Scalar Laplacian on Sasaki-Einstein Manifolds Y^{p,q}

TL;DR

This paper analyzes the scalar Laplacian spectrum on five-dimensional Sasaki-Einstein manifolds Y^{p,q}, linking eigenstates to BPS states and exploring solutions of Heun's equation, including polynomial solutions in the near BPS limit.

Contribution

It provides a detailed spectral analysis of the scalar Laplacian on Y^{p,q} manifolds, connecting mathematical solutions to physical BPS states in dual gauge theories.

Findings

Ground states correspond to BPS chiral primary operators.

Eigenvalue problem reduces to Heun's equation with four singularities.

Polynomial solutions emerge in the near BPS limit.

Abstract

We study the spectrum of the scalar Laplacian on the five-dimensional toric Sasaki-Einstein manifolds Y^{p,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.