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 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
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
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
Then the T-duality operation along $S^1_B$ divides this by $R_B$:
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.
F-theory on an elliptically fibered K3 is supposed to be equivalent to heterotic string theory compactified on a 2-torus (e.g. Sen 96).
The original article is
An early survey of its relation to M-theory with M5-branes is in
Lecture notes include
F-theory lifts of orientifold backgrounds were first identified in
Ashoke Sen, F-theory and Orientifolds (arXiv:hep-th/9605150)
Ashoke Sen, Orientifold Limit of F-theory Vacua (arXiv:hep-th/9702165)
with more details including
This is further expanded on in
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)
A large body of literature is concerned with particle physics string phenomenology modeled in the context of F-theory.
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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)