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 is the underlying vector space of the Super-Poincaré algebra?

+ 4 like - 0 dislike
56 views

The Poincare algebra $P$ is the direct sum $$P=\mathbb{R}^4\oplus so(1,3)$$ which is a real vector space of dimension $10$.

The $\mathcal{N}$-extended supersymmetry algebra is a graded Lie algebra, which enhances the Poincare algebra with fermionic generators $$Q_a^I, \quad \bar{Q}^J_{\dot{a}}$$ which transform under Lorentz transformations in the $(1/2,0)$ and $(0,1/2)$ representations respectively. The indices $a,\dot{a}$ both take values $1,2$ and $I,J$ take values $1,2,\dots,\mathcal{N}$.

The full vector space is then the graded vector space $$P\oplus g_1$$ where $$g_1={\rm span}_\mathbb{R} \{Q_a^I, \bar{Q}^J_{\dot{a}}\}$$ is the odd part.

Question: What is the dimension of this vector space? Naively I would guess $4\mathcal{N}$ but the $\bar{Q}$ are related to the $Q$ by conjugation so perhaps just $2\mathcal{N}$? However, in $\mathcal{N}=4$ SYM I have know that the superconformal algebra is $psu(2,2|4)$, implying the dimension of the odd part is $4$, and not $8$ as my reasoning suggests.


This post imported from StackExchange Physics at 2016-10-20 14:04 (UTC), posted by SE-user ryanp16

asked Oct 10, 2016 in Theoretical Physics by ryanp16 (90 points) [ revision history ]
edited Oct 20, 2016 by Dilaton

I think the answer is $4\mathcal{N}$. The superconformal algebra is not simply the super-Poincaré algebra, it also includes other generators such as dilations and special conformal transformations (also the precise nature of the conformal transformations depends on dimension), so this might be a source of discrepancy.

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$\hbar$ysicsOverfl$\varnothing$w
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
...