On rank two theories with eight supercharges part II: Lefschetz pencils

TL;DR

This paper investigates the global Seiberg-Witten geometries for rank two theories with eight supercharges, using Lefschetz pencils and mapping class group factorizations to understand IR physics of 4d SCFTs.

Contribution

It introduces a combinatorial approach to analyze rank two theories via Lefschetz pencils and MCG factorizations, extending methods to higher rank theories.

Findings

Solved MCG factorization problems for most rank two theories.

Provided a framework to determine IR physics of 4d SCFT vacua.

Generalized approach applicable to higher rank theories.

Abstract

The global Seiberg-Witten (SW) geometries for rank two theories with eight supercharges are studied. The theory is deformed generically so that there are only simplest $I_1$ or $\tilde{I}_1$ singularities on the Coulomb branch, which then geometrically gives the so-called Lefchetz pencils, The local singularity was shown to be determined by the conjugacy class of mapping class group (MCG); The global study is then reduced to the questions about MCG: a) Find the factorization of the MCG element of the singular fiber into positive products of Dehn twists (which gives the $I_1$ singularity or $\tilde{I}_1$ singularity); b) Find the factorization of identity element in terms of Dehn twists. We solved above two MCG problems for most rank two theories.The results are very helpful in determining IR physics for all vacua of 4d SCFTs. Our approach is combinatorial and many aspects can be straightforwardly generalized to the study of higher rank theory.