SUSY N=2 hyperelliptic curve from N=1 effective potential

TL;DR

This paper derives hyperelliptic curves for N=1 supersymmetric gauge theories with generalized couplings, connecting them to known N=2 curves and analyzing gaugino condensation dependence on couplings.

Contribution

It provides a derivation of hyperelliptic curves for N=1 theories with generalized couplings, extending the understanding of their relation to N=2 curves and gaugino condensation.

Findings

Derived singularity conditions for N=1 hyperelliptic curves with generalized couplings.

Reproduced known N=2 curve forms when couplings reduce to N=2 values.

Explicitly expressed the classical vacuum expectation value in terms of couplings.

Abstract

We derive the singularity conditions of the N=1 generalized (general yukawa couplings and quark masses) form of hyperelliptic curves of SU(N_c) with N_f flavors. The results reproduce the known form of N=2 curves when the yukawa couplings and the quark masses reduce to those of N=2. We obtained these curves by determining the dependence of the unbroken SU(2) gaugino condensation on the couplings in the moduli source terms which break N=2 SQCD to N=1 SU(N_c) gauge theory with the quarks and the adjoint matter, \Phi. The degenerate component of the diagonalized classical vacuum expectation value of \Phi is shown to be explicitly written in terms of these couplings, which enables us to determine the form of the gaugino condensation.