F-Theory

$F$-Theory was formulated by C. Vafa, looking for an analog theory to M-Theory for describing non-perturbative compactifications of Type IIB theory (for a review see [24,20]). Usually in perturbative compactifications the parameter $\lambda = a + i exp(-\Phi/2)$ is taken to be constant. $F$-theory generalizes this fact by considering variable $\lambda$. Thus $F$-theory is defined as a twelve-dimensional theory whose compactification on the elliptic fibration $T^2 - {\cal K} \to B$, gives the Type IIB theory compactified on $B$ (for a suitable space $B$) with the identification of $\lambda(\vec{z})$ with the modulus $\tau(\vec{z})$ of the torus $T^2$. These compactifications can be related to the $M$-theory compactifications through the known $S$ mapping ${\cal S}: IIA \to M/{\bf S}^1$ and the ${\cal T}$ map between Type IIA and IIB theories. This gives

$$F/{\cal K}\times {\bf S}^1 \Longleftrightarrow M/{\cal K}.$$ (57)

Thus the spectrum of massless states of $F$-theory compactifications can be described in terms of $M$-theory. Other interesting $F$-theory compactifications are the Calabi-Yau compactifications

$$F/CY \Leftrightarrow H/K3.$$ (58)

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