John Baez
Public
Spacetime crystals

You know about crystals in space.  What's a crystal in spacetime?    It's a repetitive pattern that has a lot of symmetries including reflections, translations, rotations and Lorentz transformations.   Rotations mix up directions in space.  Lorentz transformations mix up space and time directions.

We can study spacetime crystals mathematically - and the nicest ones are described by gadgets called hyperbolic Dynkin diagrams, which play a fascinating role in string theory.

How do these diagrams work?

Each dot stands for a reflection symmetry of our spacetime crystal.  Dots not connected by an edge are reflections along axes that are at right angles to each other.  Dots connected by various differently labelled edges are reflections at various other angles to each other.  To get a spacetime crystal, the diagram needs to obey some rules.

The number of dots in the diagram, called its rank, is the dimension of the spacetime the crystal lives in.  So, the picture here shows a bunch of crystals in 5-dimensional spacetime.

Victor Kac, the famous mathematician who helped invent these spacetime crystals, showed they can only exist in dimensions 10 or below.  He showed that:

there are 4 in dimension 10
there are 5 in dimension 9
there are 5 in dimension 8
there are 4 in dimension 7

In 1979, two well-known mathematicians named Lepowsky and Moody showed there were infinitely many spacetime crystals in 2 dimensions... but they classified all of them.

In 1989, Saclioglu tried to classify the spacetime crystals in dimensions 3 through 6.  He got a list of 118.

But he left a bunch out!  A more recent list, compiled very carefully by a big team of mathematicians, gives 220:

there are 22 in dimension 6
there are 22 in dimension 5
there are 53 in dimension 4
there are 123 in dimension 3

If they're right, there's a total of 238 spacetime crystals with dimensions between 3 and 10.

I think it's really cool how 10 is the maximum allowed dimension, and the number of spacetime crystals explodes as we go to lower dimensions... becoming infinite in dimension 2.

String theory lives in 10d spacetime, so it's perhaps not very shocking that some 10-dimensional spacetime crystals are important in string theory - and also supergravity, the theory of gravity that pops out of superstring theory.    The lower-dimensional ones seem to appear when you take 10d supergravity and 'curl up' some of the space dimensions to get theories of gravity in lower dimensions.

Greg Egan and I have been playing around with these spacetime crystals.  I've spent years studying crystal-like patterns in space, so it's fun to start looking at them in spacetime.  I'd like to say a lot more about them - but my wife is waiting for me to cook breakfast, so not now!

Nobody calls them 'spacetime crystals', by the way - to sound smart, you gotta say 'hyperbolic Dynkin diagrams'.  Here's the paper by that big team:

• Lisa Carbone, Sjuvon Chung, Leigh Cobbs, Robert McRae, Debajyoti Nandi, Yusra Naqvi and Diego Penta, Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits, http://arxiv.org/abs/1003.0564.

+J Gregory Moxness created nice pictures of all 238 hyperbolic Dynkin diagrams and put them on Wikicommons:

https://en.wikipedia.org/wiki/User:Jgmoxness

and that's where I got my picture here!

#spnetwork arXiv:1003.0564 #symmetry #KacMoody #Dynkin #geometry

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