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

+ 4 like - 0 dislike
127 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

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.
 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): Email me at this address if my answer is selected or commented on: 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$\hbar$y$\varnothing$icsOverflowThen 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.