Lectures on String Theory

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Thus, Q|χ〉 = 0 is equivalent to the condition that K i |χ〉 = 0, i.e. it is invariant under
the action of the Lie algebra. On the other hand, this state cannot be represented as
|χ〉 = Q|λ〉 for some |λ〉 because the ghost number of |λ〉 should be equal −1 which
is impossible.
Let us apply this general construction to the string case. The Lie algebra in this
case is the Virasoro algebra and we supply it with the ghosts c m and ant-ghosts b m .
The BRST operator is now
Q =
+∞∑
−∞
L X −mc m − 1 2
∞∑
(m − n) : c −m c −n b m+n : −ac 0 ,
−∞
where a is the normal-ordering ambiguity constant for L 0 . It turns out that this
expression can be written as
Q =
+∞∑
−∞
:
(
L X −m + 1 2 Lgh −m − aδ m,0
)c m :
The ghost-number operator is
U =
+∞∑
−∞
: c −m b m :
We would like to investigate the fulfilment of the relation Q 2 = 0 in quantum theory.
We find
Q 2 = 1 +∞
{Q, Q} =
2
∑
n,m=−∞
([L m , L n ] − (m − n)L m+n
)
c −m c −n .
Here L m = L X m + L gh
m − aδ m,0 is a total Virasoro operator. Thus, Q 2 = 0 for d = 26
and a = 1 as the consequence of vanishing of the total central charge!
Inverse statement is also true: from Q 2 = 0 it follows that the central charge of
the Virasoro algebra vanishes. Indeed, we first note that
From here
L m = {Q, b m }
[L m , Q] = [{Q, b m }, Q] = (Qb m + b m Q)Q − Q(Qb m + b m Q) = [b m , Q 2 ] = 0
as Q 2 = 0. Therefore, we see that
[L m , L n ] = [L m , {Q, b n }] = {[L m , Q], b n }
} {{ }
=0
+ {Q, [L m , b n ]} = (m − n){Q, b m+n } = (m − n)b m+n .