# Questions tagged [spinor]

The spinor tag has no usage guidance.

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### Is there a purely topological definition of $\text{Spin}(p,q)$?

I'm cross-posting this question from Math.SE, as it didn't get much attention there (even after a bounty).
A common way to define the group $\text{Spin}(p,q)$ is via Clifford algebras. However, $\text{...

2
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### Understanding the Seiberg-Witten equations in dimension $3$

I am trying to understand the dimensional reduction of Seiberg-Witten equations from dimension $4$ to $3$, more specifically my concern is about ellipticity of the new equations in dimension $3$ under ...

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### Unitary operators with the same inner product as vectors

Suppose we have a set of real unit vectors $v_1,\ldots,v_m \in \mathbb{R}^n$. We can always find a set of unitary operators $U_1,\ldots,U_m$ acting on $\mathbb{C}^N$ (for $N$ that is possibly much ...

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### Uniqueness of spinor representation

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$I asked a similar question on math stack exchange here, but I wonder if it may be better received here.
Let $n$ be ...

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### Some elementary geometric inequalities of spinorial origin

Let there be given $\psi_a \in S^3 \subset \mathbb{C}^2$ and $v_a \in S^2 \subset \mathbb{R}^3$ for $a = 1, \ldots, 4$, which are assumed to satisfy the following relations:
$$ \psi_a \psi_a^* = \frac{...

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### Action of volume form on spinors in odd dimension

We know that for a smooth orientable manifold of dimension $2n, i^n$ times the volume form acts as identity on the positive spinors and acts as minus identity on the negative spinors via Clifford ...

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### Norm of Killing spinor

A Killing spinor on a Riemannian spin manifold is a section of the spinor bundle satisfying the equation:
\begin{align*}
\nabla_X\phi=\lambda X\cdot\phi
\end{align*}
Here $X$ is a vector field and $\...

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### What are the applications of spin geometry? [closed]

What are applications of spin geometry to physics? Does it have something to do with gravity?

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### Killing spinor on $S^6$

It is known that $S^6$ has (real) killing spinor. I am looking for an explicit description of such a spinor. More generally, we can choose an almost complex structure say, $J$ on $S^6$. Now, we can ...

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### Need to prove $uD\psi=\frac{n}{2} \text{grad}(u)\cdot\psi$

I'm reading Thomas Friedrich's article "On the conformal relation between twistors and Killing spinors" and I'm stuck in a part of the proof of proposition 3 (page 64), which states:
...

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### Spinors in dimension 6

From the representation of $\operatorname{Spin}(6)\cong \operatorname{SU}(4)$, one can deduce that on a $6$-dimensional manifold we get the postive spinor bundle from the usual $4$-dimensional ...

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### Non-associative Clifford algebra

Let $V$ be a finite-dimensional $\mathbb{R}$ vector space equipped with a symmetric, bilinear form $b : V \times V \to \mathbb{R}$.
My question is if there exists an analog of a Clifford algebra in ...

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149
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### On a family of maps relating vectors to spinors, which includes a smooth map from $S^3$ to $Q^4$

Let $Cl_n$ denotes the Clifford algebra of $\mathbb{R}^n$ with its euclidean inner product $g$, where $n \geq 3$. We use the following convention:
$$ v.w + w.v = 2g(v,w)1.$$
In particular, note that ...

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1
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310
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### Understanding the quadratic part in Seiberg Witten equation

Lets take a closed four manifold $M:=\Sigma_1\times \Sigma_2,$ where $\Sigma_i$s are compact Riemann surfaces. Now if $V$ and $W$ are Spin$^\mathbb{C}$ bundles on $\Sigma_1$ and $\Sigma_2$ ...

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337
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### Pull back of Spin$^{\mathbb{C}}$ bundle

Let $M$ be a closed $4$-d Riemannian manifold and $Z$ be its twistor space of $M$, i.e., the bundle of almost complex structures on $M$. Let $V$ be a Spin$^{\mathbb{C}}$ bundle, $V_+$ denote the ...

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### Identifying a $4$-form on a $6$-dimensional manifold

Let $M$ be a closed $6$-dimensional Riemannian manifold with a spin$^{\mathbb{C}}$ structure. It is known that real $4$-forms on $M$ act on the positive-spinors as trace-free hermitian endomorphisms ...

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### Linear independence of complex polynomials and a "sum of squares" conjecture

This will take me some time to explain. Let $n \geq 2$ be a fixed integer. Let $p_i(z)$, for $i = 1,\ldots,n$ be $n$ nonzero complex polynomials of degree at most $n-1$. I am interested in ...

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131
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### Clifford multiplication formula for $d\beta$ where $\beta$ is a $3$-form

Say $X$ be a $6$-dimensional compact Riemannian manifold which admits a $Spin^{\mathbb{C}}$ structure. Now I want to have a Clifford multiplication formula by $d\beta$ in terms of $\beta$ where $\beta$...

2
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1
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### Help about "Varieties with small Dual Varieties" by L.Ein

I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...

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### Clifford multiplication formula on an almost complex manifold

$\DeclareMathOperator\End{End}$Following the deduction by John W. Morgan in his book The Seiberg–Witten equations and applications to the topology of smooth four manifolds, an almost complex manifold $...

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### Is there a "natural" proof of the equality $4^2=2^4$?

This question, or rather any answer that it might receive, would probably belong to the realm of Awfully sophisticated proof for simple facts. Still, I claim that I have quite serious motivation for ...

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### Spin networks as functionals on the moduli space of connections modulo gauge transformations on a graph

I have just read a big part of John Baez's nice article Spin network states in Gauge theory. The definitions are quite clear in that article. However, there is a part which is not explained explicitly ...

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### Bryant-Salamon $G_2$ manifold on the spinor bundle over $S^3$

I am trying to understand the spaces constructed in R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy.
My first problem is, essentially, about ...

2
votes

2
answers

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### Induced action by an involution on spinor bundle and Dirac operator

Let $M$ be a $4n$-dimensional spin manifold with a fixed Riemannian metric $g$. Let $S$ be a spinor bundle over $M$ and fix the Riemannian connection on it. There is a decomposition $S=S^+\oplus S^-$, ...

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### Explicit computation of spinor norm

I've asked this on math.stackexchange, unsuccessfully. I hope this question is appropriate for mathoverflow.
Let $V$ be a finite-dimensional vector space over a field $K$ with $\operatorname{char}K\...

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### Representation R where the center of Spin group acts trivially on R

For the following groups, I would like to know the given this group G and its representation R such that the center of G acts trivially (i.e. acts nothing) on R.
Let us denote $\operatorname{Spin}(n,\...

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2
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### Is Witten's proof of the positive mass theorem rigorous?

I noticed that the only official reason given for awarding Edward Witten the Fields medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully ...

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### What is a formal definition of a Fermionic quantum field?

I could not locate a definition of Fermionic quantum field (like for an electron!) in even Kevin Costello's book, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.4961&rep=rep1&...

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### $spin_{\mathbb{C}}$ Connection and Charge Parity

From the paper "Gapped Boundary Phases of Topological Insulators via Weak Coupling" on page 11,
https://arxiv.org/abs/1602.04251
the authors states that on a curved manifold with a $spin_{\mathbb{C}}...

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### Spinor bundle tensored with certain line bundle gives the dual spinor bundle

Let $E$ be a $spin^c$ bundle and $L_E$ be a (complex) line bundle defined using transition functions $\nu \circ g_{U,V}$ where $\nu:spin^c(n) \to \mathbb{T}$ is map such that $\ker \nu=spin(n)$ and $...

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### Action of the spin covariant derivative on gamma matrices?

How does the spin covariant derivative $\nabla^S_{\mu}$ act on gamma matrices satisfying: $\{\gamma^{\mu},\gamma^{\nu}\} = g^{\mu\nu}$, i.e.
$$\nabla^S_{\mu}\gamma^{\nu} = ?$$
where $\nabla^S := \...

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### Section of the spinor bundle over $S^{1}$ that extend to sections of the spinor bundle over $D^{2}$

Let $\mathbb{S} \rightarrow S^{1}$ be the spinor bundle associated to the connected double cover $\text{Spin}(S^{1}) \rightarrow S^{1}$. Let $\mathbb{D} \rightarrow D^{2}$ be the spinor bundle ...

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### The first eigenfunction of Dirac operator for surface

Let $M$ be a spherical oriented surface with Riemannian metric and with trivial spin structure. We know that the equation $$D \phi = \rho \phi$$, where $\rho: M\rightarrow \mathbb{R}$ is a real scalar ...

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### General questions on the eigenfunctions of Laplacian and Dirac operators

We know that the eigenvalues of the Laplacian contains a lot of information of a Riemannian manifold, but they do not determine the full information ( Hearing the shape of a drum). And the ...

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### Spinors on Quaternionic Kähler manifolds

Let $M$ be a quaternionic Kähler manifold, by which I understand a Riemannian manifold for which the holonomy group of its Levi-Civita connection is a subgroup of $Sp(1)Sp(n)$. Its complexified ...