Hyperelliptic families and 4d $\mathcal{N}=2$ SCFT

TL;DR

This paper classifies 4d $ ext{N}=2$ SCFTs with hyperelliptic Seiberg-Witten geometries, providing a systematic framework for understanding their structures and extending known results to higher ranks and new gauge theories.

Contribution

It introduces a classification scheme for hyperelliptic families of SW geometries in 4d $ ext{N}=2$ SCFTs, including new infinite sequences and quotient constructions.

Findings

Recovered known rank one and two results

Provided infinite sequences of theories at arbitrary ranks

Constructed $B$ and $D$ type conformal gauge theories

Abstract

We classify four dimensional $\mathcal{N}=2$ SCFTs whose Seiberg-Witten (SW) geometries can be written as hyperelliptic families. By using special K\"ahler condition of SW geometry, we reduce the problem to one parameter quasi-homogeneous hyperelliptic families $y^2=f(x,t)$. The classification is given by further demanding that the complex algebraic surface defined by $y^2=f(x,t)$ has an isolated singularity. We then write down the full SW geometry by looking at mini-versal deformations of the one parameter family, and the SW differential is also written down. The detailed physical data for these theories are found by matching the theory with other known construction. Our solutions recover the known rank one and rank two results, and give some infinite sequences valid at arbitrary ranks. We also studied $Z_2$ quotient of above hyperelliptic families which give rise to $B$ type and $D$ type conformal gauge theory, and further generalizations.