Lectures on String Theory

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only are called analytic. A holomorphic tensor with p lower and q upper indices has,
by definition, the rank n = p − q:
z } .{{ . . z}
q
V z } .{{ . . z(z, ¯z)
}
⇐ holomorphic tensor of rank n = p − q;
p
V z...z z...z(z) ⇐ analytic tensor;
Under analytic coordinate transformations z → f(z) a holomorphic tensor of rank n
transforms as follows
( ) n ∂f(z)
V (z, ¯z) →
V (f(z),
∂z
¯f(¯z)) .
Denote by V (n) the space of all holomorphic tensors of rank n. This space can be
supplied with the scalar product
∫
(V (n)
1 |V (n)
2 ) =
d 2 z √ h (h z¯z ) n (
V (n)
1
) ∗V
(n)
2 , V (n)
1 , V (n)
2 ∈ V (n)
and the associated norm ||V (n) || 2 = (V n |V (n) ). This scalar product is Weyl-invariant
for n = 1 only.
In the analytical coordinate system the Christoffel connection has only two nonvanishing
components
Γ z
zz = ∂φ ,
Γ ¯z
¯z¯z = ¯∂φ .
This connection allows to define two covariant derivatives
∇ (n)
z : V (n) → V (n+1) , ∇ (n)
z T (n) (z, ¯z) = (∂ − n∂φ)T (n) (z, ¯z)
∇ z (n) : V (n) → V (n−1) , ∇ z (n)T (n) (z, ¯z) = h z¯z ∇¯z T (n) (z, ¯z) = h z¯z ¯∂T (n) (z, ¯z) .
These two differential operators defined on holomorphic tensors of a fixed rank n
commute with the analytic coordinate transformations z → f(z). One can compute
an adjoint of ∇ (n)
z and find that
(∇ (n)
z ) † = −∇ z (n+1) .
The elements of the complex geometry we introduced above allows one to obtain
some information about the moduli space. Consider an arbitrary infinitesimal change
of the metric
δh αβ = Λh αβ + ∇ α V β + ∇ β V α + ∑ } {{ } } {{ }
i
Weyl
diff
δτ i
∂
∂τ i
h αβ
} {{ }
moduli
.