Picard-Fuchs Ordinary Differential Systems in N=2 Supersymmetric Yang-Mills Theories

TL;DR

This paper explores the properties of Picard-Fuchs ordinary differential equations in N=2 supersymmetric Yang-Mills theories, revealing new relations among periods, moduli, and the QCD scale parameter, with specific results for SU(2) and SU(3) cases.

Contribution

It introduces a novel approach to Picard-Fuchs systems using the QCD scale as a variable, deriving new relations and solutions, including a previously unknown fifth solution for SU(3).

Findings

Existence of a Wronskian producing new period relations.

Derivation of a fifth solution for SU(3) Picard-Fuchs equations.

Connection between Picard-Fuchs equations and hypergeometric systems.

Abstract

In general, Picard-Fuchs systems in N=2 supersymmetric Yang-Mills theories are realized as a set of simultaneous partial differential equations. However, if the QCD scale parameter is used as unique independent variable instead of moduli, the resulting Picard-Fuchs systems are represented by a single ordinary differential equation (ODE) whose order coincides with the total number of independent periods. This paper discusses some properties of these Picard-Fuchs ODEs. In contrast with the usual Picard-Fuchs systems written in terms of moduli derivatives, there exists a Wronskian for this ordinary differential system and this Wronskian produces a new relation among periods, moduli and QCD scale parameter, which in the case of SU(2) is reminiscent of scaling relation of prepotential. On the other hand, in the case of the SU(3) theory, there are two kinds of ordinary differential equations, one of which is the equation directly constructed from periods and the other is derived from the SU(3) Picard-Fuchs equations in moduli derivatives identified with Appell's $F_4$ hypergeometric system, i.e., Burchnall's fifth order ordinary differential equation published in 1942. It is shown that four of the five independent solutions to the latter equation actually correspond to the four periods in the SU(3) gauge theory and the closed form of the remaining one is established by the SU(3) Picard-Fuchs ODE. The formula for this fifth solution is a new one.