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 enocdes the axio-dilaton (the coupling constant and the degree-0 RR-field) tranforming under the $\mathrm{SL}\left(2,ℤ\right)$ S-duality/U-duality.

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 ${S}_{A}^{1}$ – to obtain type IIA supergravity – and then applying T-duality along another circle ${S}_{B}^{1}$ to obtain type IIB supergravity.

To obtain type IIB sugra in noncompact 10 dimensions this way, also ${S}_{B}^{1}$ is to be compactified (since T-duality sends the radius ${r}_{A}$ of ${S}_{A}^{1}$ to the inverse radius ${r}_{B}={\ell }_{s}^{2}/{R}_{A}$ of ${S}_{B}^{1}$). Therefore type IIB sugra in $d=10$ is obtained from 11d sugra compactified on the torus ${S}_{A}^{1}×{S}_{B}^{1}$. 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 $d=12$
$↓$ KK-reduction along elliptic fibration$↓$ axio-dilaton is modulus of elliptic fibration
IIA string theory in $d=9$$←$T-duality$\to$IIB string theory in $d=10$

In the simple case where the elliptic fiber is indeed just ${S}_{A}^{1}×{S}_{B}^{1}$, the imaginary part of its complex modulus is

$\mathrm{Im}\left(\tau \right)=\frac{{R}_{A}}{{R}_{B}}\phantom{\rule{thinmathspace}{0ex}}.$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 theory:

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

${g}_{\mathrm{IIA}}=\frac{{R}_{A}}{{\ell }_{s}}\phantom{\rule{thinmathspace}{0ex}}.$g_{IIA} = \frac{R_A}{\ell_s} \,.

Then the T-duality operation along ${S}_{B}^{1}$ divides this by ${R}_{B}$:

$\begin{array}{rl}{g}_{\mathrm{IIB}}& ={g}_{\mathrm{IIA}}\frac{{\ell }_{s}}{{R}_{B}}\\ & =\frac{{R}_{A}}{{R}_{B}}\\ & =\mathrm{Im}\left(\tau \right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B} \\ & = \frac{R_A}{R_B} \\ & = Im(\tau) \end{aligned} \,.

References

General

The original article is

Lecture notes include

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

Phenomenology and model building

A large body of literature is concerned with particle physics 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 February 24, 2012 12:26:27 by Urs Schreiber (89.204.155.170)