# Questions tagged [spinor]

The spinor tag has no usage guidance.

21
questions

**3**

votes

**1**answer

205 views

### 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 ...

**4**

votes

**1**answer

229 views

### 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 ...

**0**

votes

**1**answer

108 views

### 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 ...

**0**

votes

**0**answers

96 views

### 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**

votes

**1**answer

205 views

### 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 ...

**2**

votes

**1**answer

223 views

### 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 $...

**16**

votes

**0**answers

679 views

### 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 ...

**8**

votes

**1**answer

111 views

### 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 ...

**2**

votes

**0**answers

156 views

### 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

175 views

### 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 $S=S^+\oplus S^-$, ...

**6**

votes

**1**answer

372 views

### 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\...

**3**

votes

**0**answers

110 views

### 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,\...

**20**

votes

**1**answer

2k views

### 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 ...

**7**

votes

**2**answers

472 views

### 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&...

**4**

votes

**2**answers

574 views

### $spin_{\mathbb{C}}$ 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}}...

**6**

votes

**1**answer

107 views

### 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 $...

**6**

votes

**2**answers

538 views

### 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 := \...

**4**

votes

**0**answers

68 views

### 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 ...

**4**

votes

**1**answer

200 views

### 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 ...

**7**

votes

**2**answers

426 views

### 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 ...

**2**

votes

**0**answers

149 views

### 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 ...