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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23 views

“enlarged” form and Aut(Spin(8))

In https://en.wikipedia.org/wiki/SO(8)#Spin(8), it says that "Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a ...
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26 views

Why no Majorana spinors in Spin(4)?

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$ We know that $$ \Spin(1,3)=\SL(2,\mathbb C) $$ and $$ \Spin(4)=\SU(2) \times \SU(2). $$ It is also said $\Spin(1,...
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36 views

$16 \otimes_s 16 = [1] \oplus [5]^+$ from spinor representation

A teacher says that in the spinor representation of $Spin(10)$, there is $$ 16 \otimes_s 16 = [1] \oplus [5]^+ $$ here $16$ is irreducible spinor representation of $Spin(10)$. And $[n]$ denotes the $n$...
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41 views

Invariant bilinear forms on Dirac spinors

Let $V$ be an even dimensional complex vector space $\dim V =2m$. Equip $V$ with non-degenerate symmetric bilinear form $g$, then Clifford algebra $\operatorname{Cl}(V,g)$ is simple and thus has ...
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54 views

Formulas for the Spinor Representation Product Decompositions $2^{[\frac{N-1}{2}]} \otimes 2^{[\frac{N-1}{2}]}=?$ and …

We know that given the dimension $N$, we can construct the corresponding spinors for the $Spin(N)$ group (which has $Spin(N)/\mathbb{Z}_2=SO(N)$ so $Spin(N)$ is a double cocver of the spatial ...
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31 views

Embedding the full $U(n) \subset Spin(2n)$?

We can show that $U(n) \subset SO(2n)$. For example, we can see for a $n$-dimensional complex $\mathbb{C}$ vector: $$ Z_j=X_j +i Y_j $$ for $j=1,\dots, n$ being acted by the rank-$n$ $U(n)$ matrix ...
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80 views

Contrast between SO(n) and Spin(n) representation

Earlier I asked this Comparison between SO(n) and Spin(n) representation theory which is closed. I think the question is certainly valid and a good one. But my comments are too many and too long, so ...
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92 views

Comparison between SO(n) and Spin(n) representation theory [closed]

We know that $Spin(n)/\mathbb{Z}_2=SO(n)$. The $SO(n)$ and $Spin(n)$ have the same Lie algebra. When it comes to the representation of $SO(n)$ and $Spin(n)$, does it make any difference? $Spin(2n)$ ...
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1answer
61 views

How to prove the first two Stiefel-Whitney classes are obstruction to the existence of spin structure of manifolds [closed]

How to prove this as in the title? Is there any Čech cohomology version of characteristic classes?
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1answer
26 views

How does spinors on manifold transformed with coordinate

Everyone knows that $SU(2)$ is a double cover of $SO(3)$. $SL(2,\mathbb{C})$ is a double cover of Proper Lorentz Group $L^\uparrow_+$. These groups are all associated with the spinors and tensors in ...
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How are spin network edges related to anti-symmetric projectors on the Hilbert space of the fundamental rep of SU(2)?

In the paper here https://arxiv.org/pdf/gr-qc/9905020.pdf we see an introduction to Spin-networks of the original Penrose type i.e an undirected open graph whose edges have labels that are irreducible ...
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1answer
33 views

Spin$(n)$ is connected for $n\geq 2$

I am reading the proof that Spin$(n)$ is connected for $n\geq 2$ from the book Dirac Operators in Riemannian Geometry. I want to understand why it's sufficient to find a path between $-1$ and $1$, ...
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30 views

Weird ${\rm Spin}_{\mathbb C}(4)$ isomorphism

In T. Friedrich, "Dirac Operators in Riemannian Geometry", the author gives the following exercice (p34) : Prove that the group ${\rm Spin}_{\mathbb C}(4)$ is isomorphic to the following ...
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25 views

Identity with Representation of Clifford Algebras

I'm not sure how it works with Clifford algebras. For example, take $c : \: Cl(TM) \rightarrow \text{End}(\sigma)$ be the representation of the Clifford algebra on the bundle of spinors $\sigma$, so ...
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20 views

Vector bundles with structure groups $G_2$ and $\operatorname{Spin}(7)$

As is stated for example in this post (Structures on Vector Bundles with Reduced Structure Group), we can impose interesting restrictions on real vector bundles by demanding their transition functions ...
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10 views

Simplifying the Dirac Equation with a Perturbed Metric

I think I heard it mentioned that one could simplify the Dirac equation by taking the metric to be the perturbation of some simple metric (for example, a perturbation of the Schwarzchild metric): $g_{...
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29 views

Invariance of Clifford multiplication

I am checking some basic properties of Clifford multiplication while reading the book "Twistors and Killing spinors on Riemannian manifolds" by Baum et al. It is said that Clifford multiplication is ...
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19 views

Representations of the Spin(N) group in terms of differential operators?

The group $SO(n)$ has a representation in terms of infinitessimal generators on $\mathbb{R}^n$ $$M^{\mu\nu} = x^\mu \partial_\nu - x^\nu \partial_\mu$$ Does the $Spin(n)$ have a representation in ...
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35 views

Is there a geometrical interpretation of a spinor? [duplicate]

To give a geometrical interpretation of a vector one can associate a vector with two points in space $(A,B)$. Any vector can be thought of as an equivalence class of pairs of points. Like wise a ...
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15 views

Explicit two-cocycle for central extension of $\mathbb{Z}_2 \hookrightarrow Spin(n) \twoheadrightarrow SO(n)$?

I was wondering if there was a simple expression for a two-cocycle representing the central extension central extension of $\mathbb{Z}_2 \hookrightarrow Spin(n) \twoheadrightarrow SO(n)$. It would ...
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19 views

What's the simplest generalisation of graphs that has orientation entanglement?

Looking at this intuitive description of spinors. Is there a way to add something like this to a network graph? It looks like simple graphs can't model this as it needs the concept of lines that ...
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90 views

Integrable in $\mathcal{L}^1(\mathbb{R}^n\times\mathbb{R}^n)$ from Spin Geometry's Book

For $\xi \in \mathbb{R}^n$ we have $(1+|\xi|)^{-t}$ is integrable in $\mathcal{L}^1(\mathbb{R}^n)$ for $t>n$. Now, for $\xi,\eta \in \mathbb{R}^n$, i need to prove that for every $d\in \mathbb{R}$ ...
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73 views

What does “transform like” mean?

I read on a pdf that considering $SU(2)$ the spinor $(\xi_1, \xi_2)^T$ transform the same way as $(-\xi_2^*, \xi_1^*)^T$. What does it mean that they transform the same way? I don't know what's the ...
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29 views

The magic properties of the quotient space from $U(2^{l-1})/{\rm Spin}(2 l)$

Inspired by a previous post, Embed a Spin group to a special unitary group I am wondering what are the magic properties of the quotient space from $$U(2^{l-1})/{\rm Spin}(2 l)$$ that makes such an ...
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1answer
108 views

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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37 views

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) ...
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83 views

Curvature of canonical connection on 4-manifold with self-dual harmonic 2 form

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 $S_+$ ...
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83 views

When is the orientable double cover of a product of non-orientable surfaces spin?

Let $M_{k,l}$ denote the orientable double cover of the non-orientable four-manifold $k\mathbb{RP}^2\times l\mathbb{RP}^2$; here $k\mathbb{RP}^2$ denotes the connected sum of $k$ copies of $\mathbb{RP}...
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22 views

$K_n=R,C$ or $H$ is the maximal commuting subalgebra for an irreducible real representation of $Cl_n$?

Let $Cl_n$ denotes the clifford algebra generated by quadratic form of signature $(n,0)$. Since $Cl_n$ is either of the form $K(2^m)\oplus K(2^m)$ or $K(2^m)$ where $K=R,C,H$, real, complex and ...
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20 views

How to produce isomorphism $Cl_{r,s+1}\cong Cl_{s,r+1}$

Let $q$ be a quadratic form of signature $(r,s+1)$. Then $Cl_{r,s+1}$ denotes the clifford algebra associated to $q$ over $R^{r+s+1}$ where $R$ is real number. "$Cl_{r,s+1}\cong Cl_{s,r+1}$" $\...
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17 views

If $Cl_{r,s}$ has representation with image of $-1$ non-trivial,such representation are not induced from $O_{r,s}$ or $SO_{r,s}$?

Consider $R^{r+s}$ associated to quadratic form $q=\sum_{1\leq i\leq r}x_i^2-\sum_{1\leq i\leq s}x_{i+r}^2$. Then one can associate $Cl_{r,s}$ Clifford algebra to $R^{r+s}$. There is twisted ...
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32 views

$SO(V,q)$ is often almost a simple group?

Let $Ad:Pin(V,q)\to O(V,q)$ be defined as the following where $q$ is a non-degenerate quadratic form over a finite dimensional vector space $V$. Let $a:V\to V$ be $v\to -v$ map and induce $a:Cl(V,q)\...
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72 views

4th Stiefel whitney class of a 7-dimensional Spin manifold

In Massey's paper "On the Stiefel Whitney classes of a manifold I" he shows that manifolds of dimension n = 4s + 3 have $w_n = w_{n-1} = w_{n-2} = 0$. Where $w_i$ is a mod 2 Stiefel-Whitney class ...
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1answer
67 views

A doubt from book Spinors and Calibrations

In chapter 14, the author shows that the octonionic projective plane is the quotient of exceptional Lie group $F_{4}$ by the group $Spin(9)$ (Theorem 14.99). When he is leading with an element that ...
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44 views

Index of Dirac operator and Chern character of symmetric product twisting bundle

I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text (see image below). We are twisting the spinor bundle $\Sigma$ with an ...
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1answer
22 views

Spin structure on twice an oriented bundle

A spin structure on a principal $SO(n)$-bundle $E$ is a cohomology class in $H^1(E, \mathbb{Z}/2)$ that restricts to a generator of $H^1(SO(n), \mathbb{Z}/2)$. It is well-known that an oriented ...
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30 views

Example of surjective sheaf morphisms which are not surjective on sections

Let $X$ be a topological space and $\varphi \colon \mathcal{F} \to \mathcal{G}$ be a sheaf morphism on $X$.I know that surjectiveness of $\varphi$ means surjective stalk homomorphisms $\varphi_x \...
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78 views

Dirac Operator on $\mathbb{R}^2$

I have a doubt regarding Pag. 119/120 of Spin Geometry by Lawson and Michelson. Namely, I am trying to calculate the Dirac Operator on the Euclidean space $\mathbb{R^2}$. The Clifford Algebra is $...
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23 views

Write the $Z(SO(n))=\mathbb{Z}/2$ center subgroup in terms of the matrix of spinor representation of $Z(Spin(n))=\mathbb{Z}/4$ for $n = 2 \mod 4$

My question concerns writing the $Z(SO(n))=\mathbb{Z}/2$ center subgroup inside the spinor representation of $Spin(n)$ which has $Z(Spin(n))=\mathbb{Z}/4$ for $n = 2 \mod 4$ and $n>2$. Let us take $...
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85 views

The center of the Spin group written in the matrix of Spinor representation

My question concerns how to write the center of the Spin group written in the matrix of Spinor representation. (1). For example, for $Spin(3)=SU(2)$, we have that the center $$Z(Spin(3))=Z(SU(2))=...
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1answer
35 views

The spiral equation in system of coordinates

I have a dependence $r(t)=\sqrt[3]{a t^2}$, where r is the distance and t is the time (which is cartesian coordinate, argument from ox axis). The initial distance is $r_{0}$ and it is decreasing over ...
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1answer
57 views

spin structure on $\Gamma \backslash S^n$

Let $Γ \subset SO(n + 1)$ be a finite group acting freely on the unit sphere $S^n \subset \mathbb{R}^{n+1}$. Show that the quotient $\Gamma \backslash S^n$ admits a spin structure if and only if there ...
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14 views

Defining Clifford bundle through adjoint representation

I am working my way through Lawson's book Spin geometry. Presently I am having problems understanding the construction of $ Cl (E) $ via the adjoint representation (page 97). The author says that the ...
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54 views

Spin structure on a torus

Definition: A spin structure on a Riemannian manifold $M$ is a complex vector bundle S→M together with a isomorphism of algebra bundles Cl(M)→End(S). Here Cl(M) denotes the Clifford bundle over the ...
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16 views

Equivalent definitions for spin structure

Definition A: A spin structure on a Riemannian manifold $M$ of dimension $n$ is a $Spin(n)$-principal bundle $\pi: P \to M$ together with a bundle map $P \to SO(M)$ that reduces in each fiber to the ...
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1answer
41 views

Locus of all points

What is the locus of all points equidistant from a fixed point and a fixed circle on a sphere? (By examining an "extreme" case, i.e. the fixed point being the North Pole and the fixed circle being a ...
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1answer
92 views

Existence of Harmonic Spinors

The harmonic spinor equation states that $D \psi=0$, where $D$ is a Dirac operator and $\psi$ is a spinor. A spinor satisfying this equation is said to be harmonic. Is there some result that one ...
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20 views

Trying to understand Catmull-Rom curve

This is in 3D world space. So, I have 4 points P0, P1, P2, P3. I have to create a Catmull curve between points P1 and P2. I need 200 points between P1 and P2 to create a smooth curve. Wanted to know ...
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1answer
57 views

Is the 3-sphere isomorphic to Spin(3)?

I am trying to learn some spin geometry stuff and getting a bit confused. The unit quaternions can be thought of as a group structure on $S^3$ which gives the group $\text{Spin}(3)$. Is there some ...
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39 views

Spinors and derivatives of spinors

Spinors $\psi$ rotate vectors. For instance $a'=\psi a\psi^{-1}$. In physics, one often encounters Lagrangians with spinors. For instance, the Weyl Lagrangian is $$ L=i\psi^t _R \sigma^u \partial_u ...