# About the notation OSP(8|4) etc.

1. May 13, 2009

### isospin

Are there any books or papers which explain these notations like OSP(N|4), sl(m|n)? It seems they are all considered as superalgebras, but how is this kind notation generally defined?
By the way, I know for a 3-d superconformal field theory, OSP(8|4) means a SO(8) R-symmetry and a Sp(4) conformal symmetry, which is isomorphic to SO(2, 3).
Thanks.

2. May 13, 2009

m|n denotes the number of bosonic (ordinary, even) dimensions and the number of fermionic (odd) dimensions. Thus, sl(m|n) is the special linear algebra acting on a superspace with m even and n odd dimensions. OSP stands for ortho-symplectic, because it mixes aspects of the orthogonal and symplectic groups. Recall that the orthogoal group preserves the even metric, and the symplectic group preserves the odd symplectic metric; the OSP group preserves a metric which symmetry depends on Grassmann parity.

3. May 13, 2009

### xepma

By any chance, do you know a good book or reference paper on these matters?

4. May 13, 2009

### Haelfix

Try the book Supermanifolds by DeWitt