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We construct nonminimal and irreducible solutions to the Ginzburg–Landau equations on closed manifolds of arbitrary dimension with trivial first real cohomology. Our method uses bifurcation theory where the "bifurcation points" are characterized by the eigenvalues of a Laplace-type operator. To our knowledge these are the first such examples on nontrivial line bundles. [ABSTRACT FROM AUTHOR]

We study solutions to Nahm's equations with continuous symmetries, and under certain (mild) hypotheses, we classify the corresponding Ansätze. Using our classification, we construct novel Nahm data and prescribe methods for generating further solutions. Finally, we use these results to construct new BPS monopoles with spherical symmetry. [ABSTRACT FROM AUTHOR]

We study solutions to the Kapustin–Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is R 4 or R 3 × S 1 and the structure group is SU (2) . [ABSTRACT FROM AUTHOR]

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator, these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on R 3 is a Kähler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold—an A-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin–Witten monopoles for which we have showed a vanishing theorem in Nagy and Oliveira (Kapustin–Witten equations on ALE and ALF Gravitational Instantons, 2019). [ABSTRACT FROM AUTHOR]