# Elliptic Fibrations and Singularities to Anomalies and Spectra 3 of 4

Date: Wed, Aug 25, 2021

Location: PIMS, University of Saskatchewan, Zoom, Online

Conference: 2nd PIMS Summer School on Algebraic Geometry in High Energy Physics

Subject: Mathematics, Algebraic Geometry, Physics, Particle Physics and Quantum Field Theory

Class: Scientific

### Abstract:

Throughout my lectures I will explain the geometry of elliptic fibration which can gåve rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.