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.
8v transforms normally as
v↦Mv
where
MMT=1 is the
8×8 real orthogonal
SO(8) matrix. The spinor reps
8s⊕8c 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+iωijJij where ω are the angle parameters and J are the generators.
In the Dirac spinor representation, Jij is written as
Jij=γiγj−γjγi4
where
γ are the Dirac matrices that may be written as tensor products of Pauli matrices and the unit matrix and that obey
γiγj+γjγi=2δij⋅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
2n. For
n=4 we get
24=16.
This 16-dimensional spinor representation is real and may be split, according to the eigenvalue of the Γ9 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 S3 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.
This post imported from StackExchange Physics at 2014-03-07 16:30 (UCT), posted by SE-user Luboš Motl