Geometric Engineering of N=2 CFT_{4}s based on Indefinite Singularities: Hyperbolic Case

TL;DR

This paper explores the geometric engineering of four-dimensional N=2 superconformal field theories using indefinite singularities, establishing a link with Kac-Moody algebra classification and deriving explicit mirror geometries for hyperbolic cases.

Contribution

It introduces a novel class of N=2 CFT4s based on indefinite singularities, connecting their beta function conditions with Kac-Moody algebra classifications and providing explicit geometric solutions.

Findings

Vanishing beta function condition matches Kac-Moody classification

Derived mirror geometries for CY threefolds with hyperbolic singularities

Established a new link between geometric engineering and indefinite singularities

Abstract

Using Katz, Klemm and Vafa geometric engineering method of $\mathcal{N}=2$ supersymmetric QFT$_{4}$s and results on the classification of generalized Cartan matrices of Kac-Moody (KM) algebras, we study the un-explored class of $\mathcal{N}=2$ CFT$_{4}$s based on \textit{indefinite} singularities. We show that the vanishing condition for the general expression of holomorphic beta function of $\mathcal{N}=2$ quiver gauge QFT$_{4}$s coincides exactly with the fundamental classification theorem of KM algebras. Explicit solutions are derived for mirror geometries of CY threefolds with \textit{% hyperbolic} singularities.