Questions tagged [spinor]

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20
votes
1answer
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
2answers
380 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 ...
7
votes
1answer
306 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&...
6
votes
2answers
427 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 := \...
6
votes
1answer
98 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 $...
4
votes
1answer
145 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\...
4
votes
1answer
193 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 ...
4
votes
0answers
62 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 ...
3
votes
2answers
470 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}}...
3
votes
0answers
86 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,\...
2
votes
0answers
140 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 ...