Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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.

373 questions
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Is there Geometric Interpretation of Spinors?

Usually in Physics we define a spinor to be an element of the $\left(\frac{1}{2},0\right)$ representation space of the Lorentz group. Essentially this boils down to the 'n-tuple of numbers that transforms like a spinor' definition that physicists…
gautampk
  • 542
23
votes
2 answers

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the above is true? Note. More precisely, from Wikipedia:…
wonderich
  • 6,059
23
votes
2 answers

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 other places, too. Being a novice to the subject,…
user569579
  • 609
21
votes
3 answers

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions (for example, the use of half-angles for…
JCW
  • 718
21
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0 answers

Do Hopf bundles give all relations between these "composition factors"?

Write a fiber bundle $F\to E\to B$ in short as $E=B\ltimes F$ (in analogy with groups). (This is not necessary, but: given another bundle $X\to B\to Y$, we can write $E=(Y\ltimes X)\ltimes F$, but we may also compose $E\to B$ with $B\to Y$ to get…
anon
  • 155,259
20
votes
3 answers

A space more fundamental than Euclidean space

Summary: The mathematical physicist Paolo Budinich attributes to Élie Cartan the statement that the geometry of pure spinors is "more elementary" or more "fundamental" than Euclidean geometry, which is "more complicated". This raises several…
TROLLHUNTER
  • 8,875
14
votes
1 answer

Reference for rigorous treatment of the representation theory of the Lorentz group

When I studied representation theory for the first time it was only focused on finite groups. It was the second half of a one semester course in group theory, and the book employed was "Representation Theory: A First Course" by Harris and…
Gold
  • 28,141
12
votes
0 answers

Conflicting definitions of a spinor

I've come across two definitions of "spinors" that I'm having a hard time reconciling: Spinors are the "square root" of a null vector (see here, and also Cartan's book "The Theory of Spinors") Spinors are minimal ideals in a Clifford algebra (see…
eigenchris
  • 2,182
12
votes
2 answers

References for learning real analysis background for understanding the Atiyah--Singer index theorem

I am interested in learning the Atiyah--Singer theorem, and its version for families of operators. For this purpose, I have tried to read the recent book by D.Bleecker et.al.. However I have difficulty understanding some proofs due to my weak…
user90041
  • 5,628
12
votes
2 answers

Intuition behind definition of spinor

Some time ago I searched for the definition of spinors and found the wikipedia page on the subject. Although highly detailed the page tries to talk about many different constructions and IMHO doesn't give the intuition behind any of them. As far as…
Gold
  • 28,141
11
votes
1 answer

Is a "spinor" an element of the Spin group, or an object that transforms under the Spin group—or both?

I've been researching spinors, and I'm a bit confused by some of the terminology. In some cases, spinors seem to be presented as elements of the Spin group, whereas in others they seem to be presented as "vector-like" objects that transform under…
TheMac
  • 195
10
votes
4 answers

How to prove the tensor product of two copies of $\mathbb{H}$ is isomorphic to $M_4 (\mathbb{R})$?

How to prove the tensor product over $\mathbb{R}$ of two copies of the quaternions is isomorphic to the matrix algebra $M_4 (\mathbb{R})$ as algebras over $\mathbb{R}$? More precisely, the problem is to show the isomorphism $\mathbb{H}…
Lao-tzu
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10
votes
1 answer

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a spinor. One is representation theoretic. In this point…
Mozibur Ullah
  • 7,161
10
votes
1 answer

Can one reformulate tensor methods and young tableaux to account for spinor representations on $\operatorname{SO}(n)$?

Standard tensor methods and Young tableaux methods don't give you the spinor reps of $\operatorname{SO}(n)$. Is this because spinor representation are projective representations? If so, where does this caveat of projective representations enter…
DJBunk
  • 221
10
votes
1 answer

Do oriented null cobordant manifolds admit spin structures?

Let $M$ be an oriented null cobordant manifold. Since $M$ is oriented its first Stiefel-Whitney class vanishes. Since $M$ is null cobordant all of its Stiefel-Whitney numbers vanish. Is it known if this implies that the second Stiefel-Whitney class…
Flaccid Sheaf
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