Geometry/topology Seminar
Monday, September 9, 2019, 3:15pm, 119 Physics
Akos Nagy (Duke University, Mathematics)
Complex monopoles

Abstract:

The Bogomolny monopole equation can be complexified in many inequivalent ways, but there are two obvious options: One can extend Hodge duality in either a complex linear fashion, or in a conjugate linear one. The two cases result in two very different equations. These equations are also the dimensional reductions of the 4-dimensional Haydys and Kapustin-Witten equations, respectively. Thus we call solutions of the first equations Haydys monopoles, while solutions of the second equations called Kapustin-Witten monopoles. We find a stark contrast between the two cases: On one hand, we construct an open neighborhood of the Bogomolny moduli space within the Haydys moduli space, and show that this neighborhood is a smooth, hyperkahler manifold of dimension twice that of the Bogomolny moduli space. On the other hand, we prove that a (finite energy) Kapustin-Witten monopole is necessarily a Bogomolny monopole when the structure group is SU(2). (Joint work with Goncalo Oliveira)

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