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A lift of the Seiberg-Witten equations to Kaluza-Klein five-manifolds. (English) Zbl 1462.81079

Summary: We consider Riemannian four-manifolds \(\left(X, g_X\right)\) with a \(spin^c\)-structure and a suitable circle bundle \(Y\) over \(X\) such that the \(spin^c\)-structure on \(X\) lifts to a spin-structure on \(Y\). With respect to these structures, a spinor \(\varphi\) on \(X\) lifts to an untwisted spinor \(\psi\) on \(Y\) and a \(\operatorname{U} \left(1\right)\)-gauge field \(A\) for the \(spin^c\)-structure can be absorbed into a Kaluza-Klein metric \(g_Y^A\) on \(Y\). We show that irreducible solutions \(\left(A, \phi\right)\) to the Seiberg-Witten equations on \(\left(X, g_X\right)\) for the given \(spin^c\)-structure are equivalent to irreducible solutions \(\psi\) of a Dirac equation with cubic non-linearity on the Kaluza-Klein circle bundle \(\left(Y, g_Y^A\right)\). As an application, we consider solutions to the equations in the case of Sasaki five-manifolds, which are circle bundles over Kähler-Einstein surfaces.
©2021 American Institute of Physics

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81R25 Spinor and twistor methods applied to problems in quantum theory
35Q55 NLS equations (nonlinear Schrödinger equations)
53C27 Spin and Spin\({}^c\) geometry
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57K41 Invariants of 4-manifolds (including Donaldson and Seiberg-Witten invariants)
83E15 Kaluza-Klein and other higher-dimensional theories
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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