String Theory Demystified

CHAPTER 5 Conformal Field Theory Part I
Using w= z/( 1 + az)
we get
z
w
T w
az
b ( ) = =
1+
1+
bw ⎛ z ⎞
1+
b
⎝
⎜
+ az⎠
⎟
1
z
=
⎛ ⎛ z ⎞⎞
( 1+ az) 1+
b
⎝
⎜
1+
⎠
⎟
⎝
⎜
az ⎠
⎟
z z
= = = Ta+ b(
z
1+ az + bz 1+
( a+ b) z
Hence, T () z = z/( 1+ az)
satisfi es the group composition property.
a
)
103
EXAMPLE 5.3
Let Ta () z = z/( 1 + az)
and suppose that a is real and a = 1. Determine the generators
of this transformation.
SOLUTION
n+1
Recall that the generators have the form n =−z ∂zand
similarly for the complex
conjugate. This form holds for an infi nitesimal transformation parameter
n
ε( z) =−∑ anz +1 . So we can deduce the expressions for the generators by writing
the transformation as a series.
Consider the following series
Now multiply by r to give
s= 1−
r+ r − r +
2 3
2 3 4
rs = r − r + r − r +
Now add both series. On the left side we obtain s+ rs= s( 1 + r)
. On the right, we
have
2 3 2 3 4
s+ rs= 1− r+ r − r + + r− r + r − r + =
1