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Seiberg-Witten like equations on 8-manifolds with structure group Spin(7). (English) Zbl 1187.53049

Seiberg-Witten equations in 4-dimensions consist classically of two equations. The first one expresses the harmonicity condition of the spinors and this condition is linear. The second one couples the self-dual part of a 2-form with a spinor field and it is non linear. In differential geometry, the algebraic study of Clifford algebras leads classically to the study of a \(spin\)-structure, respectively a spin\(^{c}\)-structure on an oriented Riemannian \(n\)-dimensional manifold \(M\). In the thorough presentation given, the authors want to write analogues of Seiberg-Witten equations in 8 dimensions. The paper deals with 8-dimensional compact manifolds \(M\) with structure group Spin(7), i. e. such that the transition maps of the tangent bundle \(T(M)\) take their values in the group Spin(7). Hence, they consider the classical spinor bundle \(S\) on \(M\), they express the Dirac operator on it and write down the first Seiberg-Witten like equation.
For the second part, they use a kind of self-duality notion of 2-forms. If the structure group SO(8) of \(M\) reduces to Spin(7), they define a distinguished 4-form \(\Phi\) on \(M, \)called the fundamental form, which allows them to define a self-dual \(2-\)form in 8 dimensions. Some local discussions are given.

MSC:

53C27 Spin and Spin\({}^c\) geometry
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C10 \(G\)-structures
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References:

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