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

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.

The boundaries y = y 1 , y 2 are given by the two smallest roots of the cubic b − 3y 2 + 2y 3 , respectively, while the remaining root takes the value The period of α is 2πl with For the metric (1) we have the scalar Laplacian ♮ , The operatorQ R represents the Reeb Killing vector field, which corresponds to the R-symmetry of the dual gauge theory [5]. The operatorK is the second Casimir of SU(2), which is a part of the isometry SU(2) × U(1) 2 , Owing to the isometry, the eigenfunction takes the form with N φ , N ψ , N α ∈ Z. Then the equation, Φ = −EΦ, reduces tô and ♮ The spectrum of the scalar Laplacian on T p,q is examined in [11]. For that of the scalar Laplacian on the Calabi-Yau cone C(Y p,q ), some properties are studied in [7].
The quantum numbers J and Q R = 2N ψ − 1 3l N α correspond to SU(2)-spin and R-charge, respectively. The regular solutions of the first equation (11) are given by Jacobi polynomials. After some calculations we find that the second equation (12) is of Fuchsian-type with four regular singularities at y = y 1 , y 2 , y 3 and ∞, i.e. Heun's equation; where and The exponents at the regular singularities are given by ±α i at y = y i (i = 1, 2, 3), while −λ and λ + 2 at y = ∞, where we put It is convenient to transform the singularities from {y 1 , y 2 , y 3 , ∞} to {0, 1, a = y 1 −y 3 y 1 −y 2 , ∞}. This is achieved by the transformation together with the rescaling which transforms (13) to the standard form of Heun's equation: Here, Heun's parameters are given by and k, which is called as the "accessory" parameter, is Note that the parameter a satisfies the inequality a > 1 reflecting p > q.
The regularity of the eigenfunction in the range 0 ≤ x ≤ 1, which corresponds to y 1 ≤ y ≤ y 2 in the original variable, requires that h(x) should be a Heun function in the sense of [12]; h(x) has the zero exponents at the both boundaries x = 0 and 1.
Simple such an example is a polynomial solution. The polynomial condition on Heun's parameters is non-trivial (see (27)), although the necessary condition is simply given by α ∈ Z − = {0, −1, −2, −3, · · · }(or β ∈ Z − ). If we write the regular local solution around then the radius of convergence is normally min (1, a). Since a > 1, the series will generically have radius of convergence 1; it will therefore only represent a local solution. If a certain condition on Heun's parameters is satisfied, the radius of convergence is increased to a, so that one can obtain a Heun function [12]. However, it is not easy to determine when the condition is satisfied. When the series breaks off at the n-th order, (26) becomes a polynomial solution of degree n. Then, the coefficients c m satisfy a set of equations, − kc 0 + aγc 1 = 0 , P m c m−1 − (Q m + k)c m + R m c m+1 = 0 (m = 1, 2, · · · , n − 1) , where First, we consider states with the quantum charges {J, Q R , N α } corresponding to the chiral operators of dual superconformal field theory. Then, Heun's parameters are completely fixed except for one parameter λ. The basic chiral primary operators of mesons are given by the three fields {S, L + , L − } with charges given in Table 1 [5,8]. One can read off these charges from the data of Y p,q quiver gauge theory. Note that the R-charges of L ± appear in the exponents α 1,2 (see (15) and (16)) at the regular singularities y 1,2 , respectively. This is consistent with the semi-classical analysis [8].
When we choose the quantum charges {J, Q R , N α } as Table 1 together with λ = λ 0 in Table 2, then Heun's parameters α and k vanish, so that Heun's equation admits a constant solution (a polynomial solution of degree 0) for each meson. These solutions represent ground states with fixed {J, Q R , N α } ♭ . Table 2: The corresponding ground energy E 0 = 4λ 0 (λ 0 + 2).
The conformal dimension ∆ of the dual operator is related to the ground energy E 0 by the formula [1] [11], Using Table 2, we obtain the equality ∆ = 3 2 Q R . In this way, we have reproduced the BPS condition of the dual operator from the gravity side (see also [7]). This is consistent with the AdS/CFT correspondence.
Next, let us consider the first excited state with λ = λ 0 + 1 and {J, Q R , N α } in Table   1. We find that it exists and the corresponding Heun function h 1 (x) is given by the polynomial solution of degree 1; The existence of the higher excited states with λ = λ 0 + n (n = 2, 3, · · · ) is indicated by numerical simulation. For them, Heun functions are not polynomials, although the parameter α is a negative integer −n. The difference ∆ n − 3 2 Q R is equal to 2n, where ∆ n = −2 + √ 4 + E n and E n = 4(λ 0 + n)(λ 0 + n + 2).
solutions reduce to those given above if we set N = 1. Even if N > 1, one can obtain the similar results to those found in the case of N = 1. Especially, when we set n = 0, the states A, B and C represent those dual to N mesonic chiral operators corresponding to S, L + and L − , respectively. The quantum charges for N meson operators are obtained by multiplying charges in Table 1 and 2 by N, Finally, taking N large, we find that the Heun functions take the form h n (x) ≃ npolynomial + O(1/N), and in the limit N → ∞, they tend to the following polynomials; where h 1 (x) is defined in (30). These polynomials are derived by using 1/N-expansion of three-term recursions (27) ⋄ . The near BPS states with n ≪ N correspond to the near BPS geodesics in [8].
In this letter, we studied the spectrum of the scalar Laplacian on Y p,q corresponding to mesonic chiral operators of the dual superconformal field theories. The states dual to the baryonic chiral operators will be extracted in the similar manner. In [13], inhomogeneous toric Sasaki-Einstein manifolds in arbitrary odd-dimensions have been constructed. It is also interesting to examine the spectrum of the scalar Laplacian on seven-dimensional toric Sasaki-Einstein manifolds and the 3-dimensional dual gauge theories. For a large class of Sasaki-Einstein spaces, their volumes are calculated in [14]. It is interesting to derive the volume of the non-toric 3-Sasakian manifolds constructed in [15]. We hope to report this point in the future. After [16], many compact Sasaki-Einstein manifolds [17] have been derived from AdS-Kerr black holes. It is important to clarify the spectrum on these manifolds. These are left for future investigations.
We are grateful to Yoshitake Hashimoto and Sean Hartnoll for discussions and useful comments. This work is supported by the 21 COE program "Constitution of wide-angle mathematical basis focused on knots" and in part by the Grant-in Aid for scientific Research (No. 17540262 and No. 17540091) from Japan Ministry of Education.