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

PO is now at the Physics Department of Bielefeld University!

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

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  How do Killing spinors arise in Calabi-Yau compactifications?

+ 2 like - 0 dislike
1223 views

How do Killing spinors arise in CY compactifications of string theory? Why should there be Killing spinors and what is special about those CYs? I understand they preserve some SUSY which is interesting but also a sphere will preserve SUSY.

asked Feb 18, 2015 in Theoretical Physics by Outlander (95 points) [ revision history ]
edited Apr 19, 2015 by dimension10

It's about preserving precisely 1 susy, no more no less.See standard references listed e.g. here.

1 Answer

+ 5 like - 0 dislike
The whole idea of compactifying at a CY manifold to obtain 4-dimensional physics dates back to the 80's and the reason why people started thinking about it is, as Urs says, that CYs only conserve part of the supersymmetry. Consider strings in the total 10-dimensional space-time $M$ which can be "decomposed" as $\mathcal{M}=M_4 \times X_6$. This space-time is always locally 10 dimensional and thus there exists (locally) a set of 10d supercharges, transforming in the spinor representation of $SO(10)$. The supersymmetry we want to achieve, in 4d, contains supercharges which are well defined globally in the $X_6$ part of the manifold $\mathcal{M}$. Now, the compact space $X_6$ is curved and as a result if we parallel transport a supercharge around $X_6$ it does not have to return to identity. Therefore, supercharges within $X_6$ are related by parallel transport with the $SO(6)$ spin connection corresponding to the metric of $X_6$. Now, let us consider a "test" local supercharge at a point $p \in X_6 \subset \mathcal{M}$. We can parallel transport this supercharge around a closed loop $C$ and come back to $p$. This action will result in a rotation of the supercharge by a  non-trivial  element $R \in SO(6)$. The important thing here is that such supercharges that get rotated after a parallel transport do not lead to globally well-defined supercharges. What does this mean for $X_6$? It means that upon string compactification on $X_6$ this compact space will preserve some of the supersymmetry if there exist 6d non-trivial spinors $\zeta(y^i)$, where $y$ is a coordinate system in $X_6$ and $i=4, \ldots 9$. These are the Killing spinors and are covariantly constant in the $X_6$, i.e. $\nabla_{X_6}\zeta$ =0. Now, here is where the holonomy group of $X_6$ plays a special role. Generally, 6d manifolds have the holonomy group $SO(6)$ and all components of the 10d spinor should get transformed under this. This means that there are no Killing-spinors and as result all supersymmetries are broken in a general manifold of such a holonomy group. But we need to consider manifolds that preserve some susy, that is manifolds with special holonomy groups which should be subgroups of the bigger $SO(6)$ group. What will work for us are manifolds with $SU(2)$ holonomy such that indeed they preserve some supersymmetry. Such manifolds are the Calabi-Yau manifolds. This gives us the connection between Killing spinors and CYs.

This gives the basic idea. Now, as Urs says, there are nice reference with more details in the nLab entry.
answered Feb 19, 2015 by conformal_gk (3,625 points) [ no revision ]

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$\hbar$ysicsOverf$\varnothing$ow
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).
Please complete the anti-spam verification




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

Your rights
...