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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.

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

  2. What do they actually mean? How do you represent something in vector/spinor notation.

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I suggest that you put an email address in your profile so that people can contact you. If someone who actually knows string theory could talk with you, they should be able to quickly identify what would be useful for you to learn next. – Mitchell Porter Apr 14 '13 at 6:48
up vote 5 down vote accepted

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:

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.

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Thanks a lot. I had learnt spinors, lie groups etc, but I didn't encounter this notation in earlier sources. Thanks for clearing my doubts. – centralcharge Apr 15 '13 at 1:13

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