nLab
F-theory

Contents

Idea

F-theory is a toolbox for describing type IIB string theoryincluding non-perturbative effects induced from the presence of D7-branes and (p,q)-strings – in terms of complex elliptic fibrations whose fiber modulus τ\tau encodes the axio-dilaton (the coupling constant and the degree-0 RR-field) tranforming under the SL(2,)SL(2, \mathbb{Z}) S-duality/U-duality. See also at duality in string theory.

More technically, F-theory is what results when KK-compactifying M-theory on a elliptic fibration (which yields type IIA superstring theory compactified on a circle 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.

Or rather, this is type IIB string theory with some non-perturbative effects included. 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.

Properties

Relation to (or motivation from) 11d supergravity

The following line of argument shows why first compactifying M-theory on a torus S 1 A×S 1 BS_1^A \times S_1^B to get type IIA on a circle and then T-dualizing that circle to get type IIB indeed only depends on the shape R AR B\frac{R_A}{R_B} 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 S A 1S^1_A – to obtain type IIA supergravity – and then applying T-duality along another circle S B 1S^1_B to obtain type IIB supergravity.

To obtain type IIB sugra in noncompact 10 dimensions this way, also S B 1S^1_B is to be compactified (since T-duality sends the radius r Ar_A of S A 1S^1_A to the inverse radius r B= s 2/R Ar_B = \ell_s^2 / R_A of S B 1S^1_B). Therefore type IIB sugra in d=10d = 10 is obtained from 11d sugra compactified on the torus S A 1×S B 1S^1_A \times S^1_B. 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 d=11d = 11F-theory in d=12d = 12
\downarrow KK-reduction along elliptic fibration\downarrow axio-dilaton is modulus of elliptic fibration
IIA string theory in d=9d = 9\leftarrowT-duality\rightarrowIIB string theory in d=10d = 10

In the simple case where the elliptic fiber is indeed just S A 1×S B 1S^1_A \times S^1_B, the imaginary part of its complex modulus is

Im(τ)=R AR B. Im(\tau) = \frac{R_A}{R_B} \,.

By following through the above diagram, one finds how this determines the coupling constant in the type II string theory:

First, the KK-reduction of M-theory on S A 1S^1_A yields a type IIA string coupling

g IIA=R A s. g_{IIA} = \frac{R_A}{\ell_s} \,.

Then the T-duality operation along S B 1S^1_B divides this by R BR_B:

g IIB =g IIA sR B =R AR B =Im(τ). \begin{aligned} g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B} \\ & = \frac{R_A}{R_B} \\ & = Im(\tau) \end{aligned} \,.

Relation to orientifold type II backgrounds

The general vacuum of type II superstring theory (including type I superstring theory) is an orientifold.

The target space data of an orientifold is a 2\mathbb{Z}_2-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 SL 2()SL_2(\mathbb{Z})-local system (being the “homological invariant” of the elliptic fibration).

An argument due to (Sen 96, Sen 97) says that the F-theory data does induce the orientifold data along the subgroup inclusion 2SL 2()\mathbb{Z}_2 \hookrightarrow SL_2(\mathbb{Z}).

The degeneration locus of the elliptic fibration is that of D7-branes and O7-planes.

Reasoning like this might suggest that in generalization to how type II orientifolds involve 2\mathbb{Z}_2-equivariant K-theory (namely KR-theory), so F-theory should involve SL 2()SL_2(\mathbb{Z})-equivariant elliptic cohomology. This was indeed conjectured in (Kriz-Sati 05, p. 3, p.17, 18). For more on this see at modular equivariant elliptic cohomology.

Relation to heterotic string theory

The duality between F-theory and heterotic string theory:

F-theory on an elliptically fibered K3 is supposed to be equivalent to heterotic string theory compactified on a 2-torus. An early argument for this is due to (Sen 96).

More generally, heterotic string theory on an elliptically fibered Calabi-Yau ZBZ \to B of complex dimension (n1)(n-1) is supposed to be equivalent FF-theory on an nn-dimensional XBX\to B with elliptic K3-fibers.

A detailed discussion of the equivalence of the respective moduli spaces is originally due to (Friedman-Morgan-Witten 97). A review of this is in (Donagi 98).

Model building and phenomenology

For F-theory a fairly advanced model building and string phenomenology has been developed. A detailed review is in (Denef 08).

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.

See also at flux compactification and landscape of string theory vacua.

F-theory KK-compactified on elliptically fibered complex analytic fiber Σ\Sigma

dim (Σ)dim_{\mathbb{C}}(\Sigma)1234
F-theoryF-theory on CY2F-theory on CY3F-theory on CY4

References

General

The original article is

An early survey of its relation to M-theory with M5-branes is in

Lecture notes include

Relation to orientifolds

F-theory lifts of orientifold backgrounds were first identified in

with more details including

  • Zurab Kakushadze, Gary Shiu, S.-H. Henry Tye, Type IIB Orientifolds, F-theory, Type I Strings on Orbifolds and Type I - Heterotic Duality, Nucl.Phys. B533 (1998) 25-87 (arXiv:hep-th/9804092)

This is further expanded on in

Relation to elliptic cohomology

A series of articles arguing for a relation between the elliptic fibration of F-theory and elliptic cohomology (see also at modular equivariant elliptic cohomology)

Relation to the heterotic string

Phenomenology and model building

A large body of literature is concerned with particle physics string phenomenology modeled in the context of F-theory.

4-Form flux and instantons

The image of the supergravity C-field from 11-dimensional supergravity to F-theory yields the G 4G_4-flux.

  • Andres Collinucci, Raffaele Savelli, On Flux Quantization in F-Theory (2010) (arXiv:1011.6388)

  • Sven Krause, Christoph Mayrhofer, Timo Weigand, G 4G_4 flux, chiral matter and singularity resolution in F-theory compactifications (arXiv:1109.3454)

  • Thomas Grimm, Denis Klevers, Maximilian Poretschkin, Fluxes and Warping for Gauge Couplings in F-theory (arXiv:1202.0285)

  • Sven Krause, Christoph Mayrhofer, Timo Weigand, Gauge Fluxes in F-theory and Type IIB Orientifolds (2012) (arXiv:1202.3138)

and with M5-brane instanton contributions:

Reviewed in

For more on this see also at F/M-theory on elliptically fibered Calabi-Yau 4-folds.

Revised on July 3, 2014 00:12:24 by Urs Schreiber (82.136.246.44)