Lectures on String Theory

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Any point in the upper-half plane is related by a modular transformation to a point
in the so-called fundamental domain F ≡ M g=1 of the modular group. It can be
chosen as
{
M g=1 =
− 1 2 ≤ Reτ ≤ 0, |τ|2 ≥ 1 ∪ 0 < Reτ < 1 }
2 , |τ|2 > 1 .
The modular group does not act freely on the upper-half plane because some of the
modular transformations have fixed points. The point τ = i is the fixed point of S:
τ → − 1 2πi
and τ = e
τ
3 = − 1 + i √ 3 is the fixed point of ST . The existence of the
2 2
fixed points implies that M g=1 is not a smooth manifold, rather it has singularities
of the orbifold type.
Since it does not matter which fundamental region to choose in order to integrate
over in the one-loop path integral the integrand the corresponding string amplitude
should be invariant under modular transformations. The requirement of the modular
invariance is one of the most important principles of string theory. In particular, it
leads to strong restrictions on the possible gauge groups for the heterotic string.
6. Classical fermionic superstring
Introduction of the world-sheet fermionic degrees of freedom requires understanding
of how spinors on curved manifolds (world-sheets) are defined. The discussion in the
next paragraph is general and can be applied to a manifold of an arbitrary dimension
d.
6.1 Spinors in General Relativity
It is not straightforward to introduce spinors in General Relativity. If we have a
tensor field T i 1...i p
j 1 ...j q
of rank (p, q) on a manifold M then under general coordinate
transformations of the coordinates x i on M: x i → x ′i (x j ), this field transforms as
follows
T ′k 1...k p
l 1 ...l q
(x ′ ) = ∂x′k 1
∂x · · · ∂x′k p
i 1 ∂x i p
∂x j 1
∂x ′l 1 · · · ∂xj q
∂x ′l q
T i 1...i p
j 1 ...j q
(x) .
Here tensor indices are acted with the matrices ∂x′i which form a group GL(d, R).
∂x j
This is a group of all invertible real d × d matrices. This group does not have spinor
representations.