Factorization of Seiberg-Witten Curves and Compactification to Three Dimensions

TL;DR

This paper proves algebraically that the factorization of Seiberg-Witten curves in N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1 is correctly obtained via the Lax matrix substitution, confirming a key conjecture.

Contribution

It provides an algebraic proof that the Lax matrix substitution yields the correct Seiberg-Witten curve factorization, supporting the integrable system approach.

Findings

Algebraic proof of curve factorization method

Confirmation of the integrable system conjecture

Independent geometric proof of the factorization

Abstract

We continue our study of nonperturbative superpotentials of four-dimensional N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1, broken to N=1 due to a classical superpotential. In a previous paper, hep-th/0304061, we discussed how the low-energy quantum superpotential can be obtained by substituting the Lax matrix of the underlying integrable system directly into the classical superpotential. In this paper we prove algebraically that this recipe yields the correct factorization of the Seiberg-Witten curves, which is an important check of the conjecture. We will also give an independent proof using the algebraic-geometrical interpretation of the underlying integrable system.