### A Categorified Supergroup for String Theory

#### Posted by John Baez

My student John Huerta is looking for a job. You should hire him! And not just because he’s a great guy. He’s also done some great work.

He recently gave a talk at the School on Higher Gauge Theory, TQFT and Quantum Gravity in Lisbon:

- John Huerta, A categorified supergroup for string theory.

This has got to be the first talk that combines tricategories and the octonions in a mathematically rigorous way to shed light on the foundations of M-theory! It’s a preview of his thesis.

What’s the idea?

Very roughly, the idea is to see how these 3 facts fit together:

- The only normed division algebras have dimensions 1, 2, 4, and 8.
- There are classical superstring Lagrangians of the simplest sort only in dimensions 3, 4, 6 and 10.
- There are classical super-2-brane Lagrangians of the simplest sort only in dimensions 4, 5, 7 and 11.

In particular, the 8-dimensional octonions lead to a special relation between vectors and spinors in 10-dimensional spacetime. John used this to construct a ‘Lie 2-supergroup’ extending the Poincaré group in 10 dimensions. A Lie 2-group is a categorified Lie group, but John is doing the supersymmetric case. Just as ordinary groups are important for describing the motion of point particles in gauge theory, this 2-supergroup is important for describing the motion of superstrings.

Similarly, in 11 dimensions John used the octonions to construct a *‘Lie 3-supergroup’* extending the Poincaré group. This is important for describing the parallel transport of super-2-branes in 11 dimensions. And *that*, in turn, is probably important for M-theory.

Actually, John used the same construction starting from all four normed division algebras to get Lie 2-supergroups extending the Poincaré group in 3, 4, 6 and 10 dimensions. He used a similar construction to get Lie 3-supergroups in dimensions 4, 5, 7 and 11. But curiously, the octonionic cases seem to be the ones that lead to sensible *quantum* theories of superstrings and super-2-branes. That’s the main reason I’m emphasizing those cases.

There are general procedures for integrating Lie $n$-algebras to get ‘stacky’ Lie $n$-groups, but John uses a special procedure that gives a very nice simple model of the Lie 2-supergroups and Lie 3-supergroups he wants.

In particular, his Lie 3-supergroup is a one-object tricategory with strict inverses for everything… *in the category of affine superschemes*. So, it’s actually what you might call an ‘affine algebraic 3-supergroup’, by analogy with the more familiar concept of ‘affine algebraic group’.

So, if you want to hire someone who knows some serious heavy-duty math and physics, he’s your man.

For a talk that focuses on the normed division algebra and Lie superalgebra aspects, not on the $n$-supergroup aspects, try:

- John Huerta, $L_\infty$-superalgebras for superstrings and 2-branes.

## Re: A Categorified Supergroup for String Theory

Hi, thanks for blogging about this! But that talk doesn’t mention tricategories at all. For now, that’s only in the thesis.