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Bott periodicity for Dirac matrices?

+ 2 like - 0 dislike
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In most of the contexts I've seen, Bott periodicity refers to the periodicity of the homotopy groups of the manifolds of classical groups. However, recently I have heard of some physicists speaking of "Bott periodicity" as a periodicity in the reality properties of Dirac matrices, as summarized in Table 3 (page 8) in this document. However, it's not clear to me how the two kinds of periodicity are related. I'm looking for some explanations for this.

Cross-posted: https://math.stackexchange.com/questions/2242385/bott-periodicity-for-dirac-matrices

asked Apr 28 in Mathematics by Jia Yiyang (2,465 points) [ no revision ]

2 Answers

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The most concrete way to express the relation between Dirac matrices and homotopy groups of classical groups is given by the following construction.

Let $(\gamma^\mu)_{\mu=1, \dots ,d}$ be $n \times n$ Dirac matrices (in Euclidean signature), i.e. satisfying

$\{ \gamma^\mu , \gamma^\nu \} = 2 \delta^{\mu \nu}$

Then the map

$\mathbb{R}^d \rightarrow M_n (\mathbb{C})$

$(x_\mu)_{\mu=1, \dots, d} \mapsto \sum_{\mu =1}^d \gamma^\mu x_\mu$

gives in restriction to the unit sphere $\sum_{\mu=1}^d (x_\mu)^2=1$ a map

$\mathbb{S}^{d-1} \rightarrow GL_n (\mathbb{C})$

(indeed, in this case: $(\sum_\mu \gamma^\mu x_\mu)(\sum_\nu \gamma^\nu x_\nu)= \sum_{\mu=1}^d (x_\mu)^2=1_n$).

Taking the homotopy class of this map, we get an element in $\pi_{d-1}(GL_n (\mathbb{C}))=\pi_{d-1}(U_n)$.

If the Dirac matrices are real, we get a map

$\mathbb{S}^{d-1} \rightarrow GL_n (\mathbb{R})$

and so an element in $\pi_{d-1}(GL_n (\mathbb{R}))=\pi_{d-1}(O_n)$.

We get a map from possible Dirac matrices to homotopy groups of unitary groups (or orthogonal groups). Up to appropriate equivalence on both sides, this is a one to one correspondence : see

http://www.maths.ed.ac.uk/~aar/papers/abs.pdf

for the precise (and original) statement. In particular, under this correspondence, Bott periodicity on the Dirac matrices side matches with Bott periodicity on the homotopy groups side.

For a physical interpretation in the context of D-branes, see https://arxiv.org/abs/hep-th/9810188

answered Apr 30 by 40227 (4,660 points) [ revision history ]

Thanks, this is almost clear, but how do I see  $\sum_{\mu =1}^d \gamma^\mu x_\mu, (x^\mu)^2=1$ can take all possible $U(n)$ matrices? In other words how can one be sure it doesn't live on a smaller submanifold of $U(n)$? 

In general, it will not take all possible $U(n)$ matrices and it will indeed live on a smaller manifold of $U(n)$. But it is not an issue. The homotopy group $\pi_{d-1}$ of a space is about the topologically different ways to map a $(d-1)$-dimensional sphere to this space. This can be non-trivial even if $(d-1)$ is smaller than the dimension of the space.

But then in the case of real Dirac matrices what can stop me from interpreting the map being in the homotopy class of $S^{d-1}\to GL_n(\mathbb{C})$?

I am not sure to understand the question. A map $S^{d-1}  \rightarrow GL_n(\mathbb{R})$ can indeed be seen as a map $S^{d-1} \rightarrow GL_n(\mathbb{C})$ and so induces elements both in $\pi_{d-1}(GL_n(\mathbb{R}))$ and in $\pi_{d-1}(GL_n(\mathbb{C})$, related by a natural map $\pi_{d-1}(GL_n(\mathbb{R})) \rightarrow \pi_{d-1}(GL_n(\mathbb{C}))$. But the element in $\pi_{d-1}(GL_n(\mathbb{R}))$ is in general more interesting than an element in $\pi_{d-1}(GL_n(\mathbb{C}))$ (there is more space in $GL_n(\mathbb{C})$ than in $GL_n(\mathbb{R})$ to contract a sphere and so to show that this sphere is topologically trivial).

I guess I'm still not seeing the logic that gives "topological Bott periodicity = Bott periodicity for reality". If the logic simply is that, because you can construct such a map (as you wrote),  which can be interpreted as in a homotopy class of $\pi_{d-1}(O_n)$, which has a period of 8 in d, and somehow this must also be the period of reality on the Dirac matrix side, then I can't see why I shouldn't just interpret it as in a homotopy class of $\pi_{d-1}(U_n)$ and conclude the period for reality must be 2, which is wrong.

(It's probably covered in the paper you cited, but at the moment it looks a bit intimidating to me.)

It is not enough to have the map to conclude non-trivial things, one needs to know that this map has good properties, in particular is one to one up to appropriate equivalences, something that I admit I have not explained, just referring to the paper. But you can probably guess that to obtain something useful about real Dirac matrices, you need some form of reality condition and so the map to $\pi_{d-1}(O_n)$ will have such good properties but not the map to $\pi_{d-1}(U_n)$ (knowing that something is trivial in $\pi_{d-1}(U_n)$ means that you can contract a sphere by going through complex matrices but not necessarily through real matrices, so it will only tell you something about complex Dirac matrices and nothing about reality conditions).

Ok thanks, maybe one day I'll gather some fortitude to attach the paper you cited.

+ 2 like - 0 dislike

This is essentially the same phenomenon; see https://en.wikipedia.org/wiki/Clifford_algebra, form which the following quote is taken:

Notably, the Morita equivalence class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends only on the signature (p − q) mod 8. This is an algebraic form of Bott periodicity.

answered Apr 29 by Arnold Neumaier (12,385 points) [ no revision ]
I find the introduction hard to comprehend, is there a watered down explanation for a physicist?

@JiaYiyang: I don't know which level you can comprehend. Perhaps this?

A new proof of the Bott periodicity theorem

Or this? 

https://arxiv.org/abs/1005.3213

You can google for more using ''Clifford'' and ''Bott periodicity''.

40227's answer looks about the right level for me, let me check if I can indeed understand that.

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