Surface defects and resolvents

We study a large class of BPS surface defects in 4d$$ \mathcal{N}=2 $$ gauge theories. They are defined by coupling a 2d$$ \mathcal{N}=\left( {2,2} \right) $$ gauged linear sigma model to the 4d bulk degrees of freedom. Our main result is an efficient computation of the effective twisted superpotential for all these models in terms of a basic object closely related to the resolvent of the 4d gauge theory, which encodes the curve describing the 4d low energy dynamics. We reproduce and extend the results of brane constructions and compute the effective twisted superpotential for general monodromy surface defects. We encounter novel, puzzling field theory phenomena in the low energy dynamics of the simplest surface defects and we propose some local models to explain them. We also study in some detail the behavior of surface defects near monopole points of the bulk theory’s Coulomb branch. Finally, we explore the effect on the defect of breaking the bulk supersymmetry from $$ \mathcal{N}=2 $$ to $$ \mathcal{N}=1 $$ and show that certain quantities are independent of this breaking.