In
theoretical physics, F-theory is a branch of
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
developed by Iranian physicist
Cumrun Vafa. The new
vacua
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
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".
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
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
. By doing so, one obtains
type IIB superstring theory in 10 dimensions. The
SL(2,Z)
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
(
elliptic fibration In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed fi ...
), i.e. a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
whose fiber is a two-dimensional torus (also called an
elliptic curve). For example, a subclass of the
K3 manifold
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field mea ...
s is elliptically fibered, and F-theory on a K3 manifold is dual to
heterotic string
In string theory, a heterotic string is a closed string (or loop) which is a hybrid ('heterotic') of a superstring and a bosonic string. There are two kinds of heterotic string, the heterotic SO(32) and the heterotic E8 × E8, abbreviat ...
theory on a two-torus. Also, the moduli spaces of those theories should be isomorphic.
The large number of semirealistic solutions to string theory referred to as the
string theory landscape, with
elements or so, is dominated by F-theory compactifications on
Calabi–Yau four-folds. There are about
of those solutions consistent with the Standard Model of particle physics.
Phenomenology
New models of
Grand Unified Theory have recently been developed using F-theory.
Extra time dimension
F-theory has the
metric signature (10,2), which includes a
second time dimension.
[Penrose, Roger. (2004). ''The Road to Reality''. Jonathan Cape. Page 915. (Penrose cites Vafa. (1996) and also Bars, I. (2000). "Survey of Two-Time Physics". https://arxiv.org/abs/hep-th/0008164 )]
See also
*
Dilaton
In particle physics, the hypothetical dilaton particle is a particle of a scalar field \varphi that appears in theories with Dimension (mathematics and physics)#Additional dimensions, extra dimensions when the volume of the compactified dimensions ...
*
Axion
An axion () is a hypothetical elementary particle postulated by the Peccei–Quinn theory in 1977 to resolve the strong CP problem in quantum chromodynamics (QCD). If axions exist and have low mass within a specific range, they are of interes ...
*
M-theory
References
String theory
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