F-theory
Encyclopedia
F-theory is a branch of string theory
developed by Cumrun Vafa
. The new vacua described as F-theory were discovered by Vafa, and it also 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".
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 diffeomorphism
s 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. (Eight dimensions are large.)
The well-known 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.
(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.
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
developed by Cumrun Vafa
Cumrun Vafa
Cumrun Vafa is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
. The new vacua described as F-theory were discovered by Vafa, and it also 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 compactifyCompactification (physics)
In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic....
this theory on a two-torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
. By doing so, one obtains type IIB superstring theory
Superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
in 10 dimensions. The SL(2,Z) S-duality
S-duality
In theoretical physics, S-duality is an equivalence of two quantum field theories or string theories. An S-duality transformation maps states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1/g in the dual theory...
symmetry of the resulting type IIB string theory is manifest because it arises as the group of large diffeomorphism
Large diffeomorphism
In mathematics and theoretical physics, a large diffeomorphism is a diffeomorphism that cannot be continuously connected to the identity diffeomorphism ....
s of the two-dimensional torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
.
More generally, one can compactify F-theory on an elliptically fibered manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
(elliptic fibration), i.e. a fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
whose fiber is a two-dimensional torus (also called an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
). For example, a subclass of the K3 manifolds is elliptically fibered, and F-theory on a K3 manifold is dual to heterotic string
Heterotic string
In physics, a heterotic string is a peculiar mixture of the bosonic string and the superstring...
theory on a two-torus. (Eight dimensions are large.)
The well-known large number of semirealistic solutions to string theory referred to as the string theory landscape
String theory landscape
The string theory landscape or anthropic landscape refers to the large number of possible false vacua in string theory. The "landscape" includes so many possible configurations that some physicists think that the known laws of physics, the standard model and general relativity with a positive...
, with
elements or so, is dominated by F-theory compactifications on Calabi-Yau four-folds.Phenomenology
New models of GUT unification of the fundamental forces have recently been developed using F-theory.Extra time dimensions
F-theory, as it has metric signatureMetric signature
The signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
(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
Multiple time dimensions
The possibility that there might be more than one dimension of time has occasionally been discussed in physics and philosophy.- Physics :Special relativity describes spacetime as a manifold whose metric tensor has a negative eigenvalue. This corresponds to the existence of a "time-like" direction...
.
However, the signature of the two additional dimensions is somewhat ambiguous due to their infinitesimal character. For example, the supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
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.