# 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.

120 questions
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### Adjoint of the Coupled Covariant Derivative on Spinors

I want to understand the proof for a $C^0$ bound of solutions to the Seiberg-Witten equations. Among other places, it can be found in Kronheimer, Mrowka: "The genus of embedded surfaces in the ...
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### Spin structure and characteristic classes

I do not know if anyone can help me with these doubts of spin structures and characteristic classes. 1) Is there an orientable manifold that is not spin? 2) Is there a finite group $G$ such that ...
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### Spin structure and bordism

I have some questions about bordism and spin structures on manifolds. If you have any examples or references I would appreciate it. Is there a 3-manifold $M$, orientable, which does not support 3 ...
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### spinor representation

I am trying to study Special unitary group of order 2 and some textbooks mention objects transform under special unitary group are called Spinors then How can we represent a spinor using matrix ?
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### Spinors - Groups and Double Cover of Lorentz Group

As part of a project, I keep coming across a small nit-picking area regarding the spinor group $SU(2)$. The Lorentz group can be thought of as the group of rotations in $SO(1,3)$. I am under the ...
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### Spinor bundle of a fibration over a circle

I have a question: I am reading an article and the author is looking at a submersion $Z \rightarrow M \xrightarrow{\pi} S^1$ with $Z$ is of even dimension. We have $TM = TZ \oplus T S^1$. Given ...
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### $W$ spin implies $\partial W$ spin

Let $M$ be a compact orientable manifold with the first two Stieffel-Whitney numbers equal to zero (this is my definition of SPIN manifold). Let $B$ be the boundary of $M$; I want to show that $B$ is ...
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### Any spin $M^3$, exists a natural induced $\text{Pin}^-$ structure on Poincare dual PD

It seems that "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural ...
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### Clifford Multiplication and Spinor Bundles

I am trying to follow the discussion of Clifford multiplication on page 384 of The Wild World of 4-Manifolds, by Alexandru Scorpan (link, although I hope this will be totally self-contained), and I'm ...
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### spin mapping class group of circles

The MCG of the circle is $\mathbb{Z}/2$, generated by an orientation-reversing diffeomorphism. The circle has two spin structures, a periodic one and an anti-periodic one. Each has a nontrivial ...
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### On Proposition 2.6 Gualtieri Thesis Generalized complex geometry

I'm working with Gualtieri's thesis about Generalized complex Geometry and I don't understand the proof of the Proposition 2.6 (p. 7). It says Every maximal isotropic subspace (maximal totally null ...
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### Obstruction theoretic approach to complex spin structures

It seems this question is extremely well-known in literature, but I have poor understanding on these stuffs, and it seems nobody asks this question before in math.stackexchange, so I decided to ask ...
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### Can somebody show an explicit example of a spinor?

I've seen a lot of articles of spinors defined using representation theory, Clifford Algebras, etc. But i want to see (if its possible) how a spinor "looks like" and how it changes sign with a 2pi ...
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### Clifford action of exterior derivative of self-dual 2-form

I am confused about a claim on page 30 of the lectures notes of Hutchings and Taubes on Seiberg-Witten theory (https://math.berkeley.edu/~hutching/pub/tn.pdf). Background: Let $M$ is a Riemannian 4-...
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### The correspondence between complex spin structures and associated clifford module structure

The usual definition of a $spin^c$ structure on a riemannian manifold $M$ is often given by (the equivalence class of) the lifting of the frame bundle $P\to M$ of the tangent bundle $TM$ via the ...
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### Spin structure and restriction to fibers

Let us assume that $E$ is an oriented Riemannian vector bundle equipped with spin structure. Therefore there is a $spin(n)$ principial bundle $spin(E)$ and an equivariant map $\eta:spin(E) \to SO(E)$ ...
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### Representations of $S O( n )$ coming from $GL( N ,\mathbb{ R})$

I would like to show that "the finite-dimensional spinor representation of $SO(N)$ does not arise from a finite-dimensional representation of $GL(N)$" , as stated here and here. Apparently we should ...
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### Swing-Twist decomposition for quaternion rotations (verification)

PRZEMYSLAW DOBROWOLSKI has written a paper that (I think) can be applied to swing-twist decompositions for quaternion rotations called "SWING-TWIST DECOMPOSITION IN CLIFFORD ALGEBRA" . I tried to ...
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### Confusion with two different notations for heighest weights of spin representations

I don't understand how the two notations for $SO(n)$ heighest weights in fact 1 and 2 are related; moreover, what are the weigths of all other irreducible spinorial representations in notation 2? ...