String Theory Demystified

102 String Theory Demystifi ed
Hence we conclude that
[ , ] = ( m− n)
+
m n m n
The generators 0, ± 1and 0,
± 1are
a special case. Consider the action of these
generators on the infi nitesimal coordinate transformations [Eq. (5.21)]. Taking
n =−101 , , we have
n=− 1:
z′ = z−ε(translation)
n= 0 : z′ = z−εz (scaling)
2
n= 1:
z′ = z−εz (special conformal transformation)
There are similar expressions for the complex conjugates. Hence −1 and −1
generate translation, 0 and 0, ( 0 + 0)
generate scaling and dilations, respectively,
i( 0 − 0)
generates rotations, and 1 and 1generate
special conformal transformations.
All together, the transformations generated by 0, ± 1 and 0,
± 1can
be written in
the form
az + b
z→ γ () z =
(5.25)
cz + d
Here, ad − bc =1 and the transformation γ () z is called a Möbius transformation.
EXAMPLE 5.2
Consider the transformation Ta () z = z/( + az)
1 . Show that Ta z () constitutes a
transformation group by examining the composition Tb( Ta( z)).
SOLUTION
This is actually a simple problem. We need to show that
z
Tb( Ta( z)) = Ta+ b(
z)
=
1+
( a+ b) z
Starting with T () z = z/( 1 + az)
, let w= z/( 1 + az)
. Now
a
w
Tb w
bw
( )= 1+