Toric Methods in F-theory Model Building

In this review article we discuss recent constructions of global F-theory GUT models and explain how to make use of toric geometry to do calculations within this framework. After introducing the basic properties of global F-theory GUTs we give a self-contained review of toric geometry and introduce all the tools that are necessary to construct and analyze global F-theory models. We will explain how to systematically obtain a large class of compact Calabi-Yau fourfolds which can support F-theory GUTs by using the software package PALP.Comment: 19 pages. Prepared for the special issue "Computational Algebraic Geometry in String and Gauge Theory" of Advances in High Energy Physics, v2: references added, typos correcte

Hemisphere Partition Function and Monodromy

We discuss D-brane monodromies from the point of view of the gauged linear sigma model. We give a prescription on how to extract monodromy matrices directly from the hemisphere partition function. We illustrate this procedure by recomputing the monodromy matrices associated to one-parameter Calabi-Yau hypersurfaces in weighted projected space.Comment: 32 pages, 4 figure

Grade restriction and D-brane transport for a non-abelian GLSM of an elliptic curve

We discuss a simple model for D-brane transport in non-abelian GLSMs. The model is the elliptic curve version of a non-abelian GLSM introduced by Hori and Tong and has gauge group U(2). It has two geometric phases, both of which describe the same elliptic curve, once realised as a codimension five complete intersection in G(2,5) and once as a determinantal variety. The determinantal phase is strongly coupled with unbroken SU(2). There are two singular points in the moduli space where the theory has a Coulomb branch. Using grade restriction rules, we show how to transport B-branes between the two phases along paths avoiding the singular points. With the help of the GLSM hemisphere partition function we compute analytic continuation matrices and monodromy matrices, confirming results obtained by different methods.Comment: 19 pages; prepared for the proceedings of the workshop GLSMs@30 at the Simons Center for Geometry and Physic

On genus-0 invariants of Calabi-Yau hybrid models

We compute genus zero correlators of hybrid phases of Calabi-Yau gauged linear sigma models (GLSMs), i.e. of phases that are Landau-Ginzburg orbifolds fibered over some base. These correlators are generalisations of Gromov-Witten and FJRW invariants. Using previous results on the structure of the of the sphere- and hemisphere partition functions of GLSMs when evaluated in different phases, we extract the I-function and the J-function from a GLSM calculation. The J-function is the generating function of the correlators. We use the field theoretic description of hybrid models to identify the states that are inserted in these correlators. We compute the invariants for examples of one- and two-parameter hybrid models. Our results match with results from mirror symmetry and FJRW theory.Comment: 36 page