good properties in the reals acts as an attractor. That’s the line I took in a recent paper of mine. Perhaps that’s why it’s hard to pull away to some discrete structure which does the same job in certain respects.

The issue of how the reals can just seem to show up came up in the nForum discussion on cohesive \infty-toposes, which Urs is promoting as a good setting for physics here and on nLab.

Examples to date have a continuum flavour to them: the general abstract smooth open ball; the general abstract infinitesimally thickened point; the general abstract super-point; the general abstract infinitesimally thickened smooth open ball. So far a MathOverflow plea for more examples has gone unanswered.

Posted by: David Corfield on October 22, 2010 9:00 AM | Permalink | Reply to this

Re: The Art of Math

I loved the FQXi article :)

Perhaps that’s why it’s hard to pull away to some discrete structure which does the same job in certain respects.

I would suggest that often the way discrete structures fail to be “nice” is in itself “nice”. So it might be misdirected to ask for discrete structures to do the “same job”, but instead, ask in what way the job they do differs.

Posted by: Eric on October 22, 2010 10:42 AM | Permalink | Reply to this

Re: The Art of Math

That is a really fantastic picture of a quadruple point on the page of FQXI.

Posted by: Scott Carter on October 22, 2010 7:13 PM | Permalink | Reply to this

Re: The Art of Math

Indeed!

Posted by: Bruce Bartlett on October 26, 2010 5:45 PM | Permalink | Reply to this

Re: The Art of Math

Duff and Ferrara recently cited the supersymmetry and division algebra papers in Four curious supergravities (arXiv:1010.3173 [hep-th]), in which they consider theories in D=4,5,7,11 which have division algebra interpretations.

It would be nice to relate this recent work on supermembrane theories to the insights provided by Iqbal, Neitzke and Vafa back in arXiv:hep-th/0111068. There, they found M-theory on T^k (dim k torus) corresponds to CP^2 blown up at k generic points (i.e., the del Pezzo surface B_k). This gives a mapping from 1/2-BPS states to rational curves in the corresponding del Pezzo. In D=11, this leads to a representation of the M2-brane as a line in CP^2 (i.e. a CP^1), and the M5-brane as a conic in CP^2.

Posted by: Mike Rios on October 25, 2010 7:42 PM | Permalink | Reply to this

super-branes

Some of this stuff is sloooowly being collected in the nnLab entry brane.

Posted by: Lab Elf on October 26, 2010 12:21 AM | Permalink | Reply to this

Post a New Comment