**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

View 53 previous comments

- +Tom Lowe Good catch, here is a simple roots matrix that fixes that too.

srH136={{1, 1, -1, -1}/2, {0, -1, 1, 0}, {-1, 1, 0, 0}, {1, -1, 1, 1}/2}

with 4D angles between each node of:

{{1 -> 2, 135.}, {2 -> 3, 120.}, {3 -> 4, 135.}, {4 -> 1, 120.},{1 -> 3, 90.}}

Norm'd Length between nodes is:

{1, Sqrt[2], Sqrt[2], 1}

Interestingly, it also gets closer to some of the 4D groups derived from split real even E8 (with either integer {+ 1, - 1,0,0} or half-integer vertices).

The srH136.Transpose[srH136] is very close to the symmetrized matrix.

The first 4 positive algebra root vectors are the Cartan matrix (as they should be).

I threw in a few projections in 2D and 3D from the tool I built.

http://theoryofeverything.org/MyToE/?p=2023Oct 30, 2014 - +J Gregory Moxness 2 -> 4 should also be orthogonal as there is no connection between them.Oct 30, 2014
- +Tom Lowe ugghh... did you find one that works?

Oct 30, 2014 - +J Gregory Moxness Well, I can only assume that it requires t^2 be negative. The Cartan matrix is negative definite (perhaps because these are hyperbolic Dynkin diagrams) so some complex roots seem inevitable. I end up with: (0, 2, 0, 0), (0, - 1, 1, 0), (0, 0, - 1, 1), (sqrt(7)i, - 1, - 1, - 3). The i representing the negative t^2, using (t, x, y, z). From linear combinations we can find a time-like vector for the 4th vector: (sqrt(7)i, - 0, 0, - 1), and it is perhaps nicer to make the time-like vector vertical with a boost in the z axis, with a few more transforms and scaling we get: sr136:H13(4)=(0,2root3,0,0),(0,0,2root3,0),(i,root3,root3,root7),(6i,0,0,0). Nicer because it contains a pure time vector and two pure space vectors. Not sure an easy way to verify that it works, but the fact that time and space are opposite rationality (one being a sqrt where the other is not) shows that no linear combination can give a null (light) vector, which is a requirement. It is also interesting that the total square lengths is 0, the three space-like vectors have square length 12 and the time vector has -36.Oct 30, 2014
- I like where you're going with that, but these numbers seem further off from the Dynkin angles than my last iteration (with only the {2->4,45. vs. 90.}.

For srH136={{0,2,0,0},{0,-1,1,0},{0,0,-1,1},{Sqrt[-7],-1,-1,-3}};

I get {{1->2,135.},{2->3,120.},{3->4,109.471},{4->1,103.633},{1->3,90.},2->4,{90.}} with Norms of {2,Sqrt[2],Sqrt[2],3Sqrt[2]}

Is that what you get? (I also tried your others, but similar issues with angles and Norms.

So far I've been finessing the numbers logically, but I am going to put my computer to work and do it brute force with all + / - 1 and 1/2 integer permutations. I will let you know soon... :-)Oct 31, 2014 - +J Gregory Moxness moving this conversation to your page.Oct 31, 2014

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