108
6 Conformal Field Theory on the Plane
Denoting the worldsheet coordinates in Lorentzian signature by (t, σ ) with 4
t ∈ R,
σ ∈ [0, L),
σ ∼ σ + L,
(6.9)
the metric reads
ds
2
= −dt
2
+ dσ
2
= −dσ
+ dσ
− ,
(6.10)
where the light-cone coordinates
dσ
±
= dt ± dσ
(6.11)
have been introduced. It is natural to perform a Wick rotation from the Lorentzian
time t to the Euclidean time
τ = it,
(6.12)
and the metric becomes
ds
2
= dτ
2
+ dσ
2 .
(6.13)
It is convenient to introduce the complex coordinates
w = τ + iσ,
¯
w = τ − iσ
(6.14)
for which the metric is
ds
2
= dwd ¯
w.
(6.15)
Note that the relation to Lorentzian light-cone coordinates is
w = i(t + σ ) = iσ
+ ,
¯
w = i(t − σ ) = iσ
− .
(6.16)
Hence, an (anti-)holomorphic function of w ( ¯
w) depends only on σ + (σ − ) before
the Wick rotation: this leads to the identification of the left- and right-moving sectors
with the holomorphic and anti-holomorphic sectors of the theory.
The cylinder can be mapped to the complex plane through
z = e
2πw/L ,
¯
z = e
2π ¯
w/L ,
(6.17)
4 Consistently with the comments at the beginning of Chap. 5, the Lorentzian worldsheet time is
denoted by t instead of τ M .