Vector and Spinor Representation in Ramond-Neveu-Schwarz Superstring Theory I am learning Ramnond-Neveu-Schwarz Superstring theory (RNS theory). I often find the following notation, especially in the closed string spectrum etc.:
$$\mathbf{8}_s,\mathbf{8}_v $$
And it is noted that these are vector and spinor representations of something. I have two questions regarding these.


*

*What are these representations of? Are they representations of $SO(8)$?

*What do they actually mean? How do you represent something in vector/spinor notation.
 A: Yes, they're representations of $SO(8)$, more precisely $Spin(8)$ which is an "improvement" of $SO(8)$ that allows the rotation by 360 degrees to be represented by a matrix different from the unit matrix, namely minus unit matrix.
${\bf 8}_v$ transforms normally as 
$$ v\mapsto M v$$
where $MM^T=1$ is the $8\times 8$ real orthogonal $SO(8)$ matrix. The spinor reps ${\bf 8}_s\oplus {\bf 8}_c$ label the left-handed and right-handed spinor, respectively. People usually learn spinors well before they study RNS string theory.
The spinor representation transforms under $SO(8)$ in a way that is fully encoded by the transformation rules under infinitesimal $SO(8)$ transformations, $1+i\omega_{ij} J^{ij}$ where $\omega$ are the angle parameters and $J$ are the generators. 
In the Dirac spinor representation, $J_{ij}$ is written as
$$ J_{ij} = \frac{\gamma_i \gamma_j - \gamma_j\gamma_i}{4}$$
where $\gamma$ are the Dirac matrices that may be written as tensor products of Pauli matrices and the unit matrix and that obey
$$\gamma_i\gamma_j+\gamma_j\gamma_i = 2\delta_{ij}\cdot {\bf 1}$$
Each pair of added dimensions doubles the size of the Dirac matrices so the dimension of the total "Dirac" representation for $SO(2n)$ is $2^n$. For $n=4$ we get $2^4=16$.
This 16-dimensional spinor representation is real and may be split, according to the eigenvalue of the $\Gamma_9$ chirality matrix, to the 8-dimensional chiral (=Weyl) spinor representations labeled by the indices s,c.
For $SO(8)$, there are 3 real 8-dimensional irreducible representations that are "equally good" and may actually be permuted by an operation called "triality". This operation may be  seen as the $S_3$ permutation symmetry of the 3 legs of the Mercedes-logo-like $SO(8)$ Dynkin diagram. I just wrote a text about it last night:

http://motls.blogspot.cz/2013/04/complex-real-and-pseudoreal.html?m=1

If you really need to explain what a representation of a group is, you should interrupt your studies of string theory and focus on group theory – keywords Lie groups, Lie algebras, and representation theory. Without this background, you would face similar confusion too often.
