94
4 Worldsheet Path Integral: Complex Coordinates
This implies that the tangent space of the Riemann surface is decomposed into
holomorphic and anti-holomorphic vectors: 2
T T g T T
+
g ⊕ T T
−
g ,
(4.18a)
V
z ∂ z ∈ T T
+
g ,
V
¯
z ∂ ¯
z ∈ T T
−
g ,
(4.18b)
as a consequence of the existence of a complex structure. Similarly, the components
of a 1-form ω—which is the only non-trivial form on g —can be written in terms
of the real coordinates as
ω z =
1
2
(ω 0 − iω 1 ),
ω ¯
z =
1
2
(ω 0 + iω 1 )
(4.19)
such that
ω = ω 0 dσ
0
+ ω 1 dσ
1
= ω z dz + ω ¯
z d¯ z.
(4.20)
Hence, a 1-form is decomposed into complex (1, 0)- and (0, 1)-forms:
T
∗ g
1,0 (( g ) ⊕
0,1 (( g ),
(4.21a)
ω z dz ∈
1,0 (( g ),
ω ¯
z d¯ z ∈
0,1 (( g ),
(4.21b)
since both components will not mixed under holomorphic changes of coordinates
(4.7). Finally, the metric provides an isomorphism between T T +
g and 0,1 (( g ),
and between T T −
g and 1,0 (( g ), since it can be used to lower/raise an index while
converting it from holomorphic to anti-holomorphic, or conversely:
V z = g z¯ z V
¯
z ,
V ¯
z = g z¯ z V
z .
(4.22)
This can be generalized further by considering components with more indices:
all anti-holomorphic indices can be converted to holomorphic indices thanks to the
metric:
T
q + +p −
z···z
z···z
p + +q −
= (g
z¯ z )
p − (g z¯ z )
q − T
q +
z···z
q −
¯
z···¯ z
z···z
p +
¯
z···¯ z
p −
.
(4.23)
2 However, at this stage, each component can still depend on both z and ¯
z: V z = V z (z, ¯
z) and
V ¯
z = V ¯
z (z, ¯
z).