Lectures on String Theory

– 52 –
Plugging everything together we find
e −iα′ k 2τ [L m , V ] =
∞∑
p=1,p≠m
e ipτ
√
πT
: (kα m−p )V − V 0 V + :
} {{ }
+ √ eimτ
: V − V 0 (kα 0 )V + : + √ eimτ
: V − [kα 0 , V 0 ]V + : +α ′ k 2 (m − 1)e imτ : V − V 0 V + :
} πT
{{ } πT
+ √ 1
∞∑ e −ipτ
: V − V 0 V + (kα m ) : + √ : V − V 0 V + (kα m+p ) :
} πT
{{ } p=1 πT
} {{ }
Here the underlined terms are nicely combined with a single sum 13 with the range of
summation variable p form −∞ to +∞ and if we further take into account that
we will get
It remains to note that
[kα 0 , V 0 ] =
[L m , V ] = e imτ e iα′ k 2 τ
∞∑
p=−∞
k2
√
πT
V 0
e −ipτ
√
πT
: V − V 0 V + (kα p ) :
+ α ′ k 2 (m + 1)e imτ e iα′ k 2τ : V − V 0 V + :
} {{ }
V
−i∂ τ V = α ′ k 2 V + e iα′ k 2 τ
∞∑
p=−∞
e −ipτ
√
πT
: V − V 0 V + (kα p ) :
With the account of this formula we obtain
( )
[L m , V ] = e imτ − i∂ τ + α ′ k 2 m V
and, therefore, we conclude that the operator V has the following conformal dimension
∆:
∆ = α ′ k 2 .
In particular, for k 2 = 1 the conformal dimension ∆ = 1 and the vertex operator
α ′
we discuss corresponds to emission of the tachyon with the mass m 2 = − 1 .
α ′
We also see that on the zero-momentum ground state
V (k, 0)|0〉 = V − e ik µx µ |0〉 .
13 It is convenient to shift the summation variable p for p → p − m.