Gravity and Strings

336 The Kaluza–Klein black hole
ω1
ω
2
Fig. 11.1. The lattice generated in the ω plane by ω1 and ω2.
11.4.1 The 2-torus and the modular group
In our study of the global transformations of the internal torus we have not yet taken into
account the periodic boundary conditions of the coordinates, which have to be preserved
by the diffeomorphisms in the KK setting. Clearly the rescalings R do not respect the torus
boundary conditions, but they rescale ℓ. The rotations S respect the boundary conditions
only if S−1n ∈ Zn ; the matrix entries are integers, i.e. S ∈ SL(n, Z).
The case n = 2isparticularly interesting because it occurs in many instances, 25 some
(but not all of them) associated with S dualities. In the case n = 2, up to a reflection
S =−I2×2, these diffeomorphisms are known as Dehn twists and are not connected to the
identity (in fact, they constitute the mapping class group of torus diffeomorphisms) and
they constitute the modular group PSL(2, Z) = SL(2, Z)/{±I2×2}. This is the group that
acts on M.
It is convenient to relate M to the complex modular parameter τ of the torus. We start
by defining a complex modular-invariant coordinate ω on T2 by
ω = 1
2πℓ ωT · z, ω = C 2 , (11.198)
where, under PSL(2, Z) modular transformations, we assume that the complex vector ω
transforms according to
ω ′ = S ω. (11.199)
The periodicity of ω is
ω ∼ ω +ω T · n, n ∈ Z 2 . (11.200)
The lattice generated in the ω plane by ω is represented in Figure 11.1. In terms of the
modular-invariant complex coordinate, the torus metric element
ω
ds 2 Int = dz T Gdz (11.201)
takes the form
ds 2 Int = K 1 1
2 dωd ¯ω.
Im(ω1ω2) ¯
(11.202)
(Observe that Im(ω1ω2) ¯ is a modular-invariant term, and a quite important one.)
25 Owing to the isomorphisms SL(2, R) ∼ Sp(2, R) ∼ SU(1, 1) it takes several different, but equivalent, forms.