What does it mean to define a spin-structure on a manifold? [closed]

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?

closed as unclear what you're asking by ACuriousMind♦Jan 30 at 17:54

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 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 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. – zooby Jan 30 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 at 17:54
• What I mean is dirac gamma matrices like $\gamma_n$ the n index relates to the tangent-space. – zooby Jan 30 at 18:13