A lift of the Seiberg-Witten equations to Kaluza-Klein 5-manifolds

We consider Riemannian 4-manifolds \((X,g_X)\) 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 \(\phi\) on \(X\) lifts to an untwisted spinor \(\psi\) on \(Y\)...

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Vydané v:arXiv.org
Hlavný autor: Hamilton, M J D
Médium: Paper
Jazyk:English
Vydavateľské údaje: Ithaca Cornell University Library, arXiv.org 16.08.2021
Predmet:
Dirac equation
Lifts
Linearity
Manifolds
Mathematical analysis
Spin structure
ISSN:2331-8422
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Popis
Shrnutí:We consider Riemannian 4-manifolds \((X,g_X)\) 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 \(\phi\) on \(X\) lifts to an untwisted spinor \(\psi\) on \(Y\) and a U(1)-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 \((A,\phi)\) to the Seiberg-Witten equations on \((X,g_X)\) 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 \((Y,g_Y^A)\). As an application we consider solutions to the equations in the case of Sasaki 5-manifolds which are circle bundles over Kaehler-Einstein surfaces.
Bibliografia:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.1906.10108