# 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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### Demonstrate $\mathrm{Spin}(3, 3) = \mathrm{SL}(4, R)$

It seems that there is an accidental isomorphism $$\mathrm{Spin}(3, 3) = \mathrm{SL}(4, R).$$ Some facts I am aware is that $\mathrm{SL}(4, R)$ has the Lie algebra of 15 generators. the homotopy ...
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### Lie algebra of SO(V) is isomorphic to spin(V)

where $a\in Cl^2(V)$, and $\tau(a)(v)=[a,v]$ for $v\in V$. I was wondering is the first equality correct as written? Jost earlier defined the clifford algebra as the quotient of the tensor algebra by ...
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### What makes spinors mysterious?

Everyone familiar with spinors presumably knows the quote by Sir Michael Atiyah, that spinors are mysterious in spite of their algebra being formally understood. I have heard this sentiment echoed in ...
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### Can spinors be generalized from $\mathbb{Z}_2$ to arbitrary polyhedral symmetry groups via topological gluing?

The classical example of a spinor is to consider a path around the center of a mobius strip, and attach to each point in the path a perpendicular vector that locally points "up". This ...
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### Why Spin(V) is a connected group?

I want to know why $Spin(V)$ is connected. I am watching $Spin(V)$, where $V$ is a real vector space with a positive defined metric, as the products of elements of the Clifford algebra $Cl(V)$ of the ...
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### How to take the pullback of a Witt basis onto a subspace

I'm interested in how to take the pullback of a Clifford algebra onto an induced metric. Suppose we take the topological three-sphere embedded in $\mathbb{R}^{4}$ and we allow for a complexified ...
49 views

### Triangulable Spin 5-manifolds but do not admit any smooth structure

I noticed the discussion in this post, which states that there are examples of D=5 spin manifolds which are triangulable, but do not admit any smooth structure. What are some of these examples of D=...
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### How to understand the spin structures on $T^d$ and $\mathbb{RP}^d$

How to understand the spin structures on $T^d$ and $\mathbf{RP}^d$? We know that $H_1(T^d,\mathbb{Z}_2)=(\mathbb{Z}_2)^d$. It looks that there are $2^d$ choices on 1-cycle of $T^d$. But it seems ...
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### Quadratic enhancements of the intersection form and spin structures

If we search spin structures and quadratic enhancements of the intersection form, there are many papers discussing the issue. What are the relations and basic intuitions between the quadratic ...
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### What do characteristic classes of spinor bundle depend on?

Let $M^n$ be a smooth manifold. Equip $M$ with a Riemannian metric and let $S$ be a spinor bundle. We can consider characteristic classes of $S$ (or $S_+,S_-$ for when $n$ is even), for example the ...
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### Harmonic Spinors with Vanishing Christoffel Symbols

The Dirac equation can be defined as $$D\psi = \Sigma_{i=1}^3 c(e_i) \nabla_{e_i},$$ where $$\nabla_{e_i} = e_i + \frac{1}{4} \Sigma_{j,l=1}^3 \Gamma^l_{ij} c(e_j) c(e_l).$$ $c$ is the standard ...
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### Prove $SU(n)\times Spin(k) \subset \frac{{Spin}(2n)\times Spin(k)}{\mathbb{Z}/2}$ for sufficient $n$ and $k$?

We knew that $SU(n) \subset {Spin}(2n)$ is true from Is $SU(n) \subset \text{Spin}(2n)$? also $SU(n) \subset {SO}(2n)= \frac{{Spin}(2n)}{\mathbb{Z}/2}$ is true from $U(n)$ is a subgroup of $SO(2n)$...
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### Spin connection on Ricci-flat anti-self-dual 4-manifolds

During a talk I heard it was claimed (without proof) that the canonical connection $\nabla^{S^-}$ on the bundle $S^-\to X^4$ of negative chirality spinors over a spin, Ricci-flat, anti-self-dual (ASD) ...
Let $X$ be an compact oriented Riemannian 4-manifold with $b_2^+\geq1$. Let $\omega$ be a self-dual harmonic two form vanishing transversely. On $X\setminus \omega^{-1}(0)$, the spinor bundle $S_+$ ...