Ghost number 1

The 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:


Remember are defined from and similarly .
The bar over index means:

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 .