## Abstract

We derive the exact effective superpotential in 4d, N = 1 supersymmetric SU(2) gauge theories with N_{A} triplets and 2N_{f} doublets of matter superfields. We find the quantum vacua of these theories; the equations of motion (for N_{A} = 1) can be reorganized into the singularity conditions of an elliptic curve. From the phase transition points to the Coulomb branch, we find the exact Abelian gauge couplings, τ, for arbitrary bare masses and Yukawa couplings. We thus derive the result that τ is a section of an SL(2, ℤ) bundle over the moduli space and over the parameters space of bare masses and Yukawa couplings. For N_{c} > 2, we derive the exact effective superpotential in branches of supersymmetric SU(N_{c}) gauge theories with one supermultiplet in the adjoint representation (N_{A} = 1 ) and zero or one flavor (N_{f} = 0,1). We find the quantum vacua of these theories; the equations of motion can be reorganized into the singularity conditions of a genus N_{c} - 1 hyperelliptic curve. Finally, we present the effective superpotential in the N_{A}, N_{f} < N_{c} cases.

Original language | English |
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Pages (from-to) | 160-206 |

Number of pages | 47 |

Journal | Nuclear Physics B |

Volume | 459 |

Issue number | 1-2 |

DOIs | |

State | Published - 15 Jan 1996 |

### Bibliographical note

Funding Information:We thank N. Seiberg for discussions. The work of SE is supported in part by the BRF - the Basic Research Foundation. The work of AG is supported in part by BSF-American-Israel Bi-National Science Foundation, by the BRF, and by an Alon fellowship. The work of ER is supported in part by BSF and by the BRF.