# Workshop on Elliptic fibrations and F-theory, Jan, 2010

Date: 4 January (Mon) -- 8 January (Fri), 2010

Place: Prefab-B, IPMU, University of Tokyo

Announcement:

The past 2 years has seen a resurgence of interest in constructing models of low-energy physics via F-theory compactification, due to a number of successes in matching these models with a variety of aspects of real-world phenomenology. The F-theory construction in string theory is one of the most mathematically sophisticated approaches to string compactification, involving the class of ``elliptically fibered'' Calabi--Yau manifolds. Further progress in using F-theory to produce realistic phenomenological models will necessarily involve advances in our mathematical understanding of elliptically fibered Calabi--Yau manifolds.

During this focus week, we will assemble a group of mathematicians knowledgeable about elliptic fibrations in algebraic geometry, together with a group of string theorists working on F-theory compactifications, with the aim of sharing ideas between the two groups and making substantive progress in the field.

Organizers:

Alexey Bondal (Aberdeen / IPMU), Teruhiko Kawano (U.Tokyo), David Morrison (UCSB), Kyoji Saito (IPMU), Taizan Watari (IPMU).

Program: (see below for abstracts)

January 4 (Mon): Introduction to F-theory

10:00 -- 11:00 David R. Morrison (UC Santa Barbara) Part I: "String duality and elliptic fibrations"

11:00 -- 12:00 Sheldon Katz (U.Illinois Urbana-Champaign) "Gauge Theories from Geometry"

13:00 -- 14:00 David R. Morrison Part II: "Elliptic 3-folds and F-theory in dimension 6"

14:00 -- 15:00 David R. Morrison Part III: "Elliptic 4-folds and F-theory in dimension 4"

January 5 (Tue): efforts to extract physics from F-theory

10:00 -- 11:00 Taizan Watari (IPMU) "Very elementary introduction to particle physics"

11:15 -- 12:15 Johnathan J. Heckman (IAS) "Particle Physics and F-theory"

13:30 -- 15:00 Johnathan J. Heckman

January 6 (Wed): elliptic fibration

10:00 -- 12:00 Noboru Nakayama (RIMS) "On the structure of an elliptic fibration"

13:00 -- 15:00 Tetsuji Shioda (Rikkyo U. / RIMS) "Mordell-Weil Lattices and Elliptic K3 Surfaces"

15:30 -- 16:30 Antonella Grassi (U. Pennsylvania) "Toric Weierstrass models"

January 7 (Thu): Higgs bundle and physics

10:00 -- 11:00 Ron Donagi (U. Pennsylvania) Part I "Spectral covers and Higgs bundles"

11:00 -- 12:00 Tony Pantev (U. Pennsylvania) Part I

13:00 -- 14:00 Ron Donagi Part II "Model building, GUT-breaking, and Higgs bundles in local and global F-theory"

14:00 -- 15:00 Tony Pantev Part II

See below for the list of references for the lectures by Donagi and Pantev.

January 8 (Fri) : engineering geometry of the real world

Joseph D. Marsano (Enrico Fermi Institute) and Sakura Schafer-Nameki (KITP)

"From Local to Global F-theory Model Building"

10:30 -- 12:00 Part I: "From Local to Semi-local Models" (presented by Marsano)

13:30 -- 15:00 Part II: "From Semi-local to Global Models" (presented by Schafer-Nameki)

Abstracts:

David R. Morrison:

I will give a thorough introduction to both F-theory and elliptic fibrations, starting from the original motivations from string duality in the mid-1990's. This introduction is intended for both physicists and mathematicians, and will attempt to highlight the issues where both communities have something significant to contribute.

Sheldon Katz: Gauge Theories from Geometry

ADE singularities in IIA string theory or F-theory produce an effective nonabelian supersymmetric gauge theory, with additional localized matter present where the singularities get worse. This expository talk will review some of the foundational ideas in geometry and physics.

Taizan Watari: Very elementary introduction to particle physics

We begin with a review of basic consepts in physics such as natural units, mass dimensions and scale. After that, we move on to explain the Standard Model of particle physics and what people try to do in string phenomenology. This talk is to provide a common introduction to other physics talks in this workshop. This talk is clearly not for physicists.

Johnathan J. Heckman: Particle Physics and F-theory

We review recent progress on realizing detailed features of the Standard Model of particle physics in F-theory based Grand Unified Theories (GUTs). We introduce the main structures of GUTs which are realized in such models, and explain how these geometric elements combine to provide natural resolutions of traditionally vexing problems in four-dimensional models. Conversely, we explain how demands from phenomenology require the presence of specific geometric structures.

Noboru Nakayama: "On the structure of an elliptic fibration"

I would like to explain the structure of an elliptic fibration from the viewpoint of the theory of variation of Hodge structure (VHS). The VHS of weight one and rank two defines a Weierstrass model as a basic elliptic fibration, and any projective elliptic fibration is considered as a torsor of the basic elliptic fibration in some sense. In particular, the torsor is expressed as a torsion element of a suitable cohomology group defined by the VHS where we use a cohomology theory analogous to the etale cohomology of schemes.

Tetsuji Shioda: Mordell-Weil Lattices and Elliptic K3 Surfaces

First we review the basic facts on Mordell-Weil Lattice (MWL) of an elliptic surface with a section, and the well-known case of rational elliptic surfaces (RES). Then we consider the case of K3 surfaces and plan to discuss the following subjects (if time permits).

K3 with elliptic fibration with big Mordell-Weil rank,

structure of MWL for such,

various elliptic fibrations ("elliptic parameters") on a given K3,

everywhere integral sections on a RES or K3.

Antonella Grassi: Toric Weierstrass models (work in collaboration with V. Perduca)

We define a natural toric Weierstrass model for an elliptic variety X which is an hypersurface in a Fano toric variety and discuss some properties.

Ron Donagi:

Part I: "Spectral covers and Higgs bundles"

This will be an introduction to spectral covers in integrable systems, heterotic strings, and F-theory, emphasizing their relation to Higgs bundles.

References: alg-geom/9702002, hep-th/9801057, hep-th/9802093, hep-th/9802094, hep-th/9806094, hep-th/0405014

Part II: "Model building, GUT-breaking, and Higgs bundles in local and global F-theory"

This will be a survey of recent work, mostly with Martijn Wijnholt.

References: arXiv:0904.1218, 0808.2223, 0802.2969 by Ron Donagi and Martijn Wijnholt

Tony Pantev:

[ Tentatively I plan to spend the first hour discussing Higgs bundles and linearization in the context of F-theory and the second hour discussing non-holomorphic Higgs bundles and M-theory phenomenology. ]

Reference: hep-th/0607159

Joseph D. Marsano and Sakura Schafer--Nameki:

Part 1: From Local to Semi-local Models

Part 2: From Semi-local to Global Models

We discuss the constraints that arise from embedding local F-theory GUTs into globally well-defined compactifications. In the first talk we will describe the semi-local model, which captures very general constraints that have to be met irrespective of a specific geometry. In the second talk, we will construct a sample geometry, realize 3-generation SU(5) GUTs and discuss further global constraints, such as tadpole cancellation.