• 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.


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


(propose a free ad)

Site Statistics

203 submissions , 161 unreviewed
5,006 questions , 2,162 unanswered
5,341 answers , 22,655 comments
1,470 users with positive rep
815 active unimported users
More ...

  What are Killing Spinor Equations?

+ 4 like - 0 dislike

Killing spinor equations are equations that result from supersymmetric transformations. One example of those is for example

$$\nabla_{\mu}\epsilon+\frac{i}{2}A_{\mu}\gamma_*\epsilon+\frac{i}{4}Im\mathcal{N}_{AB}\gamma_{\mu\nu}G^{\mu\nu A}(Im L^{B}-i\gamma_* Re L^{B})\gamma_{\mu}\epsilon=0$$ This is the usual one present in $N=2, d=4$ Supergravity theories.

As suggested by some books and papers on the web, this is the vanishing of the gravitini supersymmetry.

My related questions are:

1- Why do we have to solve these equations known as killing spinor equations? In other words, how are their solutions important or beneficial in any way?
2- I understand that there in supergravity there is a graviton, 2 gravitini and a 'so-called' graviphoton. So if fermions here should vanish in $N=2$ supergravity theories (those coupled to vector multiplets), why don't we see vanishing of the graviton supersymmetry or gravi-photon supersymmetry but instead we see vanishing of gaugino supersymmetry? Where am I confused about here?

asked Dec 25, 2015 in Theoretical Physics by physicsoutsideborders (40 points) [ revision history ]
edited Dec 26, 2015 by physicsoutsideborders

1 Answer

+ 3 like - 0 dislike

The underlying point of the question is: how to define a supersymmetric background ? At the classical level, this is done by requiring the vanishing of the supersymmetric variations of the various fields present in the theory. The supersymmetric variation of a bosonic field is a fermionic field but if we want a Lorentz invariant background, the background values of the fermionic fields are set to zero and so the supersymmetric variations of the bosonic fields always vanish in the background and the non-trivial constraints on the possibly non trivial background values of the bosonic fields come from the requirement of vanishing of the supersymmetric variations of the fermionic fields. The Killing spinor equation is the constraint on the background coming from the vanishing of the supersymmetric variation of the gravitino.

answered Dec 28, 2015 by 40227 (5,140 points) [ revision history ]

Thanks a lot for your interesting answers @40227 . I would like to ask you about the second part of my question. I restate it in more details here.

I understand that there in $N=2$ $D=4$ supergravity coupled to vector multiplets there is a graviton, 2 gravitini and a 'so-called' graviphoton. This can be found here http://arxiv.org/pdf/math/0002122.pdf where the author says: "As we neglect here the hypermultiplets, we have to consider the basic supergravity multiplet and the vector multiplets.The supergravity sector contains the graviton, 2 gravitini and a so-called graviphoton" So if fermions here should vanish in $N=2$ supergravity theories (those coupled to vector multiplets), we should see vanishing of gravitini supersymmetry only in order to give us the first Killing spinor equation.My question is that I see the presence of both: the vanishing gravitini and gaugini supersymmetry in some papers in $N=2, D=4$ supergravity coupled to vector muliplets. Where does the gaugini supersymmetry come from?

As indicated in my answer, to obtain a supersymmetric background one needs the vanishing of the supersymmetric variation of all the fermionic fields. In particular in a supergravity theory coupled with vector multiplets one needs the vanishing of the supersymmetric variation of both the gravitini present in the gravity multiplet and of the gaugini present in the vector multiplets. The Killing spinor equation is the equation associated to the gravitini and there is another equation associated to the gaugini.

Ah thanks @40227 I got it now. Your answers are always very helpful. So, the gravitini is present in the gravity multiplet and gaugini is present in the vector multiplet. What confused me is the fact that Van Proeyen did not mention this in the attached paper, or I did not read it clearly, or even worse, I might have not understood it. He said, "The supergravity sector contains the graviton, 2 gravitini and a so-called graviphoton." So, he implied that the supergravity sector is the one containing the gravitini. So, does this mean that the supergravity sector the same thing as the gravity multiplet? 


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:
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