N=1 Curve

Understand

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.