Exact U(N_c)-> U(N_1)xU(N_2) factorization of Seiberg-Witten curves and N=1 vacua

TL;DR

This paper provides an exact solution for factorizing Seiberg-Witten curves when breaking a U(N_c) gauge group into two factors, revealing detailed structure of N=1 vacua and their dualities.

Contribution

It offers a precise, parameter-dependent solution to the curve factorization problem for U(N_c) breaking, including discrete symmetries and duality insights.

Findings

Exact factorization formula involving discrete and continuous parameters

Identification of parameter symmetries and dualities

Proof that period integrality implies factorization

Abstract

N=2 gauge theories broken down to N=1 by a tree level superpotential are necessarily at the points in the moduli space where the Seiberg-Witten curve factorizes. We find exact solution to the factorization problem of Seiberg-Witten curves associated with the breaking of the U(N_c) gauge group down to two factors U(N_1)xU(N_2). The result is a function of three discrete parameters and two continuous ones. We find discrete identifications between various sets of parameters and comment on their relation to the global structure of N=1 vacua and their various possible dual descriptions. In an appendix we show directly that integrality of periods leads to factorization.