# Questions tagged [spin-geometry]

For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}^{\pm}(n)$ 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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### Comparison between spinor representations in $\operatorname{SL}(2,\mathbb C)=\operatorname{Spin}(1,3)$ and $\operatorname{Spin}(4)$

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$We know that $$\Spin(1,3)=\SL(2,\mathbb C)$$ and $$\Spin(4)=\SU(2) \times \SU(2).$$ The $\Spin(1,3)$ is the ...
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### Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?

I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields. For example, the first Chern class of a complex line ...
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### Discrete Pin structures

It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of ...
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### Spin groups in terms of matrices and/or linear operators

Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...
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### Vanishing of K-theoretic index and positive scalar curvature

I'm confused about a seemingly basic point about a classical result on positive scalar curvature and would appreciate it if an expert could help me out. Let $M^n$ be a closed spin manifold with ...
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### Causal fermion systems fromm fractal geometry

Okay, first off- I apologise if this is a stupid question. I'm mainly a very young physics guy, but this has primarily math basis. I'm trying to build a theory that is, long story short, some ...
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### An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space

Is there a closed, smooth, orientable manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$? Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not ...
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### Regularilty of Commutative Spectral Triples

In Connes' approach to non-commutative geometry, the notion of a spectral triple is said to generalize compact Riemannian manifolds to the non-commutative setting. Motivating classical examples ...
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### Sections of “forgetful” projections between flag manifolds

Given a subset $S\subseteq\{1,\cdots,n\}$ there is an associated flag manifold $F(S)$. Whenever $A\subseteq B$ there is a "forgetful" projection $F(A)\leftarrow F(B)$ (in fact I think its fibers are ...
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### Matrix expression for elements of $\text{SO}_0(1,4)$

Denote by $\text{SO}_0(1,4)$ the identity component of the special linear isometry group $\mathrm{SO}(1,4)$ of the Lorentz-Minkowski space $\mathbb{R}_1^5$, that is, of \text{SO}(1,4)=\left\{X\in\...
Let $M^d$ be a $d$-dimensional orientable spin manifold, and $N^4$ is a closed $4$-dimensional orientable submanifold of $M^d$. Is $N^4$ always spin? If $d=5$, is $N^4$ always spin? If $N^4$ is a ...