We construct explicit G 4 fluxes in F-theory compactifications. Our method relies on identifying algebraic cycles in the Weierstrass equation of elliptic Calabi-Yau fourfolds. We show how to compute the D3-brane tadpole and the induced chirality indices directly in F-theory. Whenever a weak coupling limit is available, we compare and successfully match our findings to the corresponding results in type IIB string theory. Finally, we present some generalizations of our results which hint at a unified description of the elliptic Calabi-Yau fourfold together with the four-form flux G 4 as a coherent sheaf. In this description the close link between G 4 fluxes and algebraic cycles is manifest.

Abstract:We study the duality between M-theory on compact holonomy G 2 -manifolds and the heterotic string on Calabi-Yau three-folds. The duality is studied for K3-fibered G 2 -manifolds, called twisted connected sums, which lend themselves to an application of fiber-wise M-theory/Heterotic Duality. For a large class of such G 2 -manifolds we are able to identify the dual heterotic as well as F-theory realizations. First we establish this chain of dualities for smooth G 2 -manifolds. This has a natural generalization to situations with non-abelian gauge groups, which correspond to singular G 2 -manifolds, where each of the K3-fibers degenerates. We argue for their existence through the chain of dualities, supported by non-trivial checks of the spectra. The corresponding 4d gauge groups can be both Higgsable and non-Higgsable, and we provide several explicit examples of the general construction.

Motivated by the desire to do proper model building with D7-branes and fluxes, we study the motion of D7-branes on a Calabi-Yau orientifold from the perspective of F-theory. We consider this approach promising since, by working effectively with an elliptically fibred M-theory compactification, the explicit positioning of D7-branes by (M-theory) fluxes is straightforward. The locations of D7-branes are encoded in the periods of certain M-theory cycles, which allows for a very explicit understanding of the moduli space of D7-brane motion. The picture of moving D7-branes on a fixed underlying space relies on negligible backreaction, which can be ensured in Sen's weak coupling limit. However, even in this limit we find certain 'physics obstructions' which reduce the freedom of the D7-brane motion as compared to the motion of holomorphic submanifolds in the orientifold background. These obstructions originate in the intersections of D7-branes and O7-planes, where the type IIB coupling cannot remain weak. We illustrate this effect for D7-brane models on CP 1 × CP 1 (the Bianchi-Sagnotti-Gimon-Polchinski model) and on CP 2 . Furthermore, in the simple example of 16 D7-branes and 4 O7-planes on CP 1 (F-theory on K3), we obtain a completely explicit parameterization of the moduli space in terms of periods of integral M-theory cycles. In the weak coupling limit, D7-brane motion factorizes from the geometric deformations of the base space.

A large number of examples of compact G 2 manifolds, relevant to supersymmetric compactifications of M-Theory to four dimensions, can be constructed by forming a twisted connected sum of two building blocks times a circle. These building blocks, which are appropriate K3-fibred threefolds, are shown to have a natural and elegant construction in terms of tops, which parallels the construction of Calabi-Yau manifolds via reflexive polytopes. In particular, this enables us to prove combinatorial formulas for the Hodge numbers and other relevant topological data.

U(1) gauge symmetries in F-theory are expected to manifest themselves as codimension three singularities of Calabi-Yau fourfolds. However, some of these are known to become massive at strong coupling via the Stückelberg mechanism. In this note, we propose a geometric picture for detecting all U(1)'s, and determining which ones are massive and which ones are massless. We find that massive gauge symmetries show up as codimension three singularities that only admit small, non-Kähler, resolutions. Our proposal passes several highly non-trivial tests, including a case with a non-diagonal mass matrix.