# On F-theory Quiver Models and Kac-Moody Algebras

## TL;DR

This paper explores F-theory quiver models based on Kac-Moody algebras, analyzing their geometric and algebraic structures, and how they can incorporate fundamental matter fields in four-dimensional gauge theories.

## Contribution

It introduces a classification of F-theory quiver models using Kac-Moody algebra geometries, including hyperbolic types, and interprets anomaly cancellation conditions within this framework.

## Key findings

- Hyperbolic geometries can incorporate fundamental fields into gauge groups.
- The equations of Kac-Moody algebras relate to anomaly cancellation in F-theory quivers.
- Adding nodes to affine geometries yields models with richer matter content.

## Abstract

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four dimensional base space. We focus on the base geometry which consists of intersecting F0=CP1xCP1 Hirzebruch complex surfaces arranged as Dynkin graphs classified by three kinds of Kac-Moody (KM) algebras: ordinary, i.e finite dimensional, affine and indefinite, in particular hyperbolic. We interpret the equations defining these three classes of generalized Lie algebras as the anomaly cancelation condition of the corresponding N =1 F-theory quivers in four dimensions. We analyze in some detail hyperbolic geometries obtained from the affine A base geometry by adding a node, and we find that it can be used to incorporate fundamental fields to a product of SU-type gauge groups and fields.