arXiv:hep-th/9304011 v1 Apr 5 1993

Exercise.
Show that (3.70) is an identity by working out the second derivative of the exponential
and using the formula for T in terms of φ.
Example: The triangle functions.
The uniformization of the three-punctured sphere is explicitly known [54]. In this
case, (3.66) has three regular singular points and can therefore be transformed to the
Gauss hypergeometric equation. The mapping is given by
f(z) = N F 2(x)
F 1 (x) ,
(3.71a)
where
(
F 2 (x) = x 1−θ 1
2 F 1 2 −
1
2 (θ 1 + θ 2 + θ 3 ), 1 + 1 2 (−θ 1 + θ 2 − θ 3 ); 2 − θ 1 ; x )
(
F 1 (x) = 2 F 1 1 +
1
2 (θ 1
1 − θ 2 − θ 3 ),
2 (θ 1 + θ 2 − θ 3 ); θ 1 ; x )
N 2 = (1 − θ 1 ) 2( ∆(θ 1 − 1) ) 2 ∆ ( 2 − 1 2 (θ 1 + θ 2 + θ 3 ) ∆ ( 1
2 (−θ 1 + θ 2 + θ 3 ) )
∆ ( 1
2 (θ 1 + θ 2 − θ 3 ) ) ∆ ( 1
2 (θ 1 − θ 2 + θ 3 ) )
x = z − z 1
z − z 2
z 32
z 31
∆(y) = Γ(y)
Γ(1 − y) . (3.71b)
Fig. 5: A tesselation of the Poincaré disk: A copy of an adjacent white and black
triangle maps to the thrice-punctured sphere. The images of the triangles under
the monodromy group of the associated Fuchsian differential equation tesselate the
Poincaré disk.
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