4.1 Geometry of Complex Manifolds
93
The integration measures are related as
d
2 σ := dτ dσ =
1
2
d
2 z,
d
2 z := dzd¯ z.
(4.10)
Due to the factor of 2 in the expression, the delta function δ (2) (z) also gets a factor
of 2 with respect to δ (2) (σ )
δ
(2) (z) =
1
2
δ
(2) (σ ).
(4.11)
Then, one can check that
d
2 z δ
(2) (z) =
d
2 σ δ
(2) (σ ) = 1.
(4.12)
The basis vectors (derivatives) and one-forms can be found using the chain rule:
∂ z =
1
2
(∂ τ − i∂ σ ),
∂ ¯
z =
1
2
(∂ τ + i∂ σ ),
(4.13a)
dz = dτ + idσ,
d¯ z = dτ − idσ.
(4.13b)
The Levi-Civita (completely antisymmetric) tensor is normalized by
01 =
01
= 1.
(4.14a)
z¯ z =
i
2
,
,
z¯ z
= −2i,
(4.14b)
remembering that it transforms as a density. Integer indices run over local frame
coordinates.
The different tensors can be found from the tensor transformation law. For
example, the components of a vector V a in both systems are related by
V
z
= V
0
+ iV
1 ,
V
¯
z
= V
0
− iV
1
(4.15)
such that
V = V
0 ∂ 0 + V
1 ∂ 1 = V
z ∂ z + V
¯
z ∂ ¯
z .
(4.16)
For holomorphic coordinate transformations (4.7), the components of the vector do
not mix:
V
w
=
∂w
∂z
V
z ,
V
¯
w
=
∂ ¯
w
∂ ¯
z
V
¯
z .
(4.17)