# Important Lie group representations in various dimensions and quantum field theories

+ 3 like - 0 dislike
1189 views

I was reading a nice paper by Seiberg on 5d susy gauge theories. Just in the introduction I stumbled upon something I had never thought about: "The spinor representation of SO(4,1) is four dimensional and is pseudoreal. Since the vector of SO(4,1) is in the antisymmetric product of two spinors, the miniml SUSY algebra is generated by two charges." This is is from hep-th/9608111.

My question goes as following: How can we determine such stuff? How can I know if the spinor representation of SO(something) is something dimensional? How can I find out about the isomorphisms between various Lie groups important for QFTs? Is there a list which physicists use? Is there a general reference where we can consult on such issues for any space-time dimension and any QFT? If not would it not be useful to have one?

Maybe a vague question but being in the beginning of my grad studies this is a very relevant question for people like me. Any help would be of great appreciation.

edited Oct 31, 2014

+ 3 like - 0 dislike

The best general reference (but not specifically for QFT) from a physicists point of view is probably the book

R. Gilmore,
Lie Groups, Lie Algebras, and Some of Their Applications,
Dover Publications 2006.

For a list of exceptional Lie algebra/Lie group isomorphisms see, e.g.:

http://en.wikipedia.org/wiki/Exceptional_isomorphism

there are also corresponding isomorphisms for the real versions of the groups.

answered Oct 31, 2014 by (15,747 points)

This is an interesting text on its own and thanks for the information. Despite that, I have in mind something that is in direct connection with supersymmetry and string theory. E.g. something that would include exceptional Lie groups and their isomorphisms with other Lie groups and their relation to supersymmetry and string theory. For example in GSW there is some discussion on E8. Maybe such a generic reference does not exists though.

+ 2 like - 0 dislike

Some useful reference is the Appendix B of volume 2 of Polchinski's book on string theory. This Appendix is called "Spinors and supersymmetry in various dimensions", which seems to me to be the kind of things you are interested in. This Appendix is maybe not totally complete and does not contain many proofs but it is a good starting point.

Very nice explanations related to the subject can be found in the Lubos Motl's post  http://motls.blogspot.co.uk/2013/04/complex-real-and-pseudoreal.html ;

answered Nov 1, 2014 by (5,140 points)

Appendix B of Polchinski is very useful indeed but its main topic is supersymmetry in various dimensions and how to reduce it. Surely it covers some parts of my question. As does the blog post of Lubos you mention as does another blog post of Lubos which gives a short review on the exceptional Lie groups. This is why i said that what I am asking might not exists. It seems that we could be making a useful guide in here though.

+ 2 like - 0 dislike

Two references that I always found useful to answer such questions about supersymmetry are the Physics Reports article  by Sohnius and an article by Kugo and Townsend.

answered Nov 1, 2014 by (1,545 points)

 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$\hbar$ysicsOver$\varnothing$lowThen 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.