Lectures on String Theory

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We can now rewrite the r.h.s. by using the propagators introduces above
T (τ, σ)T (τ ′ , σ ′ ) = 1
α ′2 : ∂ −X µ (z)∂ − X µ (z)∂ − X ν (w)∂ − X ν (w) :
+ 4
α ′2 ∂x −∂ y − 〈Xµ (z)X ν (w)〉 : ∂ − X µ (z)∂ − X ν (w) :
+ 2
α ′2 ∂z −∂ w −〈X µ (z)X ν (w)〉 ∂ z −∂ w −〈X µ (z)X ν (w)〉 .
A little computation gives
T (τ, σ)T (τ ′ , σ ′ ) = 1
α ′2 : ∂ −X µ (z)∂ − X µ (z)∂ − X ν (w)∂ − X ν (w) :
− 2 zw
α ′ (z − w) 2 : ∂ −X µ (z)∂ − X µ (w) :
+ ηµν η µν
2
z 2 w 2
(z − w) 4 .
It is further convenient to redefine the stress tensor as follows
so that
Since
we have ∂ ∂z = 1
iz
T (z) =
T (τ, σ)
z 2
1
T (z)T (w) = =
α ′2 z 2 w : ∂ −X µ (z)∂ 2 − X µ (z)∂ − X ν (w)∂ − X ν (w) :
− 2 1
α ′ zw (z − w) : ∂ −X µ (z)∂ 2 − X µ (w) :
+ ηµν η µν 1
2 (z − w) . 4
∂
∂z
∂ − = 1 ( ∂
2 ∂τ − ∂ )
= 1 ( ∂z
∂σ 2 ∂τ − ∂z ) ∂
∂σ ∂z = iz ∂ ∂z ,
and, therefore, the last formula can be written as
T (z)T (w) = = 1
α ′2 : ∂ zX µ (z)∂ z X µ (z)∂ w X ν (w)∂ w X ν (w) :
+ 2 1
α ′ (z − w) : ∂ zX µ (z)∂ 2 w X µ (w) : + ηµν η µν
2
1
(z − w) 4 .
Expanding the r.h.s. around the point w = z, we will find the following most singular
z → w contribution
T (z)T (w) = d/2
(z − w) 4 + 2
(z − w) 2 T (w) + 1
z − w ∂ wT (w) + . . . (4.13)
The first term here reflects the appearance of the conformal anomaly (a purely quantum
mechanical effect). In the general setting the coefficient of this term is c/2,
where c is the central charge. The coefficients 2 of the second term coincides with
the conformal dimension of T .