2022/09/19 by Philip C. Argyres, Argyres, Philip C., Mario Martone +1 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Black Holes and Theoretical Physics #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.2209.09248
openalex publication_date 2022/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this first of a series of three papers we outline an approach to classifying 4d N=2 superconformal field theories at rank 2. The classification of allowed scale invariant N=2 Coulomb branch geometries of dimension (or rank) greater than one is a famous open problem whose solution will greatly constrain the space of N=2 superconformal field theories. At rank 2 the problem is equivalent to finding all possible genus 2 Seiberg-Witten curves and 1-forms satisfying a special Kähler condition. This is tractable because regular genus 2 Riemann surfaces can be uniformly described as binary-sextic plane curves, and the Seiberg-Witten curves are families of such curves varying meromorphically over the two-dimensional base. There are also solutions consisting of families of degenerate genus-2 Riemann surfaces given by a bouquet of two elliptic curves which are described by a different set of curves. In this paper we set up and carry out the analysis of the generic case, i.e., those whose typical fiber is a regular genus-2 Riemann surface with no extended automorphism, and find the complete answer for polynomial coefficients.