Toric Sasaki-Einstein manifolds and Heun equations

TL;DR

This paper constructs symplectic potentials for five-dimensional toric Sasaki-Einstein manifolds, analyzes the scalar Laplacian spectrum via Heun equations, and relates holomorphic functions to lattice points, connecting geometric data with dual gauge theory properties.

Contribution

It provides explicit symplectic potentials for a broad class of toric Sasaki-Einstein manifolds and links spectral data to toric geometry and dual gauge theories.

Findings

Eigenvalues of the Laplacian relate to Heun equations.

Holomorphic functions correspond to lattice points in the toric cone.

Scaling dimensions match R-charges in dual theories.

Abstract

Symplectic potentials are presented for a wide class of five dimensional toric Sasaki-Einstein manifolds, including L^{a,b,c} which was recently constructed by Cvetic et al. The spectrum of the scalar Laplacian on L^{a,b,c} is also studied. The eigenvalue problem leads to two Heun's differential equations and the exponents at regular singularities are directly related to toric data. By combining knowledge of the explicit symplectic potential and the exponents, we show that the ground states, or equivalently holomorphic functions, have one-to-one correspondence with integral lattice points in the convex polyhedral cone. The scaling dimensions of the holomorphic functions are simply given by scalar products of the Reeb vector and the integral vectors, which are consistent with R-charges of BPS states in the dual quiver gauge theories.