The Haydys monopole equation
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The Haydys monopole equation Ákos Nagy1 · Gonçalo Oliveira2
© Springer Nature Switzerland AG 2020
Abstract 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 R3 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). Keywords Monopoles · Moduli spaces · Haydys equation · Hyperkähler manifolds Mathematics Subject Classification 53C07 · 58D27 · 58E15 · 70S15
Contents 1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Preparation and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Gonçalo Oliveira [email protected] https://sites.google.com/view/goncalo-oliveira-math-webpage/home Ákos Nagy [email protected] http://akosnagy.com
1
Duke University, Durham, NC, USA
2
Universidade Federal Fluminense IME–GMA, Niterói, Brazil 0123456789().: V,-vol
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1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dimensional reduction . . . . . . . . . . . . . . . . . . . . . . . 3 Solving the Haydys monopole equation . . . . . . . . . . . . . . 3.1 The Bogomolny monopole equation and its linearization . . 3.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . 3.3 A gap theorem . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Multiplication properties of the function spaces . . . . . . . 3.5 Preparation for the proof of the Main Theorem 1 . . . . . . . 3.6 Linearization and gauge fixing . . . . . . . . . . . . . . . . 3.7 The proof of Main Theorem 1 . . . . . . . . . . . . . . . . . 4 On the geometry of the Haydys monopole moduli space . . . . . 4.1 Dimension of the Haydys moduli space . . . . . . . . . . . 4.2 Geometric structures on the Haydys monopole moduli space 4.2.1 Linear model . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Curved model . . . . . . . . . . . . . . . . . . . .
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