Toric data and Killing forms on homogeneous Sasaki-Einstein manifold $T^{1,1}$

TL;DR

This paper explores the complex and symplectic structures of the conifold over the Sasaki-Einstein manifold T^{1,1}, explicitly deriving special Killing forms using toric data and complex coordinates.

Contribution

It provides a complete set of special Killing forms on T^{1,1} by leveraging toric data and complex coordinate methods.

Findings

Explicit complex coordinates for C(T^{1,1}) derived.

Complete set of special Killing forms obtained.

Enhanced understanding of the geometric structure of T^{1,1}.

Abstract

Throughout this paper we investigate the complex structure of the conifold $C(T^{1,1})$ basically making use of the interplay between symplectic and complex approaches of the K\"{a}hler toric manifolds. The description of the Calabi-Yau manifold $C(T^{1,1})$ using toric data allows us to write explicitly the complex coordinates and apply standard methods for extracting special Killing forms on the base manifold. As an outcome, we obtain the complete set of special Killing forms on the five-dimensional Sasaki-Einstein space $T^{1,1}$.