Trivializing a Family of Sasaki-Einstein Spaces

TL;DR

This paper explicitly constructs a diffeomorphism between certain Sasaki-Einstein spaces Y^{p,q} and S^3 imes S^2 for q ≤ 2, simplifying their topology and relating them to known geometries, with implications for flux normalization and homology classes.

Contribution

It provides explicit diffeomorphisms for Y^{p,q} spaces with q ≤ 2, simplifying their topology and relating them to familiar manifolds, also connecting T^{p,q} and Y^{p,q} spaces.

Findings

Y^{p,q} spaces with q ≤ 2 are topologically S^3 × S^2.

Explicit trivialization of bundles for q=1 and q=2 cases.

Established a diffeomorphism between T^{p,q} and Y^{p,q} spaces.

Abstract

We construct an explicit diffeomorphism between the Sasaki-Einstein spaces Y^{p,q} and the product space S^3 \times S^2 in the cases q \le 2. When q=1 we express the K\"ahler quotient coordinates as an SU(2) bundle over S^2 which we trivialize. When q=2 the quotient coordinates yield a non-trivial SO(3) bundle over S^2 with characteristic class p, which is rotated to a bundle with characteristic class 1 and re-expressed as Y^{2,1}, 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 S^2 \times S^3 coordinates to those for which the Sasaki-Einstein metric is known. We check that the RR flux on the S^3 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^3 \times S^2.