N=1 Curve

TL;DR

This paper introduces the N=1 curve for four-dimensional class S theories using Cayley-Hamilton theorem, providing a geometric framework to analyze complex field theory dynamics and vacua structures.

Contribution

It develops a novel geometric construction of the N=1 curve incorporating spectral data, constraints, and differential forms, enabling detailed analysis of moduli and dualities.

Findings

Reveals intricate vacua structures involving monopole condensation

Extracts moduli space and chiral ring relations from spectral curves

Recovers non-trivial field theory phenomena like Seiberg duality

Abstract

N=1 curve is defined for four dimensional class S theory using Cayley-Hamilton theorem for two commuting matrices. The curve consists of three ingredients: 1: A set of N+1 degree N equations defining a curve; 2: a set of constraints relating the coefficients in the curve; 3: a canonically defined differential. We then extract from spectral curve various physical information such as the space of moduli fields, chiral ring relations, full moduli space, etc. Many examples are discussed, and the curve recovers the intricate vacua structure which often involves highly non-trivial field theory dynamics such as monopole condensation, dynamical generated superpotential, Seiberg duality, etc.