N=1 Curve

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.

Figures (4)

• Figure 1: Moduli space of vacua of $\mathcal{N}=1$ class ${\cal S}$ theory from 6d $A_1$ theory.
• Figure 2: Left: Type IIA brane configuration for SQCD deformed by a quartic superpotential. Right: The lift to M5 brane description: there are two irregular singularities describing left and right NS5 branes.
• Figure 3: Left: Type IIA and M5 brane configuration for $\mathcal{N}=2$ theory, here the two irregular punctures are of the same type. Middle: Type IIA and M5 brane configuration for $\mathcal{N}=2$ theory deformed by finite adjoint mass, here $\Phi_1$ and $\Phi_2$ are both singular at the rotated puncture labeled by a blue square; Right: Type IIA and M5 brane configuration for pure $\mathcal{N}=1$ theory, here $\Phi_1$ and $\Phi_2$ are only singular at one puncture respectively.
• Figure 4: Type IIA brane configuration for $\mathcal{N}=2$ theory deformed by LG superpotential.