Lectures on String Theory

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The ghosts and anti-ghosts are conformal fields. Indeed, it is easy to compute
[L gh
m , b n ] = (m − n)b m+n ,
[L gh
m , c n ] = −(2m + n)c m+n .
Comparing this with the transformation rule of the modes of a conformal operator
of dimension ∆
[L m , A n ] = ( m(∆ − 1) − n ) A m+n
we conclude that b and c are indeed the conformal fields of the conformal dimension
∆ = 2 and ∆ = −1 respectively.
Using the explicit expressions for the ghost generators L gh it is not difficult to
compute the algebra
where the central charge appears to be
[L gh
m , L gh
n ] = (m − n)L gh
m+n + c gh (m)δ m+n ,
c gh (m) = 1
12 (2m − 26m3 ) .
Now if we introduce the total Virasoro generator as
then it will satisfy the Virasoro algebra
L m = L X m + L gh
m − aδ m,0
[L m , L n ] = (m − n)L m+n + c(m)δ m+n
with
c = d 12 (m3 − m) + 1 12 (2m − 26m3 ) + 2am .
We see that the total central charge vanishes for d = 26 and a = 1. We again
found the same values for the critical dimension and the normal-ordering constant
as followed from the light-cone approach! Here these conditions on the theory follow
from the requirement of vanishing of the total central charge.
BRST operator
The concept of the BRST operator is very general. In fact, the BRST operator can
be associated to any Lie algebra and it is a useful tool to compute the Lie algebra
cohomologies.
Consider a Lie algebra with generators K i satisfying the relations
[K i , K j ] = f k ijK k .
Introduce ghost and anti-ghost fields c i and b i satisfying the anti-commutation relations
{c i , b j } = δ i j , i = 1, . . . , dimK