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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1answer
62 views

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 ...
0
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1answer
54 views

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 ...
5
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0answers
108 views

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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20 views

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 ...
1
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1answer
39 views

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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10 views

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 ...
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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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49 views

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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22 views

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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0answers
23 views

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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17 views

Differential Operator and Trivialization Change in Lawson-Michelsohn

I am currently reading the excelent book by Lawson and Michelsohn on Spin Geometry. In chapter 3 paragraph 1, they define a differntial operator of order $m$ to be a $\mathbb{C}$-linear map of $\...
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1answer
75 views

Obstruction to a Spin structure on a bundle ξ, and ξ ⊕ $n$ det ξ

In Ref, it says that: The obstruction to putting a Spin structure on a bundle $ξ (= Rn → E → B)$ is $w_2(ξ) \in H^2(B;Z/2Z)$. Pin± structures is that Pin− structures on ξ correspond to Spin ...
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1answer
47 views

Does only the group $SO(n)$ admit a double cover which is $\operatorname{Spin}(n)$?

The double cover of the group $SO(n)$ is the spin group $\operatorname{Spin}(n)$. Do any of the other groups $SU(n)$, $\operatorname{Sp}(n)$, $G_2$, $F_4$, $E_6$, $E_7$, $E_8$ have double covers? If ...
9
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1answer
163 views

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 ...
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33 views

Supergroup for SO groups vs Spin group

We know that the $\mathbb{Z}/2$ central extension of $ SO(d) $ can give a nontrivial double/universal cover of $SO(d)$ known as the $Spin(d)$ group. They have this relation $$ 1 \to \mathbb{Z}/2 \to ...
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47 views

Examples of non-spin 2-manifolds

Do we have some examples of 2d manifolds, which are non-spin (thus $w_2(M) \neq 0$)? thus they must be non-orientable $w_1(M)^2 \neq 0$ thus $w_1(M) \neq 0$. (p.s. every 2d differentiable/triangulable ...
3
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1answer
72 views

Why does the determinant of the Dirac operator on $S^n$ approach $1$ as $n\to\infty$?

Bär and Schopka (reference below) present an interesting conjecture regarding the determinant of the Dirac operator on spheres $S^n$. The conjecture is simple: in the limit $n\to\infty$, the ...
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46 views

Are these two Seiberg-Witten equations the same?

In Jurgen Jost's book on Riemannian Geometry, he describes the following equation as one of the Seiberg-Witten equations: $$F^{+}_A = \frac{1}{4}\langle e_j \cdot e_k \cdot \varphi , \varphi \rangle e^...
2
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0answers
108 views

A spinorial generalization of the Hopf map

If $V = \mathbb{R}^3$ with the Euclidean inner product $g$, and $S = \mathbb{C}^2$ is the corresponding space of spinors, then there is a quadratic map $h: S \to V^*$, which maps $\psi \in S$ to $h(\...
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28 views

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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43 views

Spinors and Klein-Gordon Equation

I'm currently working through Chapter 13 of Wald's General Relativity and spinors are being a little illusive to me. The question is pretty much: Using the Klein-Gordon equation in the form: $$\...
2
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0answers
31 views

Compare two definitions of equivariant index of Dirac operator

Let $(M^{2n},g)$ be a Riemannian manifold with spin structure. Let $G$ is a compact Lie group. Suppose that $G$ acts on $M$ smoothly and the metric is $G$-invariant. Also, assume that the $G$ action ...
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1answer
52 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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34 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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40 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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47 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 ...
4
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61 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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37 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 ...
3
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1answer
100 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 ...
4
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105 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? $Spin(2n)$ ...
0
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1answer
56 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 ...
5
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2answers
134 views

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
38 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 $1$, ...
0
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1answer
33 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 ...
2
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0answers
29 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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11 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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0answers
33 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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0answers
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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0answers
38 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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18 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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0answers
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 ...
2
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0answers
94 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 $d\in \mathbb{R}$ ...
7
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1answer
76 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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0answers
35 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 ...
4
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1answer
149 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 $SO(2n)$...
2
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0answers
42 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) ...
9
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0answers
87 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 $S_+$ ...
5
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1answer
116 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}...
0
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0answers
25 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 ...