A Conventions
373
In chiral theories, the left-moving value is written first.
The worldsheet coordinates (τ, σ ) on the cylinder are defined by
τ ∈ R,
σ ∈ [0, L),
σ ∼ σ + L,
(A.9)
where typically L = 2π . The integration over the spatial coordinate is normalized
such that the perimeter of spatial slice is normalized to 1 if L = 2π :
L =
1
2π
L
0
dσ =
L
2π
.
(A.10)
This implies that 2d action, conserved charges, etc. are divided by an extra factor of
2π .
The coordinates can be written in terms of complex coordinates as
w = τ + iσ,
¯
w = τ − iσ
(A.11)
such that the flat metric is
ds
2
= dτ
2
+ dσ
2
= dwd ¯
w.
(A.12)
Under Wick rotation, the complex coordinates are mapped to light-cone coordinates
as follows:
w = iσ
+ ,
¯
w = iσ
− .
(A.13)
The cylinder can be mapped to the complex plane through
z = e
2πw/L ,
¯
z = e
2π ¯
w/L .
(A.14)
The definition of the Levi-Civita tensor includes the
√ g factor, such that
z¯ z =
i
2
,
,
z¯ z
= −2i
(A.15)
on the complex plane with flat metric conventions in the literature are compared in
Table A.1 .