Trivializing a Family of Sasaki-Einstein Spaces

We construct an explicit diffeomorphism between the Sasaki-Einstein spaces Y p,q and the product space S × S in the cases q 6 2. When q = 1 we express the Kähler quotient coordinates as an SU(2) bundle over S which we trivialize. When q = 2 the quotient coordinates yield a non-trivial SO(3) bundle over S with characteristic class p, which is rotated to a bundle with characteristic class 1 and re-expressed as Y , reducing the problem to the case q = 1. When q > 2 the fiber is a lens space which is not a Lie group, and this construction fails. We relate the S × S coordinates to those for which the Sasaki-Einstein metric is known. We check that the RR flux on the S is normalized in accordance with Gauss’ law and use this normalization to determine the homology classes represented by the calibrated cycles. As a by-product of our discussion we find a diffeomorphism between T p,q and Y p,q spaces, which means that T p,q manifolds are also topologically S × S.