# F-theory

F-theory is a branch of string theory developed by Cumrun Vafa.[1] The new vacua described by F-theory were discovered by Vafa and allowed string theorists to construct new realistic vacua — in the form of F-theory compactified on elliptically fibered Calabi–Yau four-folds. The letter "F" supposedly stands for "Father".[2]

## Compactifications

F-theory is formally a 12-dimensional theory, but the only way to obtain an acceptable background is to compactify this theory on a two-torus. By doing so, one obtains type IIB superstring theory in 10 dimensions. The SL(2,Z) S-duality symmetry of the resulting type IIB string theory is manifest because it arises as the group of large diffeomorphisms of the two-dimensional torus.

More generally, one can compactify F-theory on an elliptically fibered manifold (elliptic fibration), i.e. a fiber bundle whose fiber is a two-dimensional torus (also called an elliptic curve). For example, a subclass of the K3 manifolds is elliptically fibered, and F-theory on a K3 manifold is dual to heterotic string theory on a two-torus. Also, the moduli spaces of those theories should be isomorphic.

The well-known large number of semirealistic solutions to string theory referred to as the string theory landscape, with ${\displaystyle 10^{500}}$ elements or so, is dominated by F-theory compactifications on Calabi–Yau four-folds.

## Phenomenology

New models of Grand Unified Theory have recently been developed using F-theory.[3]

## Extra time dimensions

F-theory, as it has metric signature (11,1), as needed for the Euclidean interpretation of the compactification spaces (e.g. the four-folds), is not a "two-time" theory of physics.

However, the signature of the two additional dimensions is somewhat ambiguous due to their infinitesimal character. For example, the supersymmetry of F-theory on a flat background corresponds to type IIB (i.e. (2,0)) supersymmetry with 32 real supercharges which may be interpreted as the dimensional reduction of the chiral real 12-dimensional supersymmetry if its spacetime signature is (10,2). In (11,1) dimensions, the minimum number of components would be 64.The superfield C being a cocycle of the ordinary 4-differential cohomology on Calabi-Yau varietes of moduli spaces of line bundles which under decomposition into various cup product associated with a divisor of the CY4, yields intermediate Jacobians and Artin-Mazur formal groups of degrees of maximum three (0,1,2).