String Theory and M-Theory

432 String geometry
fibration. Only the complex structure of the torus is specified by the modulus
τ. Its size (or Kähler structure) is not a dynamical degree of freedom.
Recall that the type IIB theory can be obtained by compactifying M-theory
on a torus and letting the area of the torus shrink to zero. In this limit
the modular parameter of the torus gives the τ parameter of the type IIB
theory. Therefore, the best interpretation is that the torus in the F-theory
construction has zero area.
A nice way of describing the complex structure of a torus is by an algebraic
equation of the form
y 2 = x 3 + ax + b. (9.207)
This describes the torus as a submanifold of £ 2 , which is parametrized by
complex numbers x and y. The constants a and b determine the complex
structure τ of the torus. There is no metric information here, so the area
is unspecified. The torus degenerates, that is, τ is ill-defined, whenever the
discriminant of this cubic vanishes. This happens for
27a 3 − 4b 2 = 0. (9.208)
Thus, the positions of the 7-branes correspond to the solutions of this equation.
To ensure that z = ∞ is not a solution, we require that a 3 and b 2 are
polynomials of the same degree.
Since there should be 24 7-branes, the equation should have 24 solutions.
Thus, a = f8(z) and b = f12(z), where fn denoted a polynomial of degree
n. The total space can be interpreted as a K3 manifold that admits a
T 2 fibration. The only peculiar feature is that the fibers have zero area.
Let us now count the number of moduli associated with this construction.
The polynomials f8 and f12 have arbitrary coefficients, which contribute
9 + 13 = 22 complex moduli. However, four of these are unphysical because
of the freedom of an SL(2, £ ) transformation of the z-plane and a rescaling
f8 → λ 2 f8, f12 → λ 3 f12. This leaves 18 complex moduli. In addition there
is one real modulus (a Kähler modulus) that corresponds to the size of the
£ P 1 base space. The complex moduli parametrize the positions of the 7branes
(modulo SL(2, £ )) in the z-plane. The fact that there are fewer
than 21 such moduli shows that the positions of the 7-branes (as well as
their monodromies) is not completely arbitrary.
Remarkably, there is a dual theory that has the same properties. The
heterotic string theory compactified on a torus to eight dimensions has 16
unbroken supersymmetries and the moduli space
¡ + × M18,2. (9.209)