$\mathcal{N}=1$ Curves on Generalized Coulomb Branches

TL;DR

This paper derives algebraic curves describing the low energy dynamics of 4D $ =1$ supersymmetric gauge theories of class $ =1$ on the Coulomb branch, connecting IR effective couplings to UV geometric data.

Contribution

It provides a novel derivation of algebraic curves for $ =1$ class $ =1$ theories and relates IR and UV descriptions via punctured Riemann surfaces.

Findings

IR curves $ ext{X}$ encode effective gauge couplings.

UV curves $ ext{C}$ are punctured Riemann surfaces with identical pole structure to $ =2$ theories.

Residues of differentials depend exactly on marginal couplings and mass parameters.

Abstract

We study the low energy effective dynamics of four-dimensional $\mathcal{N}=1$ supersymmetric gauge theories of class $\mathcal{S}_k$ on the generalized Coulomb branch. The low energy effective gauge couplings are naturally encoded in algebraic curves $\mathcal{X}$, which we derive for general values of the couplings and mass deformations. We then recast these IR curves $\mathcal{X}$ to the UV or M-theory form $\mathcal{C}$: the punctured Riemann surfaces on which the six-dimensional $\mathcal{N}=(1,0)$ ${A_{k-1}}$ SCFTs are compactified giving the class $\mathcal{S}_k$ theories. We find that the UV curves $\mathcal{C}$ and their corresponding meromorphic differentials take the same form as those for their mother four-dimensional $\mathcal{N}=2$ theories of class $\mathcal{S}$. They have the same poles, and their residues are functions of all the exactly marginal couplings and the bare mass parameters which we can compute exactly.