Defects and Quantum Seiberg-Witten Geometry

We study various protected quantities in five dimensional gauge theories on $\mathbb{R}^4\times S^1$ with eight supercharges in the presence of defects. Surface defects can be described by a certain three dimensional quiver gauge theories on $\mathbb{R}^2\times S^1$, whose twisted chiral rings we extensively study by various methods including classical parameter spaces of supersymmetric vacua and exact partition functions. Three dimensional mirror symmetry and integrability play important roles in our analysis. We compute equivariant quantum K-theory rings of the corresponding Nakajima quiver varieties using a universal gauge invariant prescription. For a coupled 5d/3d system we find a convenient description using monodromy defects along the two-plane which is orthogonal to the 3d defect. We compute the corresponding 5d ramified instanton partition function and show that it solves a system of difference equations of a certain elliptic integrable many-body system.