Scalar Laplacian on Sasaki-Einstein manifolds \(Y^{p,q}\). (English) Zbl 1247.53053
Summary: 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.
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 58J90 | Applications of PDEs on manifolds |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 83E30 | String and superstring theories in gravitational theory |
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