On the compactness problem for a family of generalized Seiberg–Witten equations in dimension 3

Abstract

We prove an abstract compactness theorem for a family of generalized Seiberg–Witten equations in dimension 3. This result recovers Taubes’s compactness theorem for stable flat $\mathbf{P}{\mathrm{SL}}_2(\mathbf{C})$-connections as well as the compactness theorem for Seiberg–Witten equations with multiple spinors by Haydys and Walpuski. Furthermore, this result implies a compactness theorem for the ${\mathrm{ADHM}}_{1,2}$ Seiberg–Witten equation, which partially verifies a conjecture by Doan and Walpuski.