Trivializing a Family of Sasaki-Einstein Spaces
Abstract
We construct an explicit diffeomorphism between the Sasaki-Einstein spaces Yp,q and the product space S3 × S2 in the cases q 2. When q=1 we express the K\"ahler quotient coordinates as an SU(2) bundle over S2 which we trivialize. When q=2 the quotient coordinates yield a non-trivial SO(3) bundle over S2 with characteristic class p, which is rotated to a bundle with characteristic class 1 and re-expressed as Y2,1, 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 S2 × S3 coordinates to those for which the Sasaki-Einstein metric is known. We check that the RR flux on the S3 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 Tp,q and Yp,q spaces, which means that Tp,q manifolds are also topologically S3 × S2.