Alloff-Wallach Space

The homogeneous coset SU(3)/U(1) space which is characterised by two integers, k and ℓ specifying the embedding of the U(1) in SU(3). If one represents SU(3) by 3×3 special unitary matrices then the U(1) subgroup can be taken to be matrices of the form

The coset spaces simply-connected with when k and ℓ are relatively prime, and these are denoted by N(k, ℓ), and the spaces N(k, ℓ), N(ℓ, k) and N(k, −k−ℓ) are topologically identical. There is an S ₃ permutation symmetry generated by these two Z ₂ operations, and all the N(k, ℓ) admit metrics of positive sectional curvature [1]. Each space N(k, ℓ) in fact admits two inequivalent Einstein metrics, except when (k, ℓ) = (0, 1) or the S ₃-related values (1, 0) or (1, −1), when there is only one [2]. Also each such metric admits a Killing spinor, except for one of the Einstein metrics with k = ℓ, which admits 3 Killing spinors . The special case k = ℓ can be viewed as an SO(3) bundle over ℂ ℙ² [3].

The Aloff-Wallach spacesare...