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

Hiroaki Kanno·Yukinori Yasui
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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 $Σ_3=W(SU(3))$ (= 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 ${\bf CP}(2)$ 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.

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Hiroaki Kanno, Yukinori Yasui. 2001-11-29. On Spin(7) holonomy metric based on SU(3)/U(1). https://doi.org/10.1016/s0393-0440(02)00018-9

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