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

In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models.

We study Euclidean M5-branes wrapping vertical divisors in elliptic Calabi-Yau fourfold compactifications of M/F-theory that admit a Sen limit. We construct these Calabi-Yau fourfolds as elliptic fibrations over coordinate flip O3/O7 orientifolds of toric hypersurface Calabi-Yau threefolds. We devise a method to analyze the Hodge structure (and hence the dimension of the intermediate Jacobian) of vertical divisors in these fourfolds, using only the data available from a type ...

We consider N=2 supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called "dessins d'enfants" or "children's drawings" on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group Gal(\bar{Q}/Q) acts faithfully on them. We ar...

Using the graphical method developed in hep-th/9908082, we obtain the full curve corresponding to the hyperk\"ahler quotient from the extended E_7 Dynkin diagram. As in the E_6 case discussed in the same paper above, the resulting curve is the same as the one obtained by Minahan and Nemeschansky. Our results seem to indicate that it is possible to define a generalized Coulomb branch such that four dimensional mirror symmetry would act by interchanging the generalized Coulomb ...

When locally engineering F-theory models some D7-branes for the gauge group factors are specified and matter is localized on the intersection curves of the compact parts of the world-volumes. In this note we discuss to what extent one can draw conclusions about F-theory models by just restricting the attention locally to a particular seven-brane. Globally the possible D7-branes are not independent from each other and the (compact part of the) D7-brane can have unavoidable int...

In this note we show that in dual N=1 string vacua provided by the heterotic string on an elliptic Calabi-Yau together with a vector bundle respectively F-theory on Calabi-Yau fourfold the number of heterotic fivebranes necessary for anomaly cancellation matches the number of F-theory threebranes necessary for tadpole cancellation. This extends to the general case the work of Friedman, Morgan and Witten, who treated the case of embedding a heterotic $E_8\times E_8$ bundle, le...

This is a PhD thesis submitted to the University of the witwatersrand. The PhD focusses on the computation of non-holomorphic corrections and the study of monopole solutions in N=2 SUSY Yang Mills theories from the perspective of threebranes in F-theory.

Tadpole cancellation in F-theory on an elliptic Calabi-Yau fourfold $X\to B_3$ demands some spacetime-filling three-branes (points in $B_3$). If moved to the discriminant surface, which supports the gauge group, and dissolved into a finite size instanton, the second Chern class of the corresponding bundle $E$ is expected to give a compensating contribution. However the dependence of D-brane charge on the geometry of $W$ and on the embedding $i: W\to B_3$ gives a correction to...

We propose a framework for treating F-theory directly, without resolving or deforming its singularities. This allows us to explore new sectors of gauge theories, including exotic bound states such as T-branes, in a global context. We use the mathematical framework known as Eisenbud's matrix factorizations for hypersurface singularities. We display the usefulness of this technique by way of examples, including affine singularities of both conifold and orbifold type, as well as...

Branes at a $\mathbb{F}_0$ singularity give rise to two different toric quiver gauge theories, which are related by Seiberg duality. We study where in the K\"ahler moduli space each of them is physically realized.