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I'm trying to think about what information I need to add to a manifold that it describes a spin structure?

I know you can have spin-structure on a 2d plane, a 2-sphere.

I also know you can define a Dirac equation on a 2-sphere.

I would think if you have two Dirac matrices $\gamma_x$ and $\gamma_y$ you would need to have some idea of an $x$-direction and a $y$-direction at each point. (which is odd because if you use latitude and longitude this would be undefined at the poles).

It seems like spin is fundamentally linked to some underlying Euclidean space which is strange.

Is there a more intuitive way to understand this?

Qmechanic
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    I don't understand this question. The definition of a spin structure defines what information you need for one to exist. Why are you talking about Dirac matrices and directions? Note also that in physics, you often need the manifold to be not only spin, but $\text{spin}^\mathbf{C}$. – ACuriousMind Jan 30 '19 at 17:28
  • @ACuriousMind Doesn't matter I worked out that you need to define a tangent-space at each point because fermions live on a tangent space. –  Jan 30 '19 at 17:51
  • Fermions don't live in a tangent space, but if you're not willing to clarify your question, I'm certainly willing to close it. – ACuriousMind Jan 30 '19 at 17:54
  • What I mean is dirac gamma matrices like $\gamma_n$ the n index relates to the tangent-space. –  Jan 30 '19 at 18:13

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