nLab
F-theory

Idea

F-theory is a toolbox for describing type IIB string theory – including 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 $\mathrm{SL}(2,\mathbb{Z})$ 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 $S_A^1\times S_B^1$. 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 $d=12$
$\downarrow$ KK-reduction along elliptic fibration		$\downarrow$ axio-dilaton is modulus of elliptic fibration
IIA string theory in $d=9$	$\leftarrow$T-duality$\to$	IIB string theory in $d=10$

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

$$\mathrm{Im}(\tau )=\frac{R_A}{R_B}.$$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 theory:

First, the KK-reduction of M-theory on $S_A^1$ yields a type IIA string coupling

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

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

$$\begin{array}{rl}{g}_{\mathrm{IIB}}& =g_{\mathrm{IIA}}\frac{{\ell }_s}{R_B}\\ & =\frac{R_A}{R_B}\\ & =\mathrm{Im}(\tau )\end{array}.$$\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

• Cumrum Vafa?, Evidence for F-theory, Nucl.Phys.B469:403-418,1996, (arXiv:hep-th/9602022)

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 string phenomenology modeled in the context of F-theory.

• Chris Beasley, Jonathan J. Heckman, Cumrun Vafa, GUTs and Exceptional Branes in F-theory - I (arxiv:0802.3391), II: Experimental Predictions (arxiv:0806.0102)

(…)

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)