# Contents

## Idea

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 $\tau$ encodes the axio-dilaton (the coupling constant and the degree-0 RR-field) tranforming under the $SL(2, \mathbb{Z})$ 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.

## Properties

### Relation to M-theory

The following line of argument shows why first compactifying M-theory on a torus $S_1^A \times S_1^B$ to get type IIA on a circle and then T-dualizing that circle to get type IIB indeed only depends on the shape $\frac{R_A}{R_B}$ 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 $S^1_A$ – 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 M-theory on $G_2$-manifolds

In order to get minimal N=1 d=4 supergravity after KK-compactification, one needs M-theory on G2-manifolds and F-theory on CY4-manifolds.

Discussion of the relation between then G2-manifold fibers for M-theory on G2-manifolds and the corresponding Calabi-Yau 4-manifold fibers in F-theory includes (Gukov-Yau-Zaslow 02, Belhaj 02).

### 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 97a) says that the F-theory data does induce the orientifold data along the subgroup inclusion $\mathbb{Z}_2 \hookrightarrow SL_2(\mathbb{Z})$. See at Sen limit.

The degeneration locus of the elliptic fibration – where the discriminant $\Delta$ 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 $\mathbb{Z}_2$-equivariant K-theory (namely KR-theory), so F-theory should involve $SL_2(\mathbb{Z})$-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.Also,the supersymmetry of compactification of the F theory on a flat background corresponds to type IIB supersymmetry if it’s spacetime signature is (10,2). We will have a number of supercharges used for dimensional reduction,or a cascade of reductions.

### Relation to heterotic string theory

The duality between F-theory and heterotic string theory:A subclass of K3 manifolds elliptically fibered.

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).Also,the modules of those theories must be equal.

More generally, heterotic string theory on an elliptically 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).

### The singular locus of the elliptic fibration and the D7-branes

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 $S^1_A$-circle fibration degenerates. Hence in F-theory this is the locus where the fiber of the $S^1_A \times S^1_B$-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.

Now an elliptically fibered complex K3-surface

$\array{ T &\longrightarrow& K3 \\ && \downarrow \\ && \mathbb{C}\mathbb{P}^1 }$

may be parameterized via the Weierstrass elliptic function as the solution locus of the equation

$y^2 = x^3 + f(z) x + g(z)$

for $x,y,z \in \mathbb{C}\mathbb{P}^1$, with $f$ a polynomial of degree 8 and $g$ of degree twelve. The j-invariant of the complex elliptic curve which this parameterizes for given $z$ is

$j(\tau(z)) = \frac{4 (24 f)^3}{27 g^2 + 4 f^3} \,.$

The poles $j\to \infty$ 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 $\Delta = 27 g^2 + 4 f^3$) there are necessarily 24 D7-branes.

(Sen 96, page 5 Sen 97b, see also Morrison 04, sections 8 and 17, Denef 08, around (3.41)). Notice that the net charge of these 24 D7-branes is supposed to vanish, due to S-duality effects (e.g. Denef 08, below (3.41)).

### F-brane scan

from M-branes to F-branes: superstrings, D-branes and NS5-branes

M-theory on $S^1_A \times S^1_B$-elliptic fibrationKK-compactification on $S^1_A$type IIA string theoryT-dual KK-compactification on $S^1_B$type IIB string theoryF-theory on elliptically fibered-K3 fibrationduality between F-theory and heterotic string theoryheterotic string theory on elliptic fibration
M2-brane wrapping $S_A^1$double dimensional reduction $\mapsto$type IIA superstring$\mapsto$type IIB superstring$\mapsto$heterotic superstring
M2-brane wrapping $S_B^1$$\mapsto$D2-brane$\mapsto$D1-brane
M2-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$$\mapsto$$p$ strings and $q$ D2-branes$\mapsto$(p,q)-string
M5-brane wrapping $S_A^1$double dimensional reduction $\mapsto$D4-brane$\mapsto$D5-brane
M5-brane wrapping $S_B^1$$\mapsto$NS5-brane$\mapsto$NS5-brane$\mapsto$NS5-brane
M5-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$$\mapsto$$p$ D4-brane and $q$ NS5-branes$\mapsto$(p,q)5-brane
M5-brane wrapping $S_A^1 \times S_B^1$$\mapsto$$\mapsto$D3-brane
KK-monopole/A-type ADE singularity (degeneration locus of $S^1_A$-circle fibration, Sen limit of $S^1_A \times S^1_B$ elliptic fibration)$\mapsto$D6-brane$\mapsto$D7-branesA-type nodal curve cycle degenertion locus of elliptic fibration (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity$\mapsto$D6-brane with O6-planes$\mapsto$D7-branes with O7-planesD-type nodal curve cycle degenertion locus of elliptic fibration (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity$\mapsto$$\mapsto$exceptional ADE-singularity of elliptic fibration$\mapsto$E6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

### S-duality operation on $(p,q)$-branes

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 $(p,q)$ acted on by $SL(2,\mathbb{Z})$. For instance the fundamental string (F1-brane) and the D1-brane combine to the $(p,q)$-string, and similarly the NS5-brane and the D5-brane combine to a $(p,q)$-5-brane.

Namely in the F-theory picture this comes from wrapping the M2-brane and the M5-brane, respectively, on either of the two cycles of the elliptic fibration (and the T-dualizing).

(e.g. Johnson 97, p. 4)

### 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.

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

$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

A more recent survey is

• Blumenhagen, Basics of F-theory from the Type IIB Perspective (arXiv:1002.2836)

Lecture notes include

Textbook accounts include

Further survey includes

Related conferences include

### Relation to orientifolds

F-theory lifts of orientifold backgrounds were first identified in

and the corresponding gauge enhancement 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 6d superconformal theory

Realization to the 6d (2,0)-supersymmetric QFT is discussed in

### Phenomenology and model building

A large body of literature is concerned with particle physics string phenomenology modeled in the context of F-theory, in particular GUTs:

### 4-Form flux and instantons

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, $G_4$ 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 August 8, 2016 08:24:22 by Flavius? (86.126.56.107)