Geometric Engineering of Seiberg-Witten Theories with Massive Hypermultiplets

TL;DR

This paper explores the geometric engineering of N=2 SU(2) gauge theories with massive hypermultiplets, deriving differential equations for periods and linking Yukawa couplings to instanton expansions.

Contribution

It derives the Picard-Fuchs equations for these theories and connects the local A-model Yukawa coupling to instanton counting, extending understanding to massive hypermultiplets.

Findings

Derived differential equations for periods of Seiberg-Witten differential.

Connected Yukawa coupling to instanton expansion in gauge theory.

Obtained asymptotic form of instanton number for massless hypermultiplets.

Abstract

We analyze the geometric engineering of the N=2 SU(2) gauge theories with $1\leq N_f\leq 3$ massive hypermultiplets in the vector representation. The set of partial differential equations satisfied by the periods of the Seiberg-Witten differential is obtained from the Picard-Fuchs equations of the local B-model. The differential equations and its solutions are consistent with the massless case. We show that the Yukawa coupling of the local A-model gives rise to the correct instanton expansion in the gauge theory, and propose the pattern of the distribution of the world-sheet instanton number from it. As a side result, we obtain the asymptotic form of the instanton number in the gauge theories with massless hypermultiplets.