# relating spinor and fundamental representation for $E_8$

+ 2 like - 0 dislike
65 views

While proving a very important relation which is satisfied both by $SO(32)$ AND $E_8$, which makes it possible to factorize the anomaly into two parts. The relation is $Tr(F^6)=\frac{1}{48}TrF^2TrF^4-\frac{1}{14400}(TrF^2)^3$, where trace is in adjoint representation.

I am able to prove this relation but while doing so, I have some identities which relates the spinor representation $128$ of $SO(16)$ to fundamental representation of $SO(16)$ which I must show but this is not working out.

The simplest one being $TrF^2=16trF^2$, where $Tr$ is in spinor representation $128$ and $tr$ is in fundamental representation. There are other relations showing the equality between $TrF^4$ and $tr(F^2)^2$ and $trF^4$. I am aware that spinor representation would be $\sigma_{ij}$ which is $128$ dimensional. While trying to prove these identities, I have noticed that if $F^2$ in the fundamental representation is diagonal with only two elements -1 and -1 and if $\sigma_{ij}^2=\frac{-I}{4}$ where $I$ is $128$ dimensional identity matrix then I can get the result. But I can not convince myself why it should be true.

Any details would be appreciated of how to prove it. The identities can be found in GSW chapter 13 last section (VOL.2).

This post imported from StackExchange Physics at 2014-08-01 20:15 (UCT), posted by SE-user user44895
 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:$\varnothing\hbar$ysicsOverflowThen 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.