I want to reproduce formula (4.29) in http://arxiv.org/abs/0804.1773v1 given by:

$$ Z=Tr(q^{L_{0}}\bar q^{\bar L_{0}})=|q|^{-2k} \prod^{\infty}_{n=2}\frac{1}{|1-q^{n}|^{2}} $$

where the trace is over an irreducible representation of the Virasoro algebra with a ground state of weight $(h,\bar h)=(-k,-k)$.

When I'm correct one can use the formula for the character of the Verma module $$ \chi_{(c,h)}(\tau)=Tr(q^{L_{0}-c/24})= q^{h-c/24} \prod_{n=1}^{\infty}\frac{1}{1-q^{n}}. $$

With $$ \chi_{(0,-k)}(\tau)=Tr(q^{L_{0}})= q^{-k} \prod_{n=1}^{\infty}\frac{1}{1-q^{n}} $$ and $$ \bar\chi_{(0,-k)}(\bar\tau)=Tr(\bar q^{\bar L_{0}})= \bar q^{-k} \prod_{n=1}^{\infty}\frac{1}{1-\bar q^{n}} $$ I get $$ Z_{my}=\chi_{(0,-k)}(\tau) \bar\chi_{(0,-k)}(\bar\tau)=|q|^{-2k} \prod^{\infty}_{n=1}\frac{1}{|1-q^{n}|^{2}} $$

1.) My product starts at $n=1$ and I do not see why I could ignore the first contribution. Where is my mistake?

2.) Why do they set $c=0$?

This post imported from StackExchange Physics at 2014-10-12 11:42 (UTC), posted by SE-user ungerade