Lectures on String Theory

– 42 –
Using α µ q α µ m+n−q = α µ m+n−qα µ q + qδ m+n δ µ µ = α µ m+n−qα µ q + qdδ m+n , where d is the
dimension of the Minkowskian space-time where string propagates. Thus, the algebra
relation is
[L m , L n ] = 1 2
∞∑
(m − n) : α pα µ µ m+n−p : + d 2 δ n+m
p=−∞
n∑
(q 2 − nq) .
q=1
Since
n∑
q=1
q 2 = 1 n∑
n(n + 1)(2n + 1) ,
6
q=1
q = 1 n(n + 1) .
2
one finds the final result
[L m , L n ] = (m − n)L m+n + d 12 m(m2 − 1)δ m+n
which is the famous Virasoro algebra. We see that it is different from the classical
Virasoro (Wit) algebra by the presence of the central term.
In a more general setting the algebra is written as
[L m , L n ] = (m − n)L m+n + c
12 m(m2 − 1)δ m+n ,
where the constant term c is known as the central charge.
If we introduce a normal ordering constant a by shifting the definition of L m as
L m → L m − δ m,0 then the linear term in m in the central term changes
[L m , L n ] = (m − n)L m+n +
( c
12 m3 + ( 2a − c ) )
m δ m+n .
12
We see that the central term has an invariant meaning and cannot be removed for
all L m by adjusting the normal ordering constant a.
Finally, we comment on the relation to semiclassics. If we restore the Plank the
algebra relations take the form
[L m , L n ] = (m − n)L m+n + 2 c
12 m(m2 − 1)δ m+n .
We see that one can define the Poisson bracket
1
{L m , L n } = lim
~→0 i [L m, L n ] = −i(m − n)L m+n ,
which coincides with the Wit algebra. The central term obviously vanishes in the
semi-classical limit.