# Questions tagged [spin-geometry]

For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}(n)$ and $\operatorname{Spin}^c(n)$. This tag should also be used for any questions about the geometry of spin manifolds, including questions involving Dirac operators and the Lichnerowicz formula.

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### Doubts on definition of spin structure

In Friedrich's book "Dirac operators in Riemannian geometry" there is this definition for spin structure. Let $X$ be a CW-complex and $(Q, \pi, X, SO(n))$ a $SO(n)$-principal bundle on $X$. ...
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### Understanding's Wikipedia's definition of a spinor

I am trying to understand spinors from a mathematical view. I've seen similar questions on this website but I'm still unclear on what they are exactly. On Wikipedia they state: Although spinors can ...
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### Topological definition of Spin$(p,q)$?

In short, How can we define Spin(p, q) without referencing Clifford algebras? The answer should be something like "Spin$(p, q)$ is the unique double cover of SO$^+(p, q)$ such that ...". ...
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### Are spinors in 3 dimension quaternions, or are they elements of $\mathbb C^2$?

In Wikipedia both statement is present, which one is true? In one hand, here stands: def 1. Thus the (real) spinors in three-dimensions are quaternions, and the action of an even-graded element on a ...
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### How does the following representation of $\mathbb{C}l(3)$ decompose into irreducibles?

Consider the three dimensional complex Clifford algebra $\mathbb{C}l(3)$ and the following representation $$S=\text{Span}\{|0\rangle,c^\dagger|0\rangle,\gamma^3|0\rangle,\gamma^3c^\dagger|0\rangle\},$$...
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### Is there any spin group $Spin(n)$ for any value of $n$ which is isomorphic to $SU(3)$?

I was wondering about the spin group and its relations with the unitary groups. Like we have $Spin(2)\simeq U(1)$, $Spin(3) \simeq SU(2)$. I was wondering whether this relation can be taken further to ...
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### Is the total space of the frame bundle of a smooth manifold $M$ always spin?

I seem to remember reading in Kobayashi that the total space of a frame bundle is always parallelizable. If I'm remembering this correctly then the total space of the frame bundle $FM$ is always spin, ...
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