Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

What are Killing spinors?

+ 6 like - 0 dislike
239 views

What are Killing spinors? How can they be motivated? Are they directly related to Killing vectors and Killing tensors and is there an overarching motivation for all three objects? Any answer is greatly appreciated but a less formal one would be preferred.

This post imported from StackExchange Physics at 2014-09-10 17:12 (UCT), posted by SE-user theriddler
asked Sep 10, 2014 in Theoretical Physics by theriddler (30 points) [ no revision ]
retagged Sep 10, 2014

1 Answer

+ 2 like - 0 dislike

There is an interesting relation, as the \(n\)lab says. I'll try to explain it as informally as possible.

Let \(\mathcal{M}\) be a pseudo-Riemannian manifold. Then, a Killing vector field on \(\mathcal{M}\) is a covariantly constant vector field on \(\mathcal{M}\), and "pairing two covariant constant spinors (parallel spinors, i.e., Killing spinors with \(\lambda=0\)) to a vector yields a Killing vector". Similarly, a Killing tensor on \(\mathcal{M}\) is a covariantly constant section of \(\mathrm{Sym}^k(\Gamma(\mathrm{T}(\mathcal{M})))\). Therefore, you may interpret ``Killing'' as being synonymous with ``covariantly constant'' (at least in these three cases).

answered Sep 12, 2014 by SDevalapurkar (285 points) [ revision history ]
edited Sep 12, 2014 by SDevalapurkar

This is not the definition given by most authors.  Killing spinors and parallel spinors verify \(\nabla_X \psi = \lambda X.\psi, \lambda \in \mathbb C\) . Parallel spinors (covariantly constant spinors) correspond to the case \(\lambda = 0\), while Killing spinors correspond to the case \(\lambda \neq 0\)

True. I've fixed that nLab entry.

I, too, have edited my answer accordingly. 

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsO$\varnothing$erflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...