Complex monopoles II: The Kapustin--Witten monopole equation

Abstract

We study complexified Bogomolny monopoles in 3 dimensions by complexifying the compact structure groups. In this paper we use the conjugate linear extension of the Hodge star operator which yields a reduction of the Kapustin-Witten equations to 3 dimensions, thus we call its solutions Kapustin-Witten monopoles. Our main result is a vanishing theorem for these monopoles showing that the only finite energy Kapustin-Witten monopoles are the ordinary Bogomolny monopoles. We prove this by analyzing the asymptotic behavior of Kapustin-Witten monopoles, and combining our results with a recent theorem of Taubes.