A Note on the Picard-Fuchs Equations for N=2 Seiberg-Witten Theories

TL;DR

This paper presents a systematic formulation of Picard-Fuchs equations for N=2 Seiberg-Witten theories across classical gauge groups, highlighting the structure of these equations for various numbers of hypermultiplets and their computational aspects.

Contribution

It provides a unified, generic form of Picard-Fuchs equations for classical groups in N=2 theories, including cases with massless hypermultiplets, and introduces a method suitable for symbolic computation.

Findings

All rank r PF equations for N_f=0 are in a generic form.

For N_f≠0, at least r-2 equations are generic.

The classical part of the equations is always generic, quantum part is manageable.

Abstract

A concise presentation of the PF equations for N=2 Seiberg-Witten theories for the classical groups of rank r with N_f massless hypermultiplets in the fundamental representation is provided. For N_f=0, all r PF equations can be given in a generic form. For certain cases with N_f\neq zero, not all equations are generic. However, in all cases there are at least r-2 generic PF equations. For these cases the classical part of the equations is generic, while the quantum part can be formulated using a method described in a previous paper by the authors, which is well suited to symbolic computer calculations.