Scalar Laplacian on Sasaki-Einstein manifolds Yp,q

We study the spectrum of the scalar Laplacian on the five-dimensional toric Sasaki-Einstein manifolds Yp,q. The eigenvalue equation reduces to Heun's equation, which is a Fuchsian equation with four regular singularities. We show that the ground states, which are given by constant solutions of Heun's equation, are identified with BPS states corresponding to the chiral primary operators in the dual quiver gauge theories. The excited states correspond to non-trivial solutions of Heun's equation. It is shown that these reduce to polynomial solutions in the near BPS limit.