Calculating the super-virasoro generators in terms of the oscillators of the string.

+ 1 like - 0 dislike
92 views

I'm doing an easy calculation. The problem is to write down the generators of the super-virasoro algebra in terms of the bosonic and fermionic oscillators. I'm following Polchinski. I'm trying to derive equation (10.2.12a)

We write $$T_{B}(z)=\sum_{n \in \mathbb{Z}}\frac{L_{n}}{z^n+2}$$

We invert to get $$L_{n}=\oint \frac{dz}{2\pi i}z^{n+2} T_{B}(z)$$

Now, $$T_{B}(z)=-\frac{1}{2}\psi \partial \psi$$+ contributions from bosons

I'm trying to calculate the $L_n$ in terms of the fermionic oscillators. I get

$$L_n=\frac{1}{4}\sum_{r\in \mathbb{Z}+\nu}(2r+1)\psi_{n-r}\psi_{r}$$

However, the correct formula is : $$L_n=\frac{1}{4}\sum_{r\in \mathbb{Z}+\nu}(2r-m)\psi_{n-r}\psi_{r}$$

Is this a typo? Or is my formula incorrect?

I think that one uses the fact that $\psi^{2}=0$ from anti-commutativity, so one can add a constant multiplied by $\psi^{2}$ and then also expand $\psi$ in a laurent series.