4
Worldsheet Path Integral: Complex
Coordinates
Abstract
In the two previous chapters, the amplitudes computed from the worldsheet
path integrals have been written covariantly for a generic curved background
metric. In this chapter, we start to use complex coordinates and finally take
the background metric to be flat. This is the usual starting point for computing
amplitudes since it allows to make contact with CFTs and to employ tools
from complex analysis. We first recall few facts on 2d complex manifolds
before briefly describing how to rewrite the scattering amplitudes in complex
coordinates.
4.1
Geometry of Complex Manifolds
Choosing a flat background metric simplifies the computations. However, we have
seen in Sect. 2.3 that there is a topological obstruction to get a globally flat metric.
The solution is to work with coordinate patches (σ 0 , σ 1 ) = (τ, σ ) such that the
background metric ˆ
g ab is flat in each patch (conformally flat gauge):
ds
2
= g ab dσ
a dσ
b
= e
2φ(τ,σ )
dτ
2
+ dσ
2
,
(4.1)
or
g ab = e
2φ δ ab ,
ˆ
g ab = δ ab .
(4.2)
To simplify the notations, we remove the dependence in the flat metric and the hat
for quantities (like the vertex operators) expressed in the background metric when
no confusion is possible.
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_4
91