Monopoles on Sasakian Three-folds

We consider monopoles with singularities of Dirac type on quasiregular Sasakian three-folds fibering over a compact Riemann surface $\mathrm{\Sigma }$, for example the Hopf fibration $S^3⟶S^2$. We show that these correspond to holomorphic objects on $\mathrm{\Sigma }$, which we call twisted bundle triples. These are somewhat similar to Murray’s bundle gerbes. A spectral curve construction allows us to classify these structures, and, conjecturally, monopoles.