Ghost number 1The elements of correspond to the global symmetry currents of the -model. There are finitely many global symmetries. We have:We will now show that is a finite-dimensional representation of , actually the adjoint representation of . How do we see this?As a representation of , is generated by the following objects:
We see that is a finite-dimensional space.
Why there are no more elements? For example, let us consider . Modulo this is same as , and using Eq. (39) this is proportional to .
Similarly, all the other covariant derivatives of every element of (50) can be expressed as a linear sum of other elements. Therefore, they form a closed representation of .