On Spin(7) holonomy metric based on SU(3)/U(1)

Abstract

We investigate the $Spin(7)$ holonomy metric of cohomogeneity one with the principal orbit $SU(3)/U(1)$. A choice of U(1) in the two dimensional Cartan subalgebra is left as free and this allows manifest ${\Sigma }_3=W(SU(3))$ (= the Weyl group) symmetric formulation. We find asymptotically locally conical (ALC) metrics as octonionic gravitational instantons. These ALC metrics have orbifold singularities in general, but a particular choice of the U(1) subgroup gives a new regular metric of $Spin(7)$ holonomy. Complex projective space $\mathbf{CP}(2)$ that is a supersymmetric four-cycle appears as a singular orbit. A perturbative analysis of the solution near the singular orbit shows an evidence of a more general family of ALC solutions. The global topology of the manifold depends on a choice of the U(1) subgroup. We also obtain an $L^2$-normalisable harmonic 4-form in the background of the ALC metric.

21 pages, Latex, Introduction slightly expanded, an error in section 6 corrected and references added, (v3) minor corrections

References in corpus (3)

• Gauge Theory at Large N and New G_2 Holonomy Metrics
• New Complete Non-compact Spin(7) Manifolds
• Supersymmetric M3-branes and G_2 Manifolds

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