Scalar Laplacian on Sasaki - Einstein Manifolds

We study the spectrum of the scalar Laplacian on the five-dimensional toric Sasaki-Einstein manifolds Y . 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. [email protected] makoto [email protected] [email protected] 1 The AdS/CFT correspondence [1] has attracted much interest as a realization of the string theory/gauge theory correspondence. It predicts that string theory in AdS5 ×X5 with X5 be Sasaki-Einstein is dual to N = 1 4-dimensional superconformal field theory. Recently in [2, 3], 5-dimensional inhomogeneous toric Sasaki-Einstein manifolds Y p,q are explicitly constructed, besides the homogeneous manifolds, S and T . The associated Calabi-Yau cones C(Y ) are toric owing to the presence of the T -action on the 4dimensional Kähler-Einstein bases. Thanks to this property, the authors of [4][5] clarified the N = 1 4-dimensional dual superconformal field theories of IR fixed points of toric quiver gauge theories. (Further developments in this subject include [6][7].) On the other hand, in the gravity side, semiclassical strings moving on the AdS5 × Y p,q geometry are shown to be useful to establish the AdS/CFT correspondence in [8]. In this letter, we study the spectrum of the scalar Laplacian on the 5-dimensional Sasaki-Einstein manifolds Y . The eigenvalue equation of the scalar Laplacian is shown to reduce to Heun’s equation after the separation of variables. Heun’s equation is the general second-order linear Fuchsian equation with four singularities. It is known that the methods to investigate hypergeometric functions with three singularities do not work for Heun’s equation. Though there exist power-series solutions, the coefficients are governed by three-term recursive relations, and thus it is generally impossible to write down these series explicitly. We clarify some eigenstates of the scalar Laplacian, i.e. solutions of Heun’s equation, which include BPS states dual to chiral primary operators of the superconformal gauge theory. The metric tensor of Y p,q parameterized by two positive integers p, q (p > q) is written as [2] ds = 1− y 6 ( dθ + sin θdφ ) + 1 w(y)q(y) dy + q(y) 9 (dψ − cos θdφ) +w(y) [dα + f(y)(dψ − cos θdφ)] , (1) with w(y)= 2(b− y) 1− y , q(y) = b− 3y + 2y b− y2 , f(y) = b− 2y + y 6(b− y2) , b= 1 2 − p 2 − 3q 4p3 √ 4p2 − 3q2 . (2) The coordinates {y, θ, φ, ψ, α} have the following ranges: y1 ≤ y ≤ y2 , 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π , 0 ≤ ψ ≤ 2π , 0 ≤ α ≤ 2πl . (3) In [9], Heun’s equation corresponding to the scalar Laplacian on the inhomogeneous manifolds constructed in [10] is examined.