# 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 $\tau$ encodes encodes the axio-dilaton (the coupling constant and the degree-0 RR-field) tranforming under the $SL(2, \mathbb{Z})$ S-duality/U-duality. See also at duality in string theory

## Properties

### Relation to (or motivation from) 11d supergravity

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 $S^1_A$ – to obtain type IIA supergravity – and then applying T-duality along another circle $S^1_B$ to obtain type IIB supergravity.

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

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

$Im(\tau) = \frac{R_A}{R_B} \,.$

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 $S^1_A$ yields a type IIA string coupling

$g_{IIA} = \frac{R_A}{\ell_s} \,.$

Then the T-duality operation along $S^1_B$ divides this by $R_B$:

\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 $\mathbb{Z}_2$-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 $SL_2(\mathbb{Z})$-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 $\mathbb{Z}_2 \hookrightarrow SL_2(\mathbb{Z})$.

Reasoning like this might suggest that in generalization to how type II orientifolds involve $\mathbb{Z}_2$-equivariant K-theory (namely KR-theory), so F-theory should involve $SL_2(\mathbb{Z})$-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

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 eliptically fibered Calabi-Yau $Z \to B$ of complex dimension $(n-1)$ is supposed to be equivalent $F$-theory on an $n$-dimensional $X\to B$ 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).

## References

### General

The original article is

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

Lecture notes include

• Timo Weigand, Lectures on F-theory compactifications and model building Class. Quantum Grav. 27 214004 (arXiv:1009.3497)

### 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)

### 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

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

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

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

Revised on April 21, 2014 02:03:19 by Urs Schreiber (89.204.138.200)