F-theory is a toolbox for describing type IIB string theory backgrounds – including non-perturbative effects induced from the presence of D7-branes and (p,q)-strings – in terms of complex elliptic fibrations whose fiber modulus encodes the axio-dilaton (the coupling constant and the degree-0 RR-field) tranforming under the S-duality/U-duality group. See also at duality in string theory.
More technically, F-theory is what results when KK-compactifying M-theory on an elliptic fibration (which yields type IIA superstring theory compactified on a circle-fiber bundle) followed by T-duality with respect to one of the two cycles of the elliptic fiber. The result is (uncompactified) type IIB superstring theory with axio-dilaton given by the moduli of the original elliptic fibration, see below.
Or rather, this is type IIB string theory with some non-perturbative effects included, reducing to perturbative string theory in the Sen limit. With a full description of M-theory available also F-theory should be a full non-perturbative description of type IIB string theory, but absent that it is some kind of approximation. For instance while the modular structure group of the elliptic fibration in principle encodes (necessarily non-perturbative) S-duality effects, it is presently not actually known in full detail how this affects the full theory, notably the proper charge quantization law of the 3-form fluxes, see at S-duality – Cohomological nature of the fields under S-duality for more on that.
The following line of argument shows why first compactifying M-theory on a torus to get type IIA on a circle and then T-dualizing that circle to get type IIB indeed only depends on the shape of the torus, but not on its other geometry.
By the dualities in string theory, 10-dimensional type II string theory is supposed to be obtained from the UV-completion of 11-dimensional supergravity by first dimensionally reducing over a circle – to obtain type IIA supergravity – and then applying T-duality along another circle to obtain type IIB supergravity.
To obtain type IIB sugra in noncompact 10 dimensions this way, also is to be compactified (since T-duality sends the radius of to the inverse radius of ). Therefore type IIB sugra in is obtained from 11d sugra compactified on the torus . More generally, this torus may be taken to be an elliptic curve and this may vary over the 9d base space as an elliptic fibration.
Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity vacuum is the input data that F-theory is concerned with.
A schematic depiction of this story is the following:
|M-theory in||F-theory in|
|KK-reduction along elliptic fibration||axio-dilaton is modulus of elliptic fibration|
|IIA string theory in||T-duality||IIB string theory in|
In the simple case where the elliptic fiber is indeed just , the imaginary part of its complex modulus is
First, the KK-reduction of M-theory on yields a type IIA string coupling
Then the T-duality operation along divides this by :
The target space data of an orientifold is a -principal bundle/local system, possibly singular (hence possibly on a smooth stack). On the other hand, the non-singular part of the elliptic fibration that defines the F-theory is a -local system (being the “homological invariant” of the elliptic fibration).
The degeneration locus of the elliptic fibration – where the discriminant vanishes and its fibers are the nodal curve – is interpreted as that of D7-branes and O7-planes (Sen 97a, (3), Blumenhagen 10, (11)), exhibiting gauge enhancement Sen 97b see below.
Reasoning like this might suggest that in generalization to how type II orientifolds involve -equivariant K-theory (namely KR-theory), so F-theory should involve -equivariant elliptic cohomology. This was conjectured in (Kriz-Sati 05, p. 3, p.17, 18). For more on this see at modular equivariant elliptic cohomology.
More generally, heterotic string theory on an elliptically fibered Calabi-Yau of complex dimension is supposed to be equivalent -theory on an -dimensional with elliptic K3-fibers.
In passing from M-theory to type IIA string theory, the locus of any Kaluza-Klein monopole in 11d becomes the locus of D6-branes in 10d. The locus of the Kaluza-Klein monopole in turn (as discussed there) is the locus where the -circle fibration degenerates. Hence in F-theory this is the locus where the fiber of the -elliptic fibration degenerates to the nodal curve. Since the T-dual of D6-branes are D7-branes, it follows that D7-branes in F-theory “are” the singular locus of the elliptic fibration.
may be parameterized via the Weierstrass elliptic function as the solution locus of the equation
The poles of the j-invariant correspond to the nodal curve, and hence it is at these poles that the D7-branes are located. Since the order of the poles is 24 (the polynomial degree of the discriminant ) there are necessarily 24 D7-branes.
|M-theory on -elliptic fibration||KK-compactification on||type IIA string theory||T-dual KK-compactification on||type IIB string theory||F-theory on elliptically fibered-K3 fibration||duality between F-theory and heterotic string theory||heterotic string theory on elliptic fibration|
|M2-brane wrapping||double dimensional reduction||type IIA superstring||type IIB superstring||heterotic superstring|
|M2-brane wrapping times around and times around||strings and D2-branes||(p,q)-string|
|M5-brane wrapping||double dimensional reduction||D4-brane||D5-brane|
|M5-brane wrapping times around and times around||D4-brane and NS5-branes||(p,q)5-brane|
|KK-monopole/A-type ADE singularity (degeneration locus of -circle fibration, Sen limit of elliptic fibration)||D6-brane||D7-branes||A-type nodal curve cycle degenertion locus of elliptic fibration (Sen 97, section 2)||SU-gauge enhancement|
|KK-monopole orientifold/D-type ADE singularity||D6-brane with O6-planes||D7-branes with O7-planes||D-type nodal curve cycle degenertion locus of elliptic fibration (Sen 97, section 3)||SO-gauge enhancement|
|exceptional ADE-singularity||exceptional ADE-singularity of elliptic fibration||E6-, E7-, E8-gauge enhancement|
The F-theory picture gives a geometric interpretation of the S-duality expected in type II string theory, by which all branes carry two integer charges acted on by . For instance the fundamental string (F1-brane) and the D1-brane combine to the -string, and similarly the NS5-brane and the D5-brane combine to a -5-brane.
(e.g. Johnson 97, p. 4)
Via the relation between supersymmetry and Calabi-Yau manifolds there is particular interest in F-theory compactied on Calabi-Yau spaces of (complex) dimension 4. For more on this see at F/M-theory on elliptically fibered Calabi-Yau 4-folds.
|F-theory||F-theory on CY2||F-theory on CY3||F-theory on CY4|
The original article is
A more recent survey is
Lecture notes include
Textbook accounts include
Further survey includes
Related conferences include
F-theory lifts of orientifold backgrounds were first identified in
and the corresponding gauge enhancement in
with more details including
This is further expanded on in
Adil Belhaj, F-theory Duals of M-theory on Manifolds from Mirror Symmetry (arXiv:hep-th/0207208)
Realization to the 6d (2,0)-supersymmetric QFT is discussed in
Gianluca Zoccarato, Yukawa couplings at the point of in F-theory, 2014 (pdf/zoccarato.pdf))
Andres Collinucci, Raffaele Savelli, On Flux Quantization in F-Theory (2010) (arXiv:1011.6388)
For more on this see also at F/M-theory on elliptically fibered Calabi-Yau 4-folds.