Lectures on String Theory

– 36 –
Let us outline the computation of this bracket.
{α − m , α− n } = n p
i α
i
m
p +
n p
i α
i
m
p +
, pj α j o
n
p + +
+ √
πT
p +
√
πT
p +
X
k≠m,0
α i m−k αi k , pj α j n
p +
n p
i α
i
m
p + , X
α j o
n−l αj −
l
l≠n,0
√
πT
+ 2πT
−im pi p i
(p + ) 2 δ n+m −2i(m − n)
(p + ) 2 pi α i n+m
| {z } | {z }
first bracket second and third brackets
√
πT
+ X
p + α j o
n−l αj =
l
l≠n,0
√
πT n p
i α
i
n
p + p + , X
α j o
m−l αj + πT n X
l
(p
l≠m,0
) 2 α i m−k αi k ,
X
α j o
n−l αj l
k≠m,0
l≠n,0
0
1
X
@−i(m
(p + ) 2 − n)
α i n+m−k αi A
k .
k≠m+n,0
(3.81)
| {z }
forth bracket
Thus, we are getting
0
1
{α − m , α− n } = −im pi p i
(p + ) 2 δ n+m + 2πT
∞X
@−i(m
(p + ) 2 − n) α i n+m−k αi A
k ,
k=−∞
where due to α i 0 = √ pi we combined the second and a third terms into one sum. If m + n ≠ 0 then the first term vanishes and we
4πT
can rewrite the last formula as
{α − m , α− n } = √
4πT
p +
−i(m − n)α −
m+n .
If n = −m then in eq.(3.81) contribution from the second and the third term vanishes and we get
0
1
{α − m , α− −m } = −im pi p i
(p + ) 2 + 2πT @−2im X √ 0 √
(p + ) 2 α i 4πT πT h p i ! 2
−k αi A
k ≡ −2im @
p
k≠0
+ p + √ + X α i k −ki 1
αi A .
4πT
k≠0
Therefore,
√ 0 √
{α − 4πT πT
m , α− −m } = −2im @
p + p +
1
X ∞ α i k αi A
−k .
k=−∞
It is therefore natural to define
α − 0
≡ √
πT
p +
∞ X
k=−∞
α i k αi −k .
With this definition we obtained a universal formula (valid for all indices m and n):
{α − m , α− n } = √
4πT
p +
−i(m − n)α −
m+n .
Also we conclude that with this definition
p − = √ πT (α − 0 + ᾱ− 0 ) .
Thus, one finds the following result
{α − m, α − n } =
√
4πT
p + (
−i(m − n)α
−
m+n
)
.
If we introduce
L n = p+
√
4πT
α − n .
we therefore find
{L n , L m } = −i(n − m)L n+m
which is the classical Virasoro algebra! Thus, in the light-cone gauge the Virasoro
algebra is carried over by the longitudinal oscillators α − n .