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
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Supergravity, torsion and diffeomorphism invariance

+ 3 like - 0 dislike
364 views

The action for $N=1$ supergravity in an $n$ spacetime dimenions is

$$ S= \int e\left( R + \overline{\psi}_a \gamma^{abc} D_b \psi_c \right) $$ Here $R$ is the scalar curvature, $e=\det(e_{a\mu})$, and $e_{a\mu}$ is the frame field. $\psi = \psi_{\mu} dx^{\mu}$ is a spinor valued one-form. The indices $a,b\ldots = 0\ldots n-1 $ are internal indices that transform under the Lorentz group. The frame field $e_{a\mu}$ can be used to 'convert' spacetime indices to internal indices, and vice-versa. The gamma matrices obey $\gamma^a \gamma^b +\gamma^b \gamma^a = \eta^{ab} $, with $\eta^{ab}$ the 'internal metric'. $\gamma^{ab\ldots z} = \gamma^{[a} \gamma^b \ldots \gamma^{z]} $ denotes an antisymmetrised product of gamma matrices. The covariant derivative is

$$ D \psi = d \psi + \frac{1}{2} \omega_{ab} \gamma^{ab} $$

My question is the following: The RS field $\psi_{\mu}$ has a spacetime index and a spinor index, yet in the above action there is no affine connection part in the covariant derivative. The contribution from the torsion-free part of the affine connection vanishes because it is symmetric in two indices which get contracted with the antisymmetrised product of gamma matrices. But that still leaves the contorsion part. In the first order formalism, the spin connection and frame field are taken to be independent variables, so in general the spin connection may have torsion. In the second order formalism, the spin connection is not torsion free due to the presence of fermions. So in either case the contorsion tensor is non-zero. This leads me to believe that not having it in the above action will mean that the action is not invariant under diffeomorphisms.

Secondly, because the torsion is not in general zero, it seems to me that the RS action should actually be split into two pieces

$$ e\left( \overline{\psi}_a \gamma^{abc} D_b \psi_c - (D_b \overline{\psi}_a) \gamma^{abc} \psi_c \right) $$

This is because a complex conjugation should send each of those terms to each other, so that the action is real. However, if you only have one of those terms and the torsion is non-zero, when you complex conjugate you have to use integration by parts to 'move the covariant deriative to the other side', whereupon you pick up torsion tensor contributions from the covariant derivative acting on $e$, and the action is not real.

I'd be grateful if anyone could shed some light on either of these issues.

This post imported from StackExchange Physics at 2014-06-01 12:06 (UCT), posted by SE-user Steven
asked May 31, 2014 in Theoretical Physics by Steven (45 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$\hbar$ysicsOver$\varnothing$low
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
...