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 encodes encodes the axio-dilaton (the coupling constant and the degree-0 RR-field) tranforming under the 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 – to obtain type IIA supergravity – and then applying T-duality along another circle to obtain type IIB supergravity.
To obtain type IIB sugra in noncompact 10 dimensions this way, also is to be compactified (since T-duality sends the radius of to the inverse radius of ). Therefore type IIB sugra in is obtained from 11d sugra compactified on the torus . 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||F-theory in|
|KK-reduction along elliptic fibration||axio-dilaton is modulus of elliptic fibration|
|IIA string theory in||T-duality||IIB string theory in|
In the simple case where the elliptic fiber is indeed just , the imaginary part of its complex modulus is
First, the KK-reduction of M-theory on yields a type IIA string coupling
Then the T-duality operation along divides this by :
The target space data of an orientifold is a -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 -local system (being the “homological invariant” of the elliptic fibration).
Reasoning like this might suggest that in generalization to how type II orientifolds involve -equivariant K-theory (namely KR-theory), so F-theory should involve -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.
The original article is
Lecture notes include
F-theory lifts of orientifold backgrounds were first identified in
with more details including
This is further expanded on in
A large body of literature is concerned with particle physics string phenomenology modeled in the context of F-theory.