String Theory Demystified

94 String Theory Demystifi ed
Complex Coordinates
A consequence of a Wick rotation is that the light-cone coordinates ( + , − ) are
replaced with complex coordinates ( z, z ) . The description of the worldsheet is
transformed into complex variables by defi ning complex coordinates ( z, z ) which
are functions of the real variables ( τ, σ ) . One way this can be done is as
follows:
z = τ + iσ z = τ −iσ
(5.4)
Let’s use this defi nition to work out a few basic quantities and show how this
simplifi es analysis. Keep the Polyakov action in the back of your mind. Using the
Euclidean metric the Polyakov action is written as
SP d d X X X X
=
1
µ
µ
∂ ∂ +∂ ∂
4πα ′ ∫ τ σ( τ τ µ σ σ µ ) (5.5)
We’re going to fi nd out that going to complex variables will simplify the form of
Eq. (5.5).
To transform coordinates we need to know how to compute derivatives with
respect to the coordinates z and z.
This is easy enough. First we invert the coordinates
Eq. (5.4):
It follows that
and so
τ = σ
+
= −
z z z z
2 2i
∂ ∂
=
∂ ∂ =
∂
∂ =
∂
∂ =−
τ τ 1 σ 1 σ 1
z z 2 z 2i
z 2i
(5.6)
∂ ∂ ∂ ∂ ∂
= +
∂ ∂ ∂ ∂ ∂ =
∂
∂ +
∂
∂ =
τ σ 1 1 1 ⎛ ∂ ∂ ⎞
− i
z z τ z σ 2 τ 2i
σ 2 ⎝
⎜
∂τ ∂σ⎠
⎟ (5.7)
The shorthand notation ∂ z =∂= 1/ 2(
∂ −i∂ )
see that
τ σ is usually used. It is also easy to
∂
∂ ∂ ∂
=∂ =∂=∂ +
∂ ∂ ∂ ∂ ∂ =
∂
∂ −
∂
∂ =
τ σ 1 1 1 ∂
+ z
z z τ z σ 2 τ 2i
σ 2 ∂τ
∂ ⎛ ⎞ 1
i = ∂ τ + i∂σ
⎝ ∂σ
⎠ 2 ( ) (5.8)