Seiberg-Witten Like Equations on Pseudo-RiemannianSpincManifolds withG2(2)∗Structure

Nülifer Özdemir,Nedim Deǧirmenci

doi:10.1155/2016/2173214

Abstract

We consider 7-dimensional pseudo-Riemannianspincmanifolds with structure groupG2(2)∗. On such manifolds, the space of 2-forms splits orthogonally into componentsΛ2M=Λ72⊕Λ142. We define self-duality of a 2-form by considering the partΛ72as the bundle of self-dual 2-forms. We express the spinor bundle and the Dirac operator and write down Seiberg-Witten like equations on such manifolds. Finally we get explicit forms of these equations onR4,3and give some solutions.

Highlights

The Seiberg-Witten theory, introduced by Witten in [1], became one of the most important tools to understand the topology of smooth 4-manifolds
The first equation is the harmonicity condition of spinor fields; that is, the spinor field belongs to the kernel of the Dirac operator
There exist various generalizations of Seiberg-Witten equations to higher dimensional Riemannian manifolds [3,4,5,6]. All of these generalizations are done for the manifolds which have special structure groups

Summary

Introduction

The Seiberg-Witten theory, introduced by Witten in [1], became one of the most important tools to understand the topology of smooth 4-manifolds. There exist various generalizations of Seiberg-Witten equations to higher dimensional Riemannian manifolds [3,4,5,6]. All of these generalizations are done for the manifolds which have special structure groups. We consider 7-dimensional manifolds with structure group G2∗(2). In order to define spinors and Dirac operator, the manifold M must have a spinc-structure. We assume that 7-dimensional pseudo-Riemannian manifold M with signature (−, −, −, −, +, +, +) has spinc-structure. We will define self-duality of a 2-form on a 7-manifold with structure group G2∗(2) by using decomposition of 2-forms on this manifold.

Advances in Mathematical Physics

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