A dessin on the base: a description of mutually non-local 7-branes without using branch cuts

We analyze exotic matter representations that arise on singular seven-brane configurations in F-theory. We develop a general framework for analyzing such representations, and work out explicit descriptions for models with matter in the 2-index and 3-index symmetric representations of SU($N$) and SU(2) respectively, associated with double and triple point singularities in the seven-brane locus. These matter representations are associated with Weierstrass models whose discrimin...

We study the geometry of elliptic fibrations given by Weierstrass models resulting from Step 6 of Tate's algorithm. Such elliptic fibrations have a discriminant locus containing an irreducible component $S$, over which the generic fiber is of Kodaira type I$^*_0$. In string geometry, these geometries are used to geometrically engineer G$_2$, Spin($7$), and Spin($8$) gauge theories. We give sufficient conditions for the existence of crepant resolutions. When they exist, we giv...

F-theory is perhaps the most general currently available approach to study non-perturbative string compactifications in their geometric, large radius regime. It opens up a wide and ever-growing range of applications and connections to string model building, quantum gravity, (non-perturbative) quantum field theories in various dimensions and mathematics. Its computational power derives from the geometrisation of physical reasoning, establishing a deep correspondence between fu...

I use matrix factorizations to describe branes at simple singularities as they appear in elliptic fibrations of local F-theory models. Each node of the corresponding Dynkin diagrams of the ADE-type singularities is associated with one indecomposable matrix factorization which can be deformed into one or more factorizations of lower rank. Branes with internal fluxes arise naturally as bound states of the indecomposable factorizations. Describing branes in such a way avoids the...

We discuss local F-theory geometries and theirs gauge theory dualities in terms of intersecting D7-branes wrapped four-cycles in Type IIB superstring. The manifolds are built as elliptic K3 surface fibrations over intersecting F_0=CP^1 \times CP^1 base geometry according to ADE Dynkin Diagrams. The base is obtained by blowing up the extended ADE hyper-Kahler singularities of eight dimensional manifolds considered as sigma model target spaces with eight supercharges. The resul...

We consider discrete K-theory tadpole cancellation conditions in type IIB orientifolds with magnetised 7-branes. Cancellation of K-theory charge constrains the choices of world-volume magnetic fluxes on the latter. We describe the F-/M-theory lift of these configurations, where 7-branes are encoded in the geometry of an elliptic fibration, and their magnetic quanta correspond to supergravity 4-form field strength fluxes. In a K3 compactification example, we show that standard...

We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q$_7(\mathscr{L},\mathscr{S})$ model. It naturally generalizes several previous models of elliptic fibrations popular in the F-theory literature. The model is also explicitly smooth, thus relevant physical quantities can be computed in terms of topological invariants in straight manner. Since the general fiber is defined by a cubic curve, basic arithmetic operations ...

In this paper we present simplifying techniques which allow one to compute the quiver diagrams for various D-branes at (non-Abelian) orbifold singularities with and without discrete torsion. The main idea behind the construction is to take the orbifold of an orbifold. Many interesting discrete groups fit into an exact sequence $N\to G\to G/N$. As such, the orbifold $M/G$ is easier to compute as $(M/N)/(G/N)$ and we present graphical rules which allow fast computation given th...

This expository paper describes sewing conditions in two-dimensional open/closed topological field theory. We include a description of the G-equivariant case, where G is a finite group. We determine the category of boundary conditions in the case that the closed string algebra is semisimple. In this case we find that sewing constraints -- the most primitive form of worldsheet locality -- already imply that D-branes are (G-twisted) vector bundles on spacetime. We comment on ex...

In this work we study F-theory compactifications on elliptically fibered Calabi-Yau n-folds which have $\mathbb{P}^1$-fibered base manifolds. Such geometries, which we study in both 4- and 6-dimensions, are both ubiquitous within the set of Calabi-Yau manifolds and play a crucial role in heterotic/F-theory duality. We discuss the most general formulation of $\mathbb{P}^1$-bundles of this type, as well as fibrations which degenerate at higher codimension loci. In the course of...