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
4,873 answers , 20,701 comments
1,470 users with positive rep
502 active unimported users
More ...

A paradoxical equation in RNS string fermionic part

+ 3 like - 0 dislike
581 views

It is well known for RNS string, $i.e.$, worldsheet supersymmetry formalism, the open string NS sector has worldsheet fermion expansion:

\begin{equation} \psi^{\mu}_{\pm} = \frac{1}{\sqrt 2} \sum_{r \in \mathbb{Z}+\frac{1}{2}} b^{\mu}_r e^{-i r (\tau \pm \sigma)} \end{equation}

The super-Virasoro algebra can be extracted from the energy momentum tensor \begin{equation} T_{++} = (\rm{bosonic ~part}) + \frac{i}{2}\psi_+ \partial_+ \psi_+ \end{equation}

These two equations can be found at Becker, Becker, Schwarz's book, page 121, 123, and is verified by myself. However, when trying to extract the fermionic part of the super-Virasoro algebra, $L_m$, there are two ways to do. One is by direct substituting the mode expansion for $\psi_+$ in to $T_{++}$ and read off the $e^{-i m (\tau+\sigma)}$ coefficent, the other is to use the integration \begin{equation} L_m = \frac{1}{\pi} \int_{-\pi}^{\pi} T_{++} e^{i m \sigma} d\sigma \end{equation}

However, when calculating, we get: \begin{equation} \begin{aligned} \frac{i}{2}\psi_+ \partial_+ \psi_+ & = \frac{i}{4} \sum_{r,s \in \mathbb{Z}+\frac{1}{2}} (-i s)b_r \cdot b_s e^{-i (r+s) (\tau + \sigma)}\\ & = \frac{1}{4}\sum_{m \in \mathbb{Z}}\sum_{r \in \mathbb{Z}+\frac{1}{2}}(m-r) b_r \cdot b_{m-r}e^{-i m (\tau+\sigma)}\\ & = \frac{1}{4}\sum_{m \in \mathbb{Z}}\sum_{r \in \mathbb{Z}+\frac{1}{2}}(m+r) b_{-r} \cdot b_{m+r}e^{-i m (\tau+\sigma)} \end{aligned} \end{equation}

and one then put normal ordering. Here $\partial_+ = 1/2(\partial_{\tau}+\partial_{\sigma})$. In the book, page 126, the coefficent is actually $(m+2r)$. I also checked GSW's book, as well as Polchinski's book, which indicates I am wrong at some point. But after long time looking for it I do not know why.

It is a very stupid and technical question, but I have been stuck to it for some time. So it would be great that someone may help solve it.


This post imported from StackExchange Physics at 2014-10-23 07:29 (UTC), posted by SE-user Kevin Ye

asked Oct 19, 2014 in Theoretical Physics by Kevin Ye (45 points) [ revision history ]
edited Oct 23, 2014 by Dilaton

1 Answer

+ 0 like - 0 dislike

The normal ordering makes the expression of $b_{-r}\cdot b_{r+m}$ symmetric around the point $r=-\frac{m}{2}$, and these symmetric parts differ by a sign. This sign difference due to normal ordering makes $m/2$ redundancy.

This post imported from StackExchange Physics at 2014-10-23 07:29 (UTC), posted by SE-user Kevin Ye
answered Oct 20, 2014 by Kevin Ye (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$\hbar$ysi$\varnothing$sOverflow
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
...