*The distinct string theory and their strong-coupling limit. The solid line (-) denotes toroidal compactification, the dashed line (–) denotes K3 compactifications and the dotted line (…) denotes Y _{3} compactifications. The fine-dotted line (…) denotes Y_{4} compactifications while the horizontal bar (-) indicates a string-string duality. The theories marked with a ‘U’ (‘8’) have a U-duality (8-duality); the strong-coupling limit of the theories marked by ‘M’ (‘F’) are controlled by M-theory (F-theory).*

The type-IIB theory in 10 spacetime dimensions is believed to have an exact SL(2, Z) quantum symmetry which acts on the complex scalar τ = e^{-2φ} + iφ’, where φ and φ’ are the two scalar fields of type-lIB theory. This fact led *Vafa* to propose that the type-lIB string could be viewed as the toroidal compactification of a twelve-dimensional theory, called F-theory, where T is the complex structure modulus of a two-torus T^{2} and the *Kähler-class* modulus is frozen. Apart from having a geometrical interpretation of the SL(2, Z) symmetry this proposal led to the construction of new, nonperturbative string vacua in lower space-time dimensions. In order to preserve the SL(2, Z) quantum symmetry the compactification manifold cannot be arbitrary but has to be what is called an elliptic fibration. That is, the manifold is locally a fibre bundle with a two-torus T^{2} over some base B but there are a finite number of singular points where the torus degenerates. As a consequence nontrivial closed loops on B can induce a SL(2, Z) transformation of the fibre. This implies that the dilaton is not constant on the compactification manifold, but can have SL(2, Z) monodromy. It is precisely this fact which results in nontrivial (nonperturbative) string vacua inaccessible in string perturbation theory.

F-theory can be compactified on elliptic *Calabi-Yau manifolds* and each of such compactifications is conjectured to capture the nonperturbative physics of an appropriate string vacua. One finds:

- The lIB string in D = 10 can be viewed as F-theory compactified on T
^{2}with a frozen Kähler modulus. - F-theory compactified on an elliptic K3 yields an 8-dimensional vacuum with 16 supercharges which is quantum equivalent to the heterotic string compactified on T
^{2}. *F-theory compactified on an elliptic Calabi-Yau threefold has 8 unbroken supercharges and is quantum equivalent to the heterotic string compactified on K3*.- The
*heterotic string*compactified on a Calabi-Yau threefold Y_{3}is quantum equivalent to F-theory compactified on an elliptic Calabi-Yau four-fold. Calabi-Yau fourfolds are Calabi-Yau manifolds of complex dimension four and holonomy group SU(4).