The shortened KK spectrum of IIB supergravity on Y p,q

Arash Arabi Ardehali 1 James T. Liu 1 Phillip Szepietowski 0 Open Access c The Authors. Article funded by SCOAP 0 Department of Physics, University of Virginia , Box 400714, Charlottesville, VA 22904, U.S.A 1 Michigan Center for Theoretical Physics, Randall Laboratory of Physics, University of Michigan , Ann Arbor, MI 48109-1040, U.S.A We examine the shortened KK spectrum of IIB supergravity compactified on Y p,q and conjecture that the spectrum we have obtained is complete. The (untwisted) shortened spectrum on S5/Z2p and on T 1,1/Zp are obtained as special cases when p = q and q = 0, respectively. Knowledge of the shortened spectrum allows us to compute the superconformal index of these theories and to find agreement with earlier calculations from the dual field theories. We also employ the shortened spectrum to perform a 1/N 2 test of AdS/CFT by holographically reproducing the difference of the central charges, ca = p/8, of the dual CFTs. Contents 1 Introduction 2 The KK spectrum of IIB supergravity on SE5 2.1 Multiplet shortening 3 The shortened spectrum of IIB supergravity on Y p,q 3.1 The shortening condition e0 = 32 |r| 3.2 The shortening condition e0 = 32 |r| + 2 4 AdS/CFT state-operator correspondence in the mesonic chiral ring 5 Extension to p = q and q = 0 6 The superconformal index for Y p,q 7 Holographic c a for Y p,q and a 1/N 2 test of AdS/CFT 8 Discussion 1 Introduction About a decade ago a new avenue was opened in the exploration of AdS/CFT with reduced supersymmetry by the discovery of an infinite family of Sasaki-Einstein five-manifolds Y p,q [1, 2] and the construction of their dual four-dimensional quiver gauge theories [3, 4]. Various checks of the duality had been performed successfully, the most notable of which being perhaps the matching of the large-N conformal anomalies and the spectrum of baryonic states [4, 5]. However, although many other properties of the family were known from the field theory side, such as the spectrum of the mesonic states [6] and the superconformal index [7], the gravitational computations were obstructed by the difficulty in obtaining the KK spectrum of IIB supergravity on the Y p,q manifolds. In particular, the scalar Laplacian on Y p,q leads to a Heun equation [810] whose exact spectrum is not known. Despite the difficulty in finding the full spectrum of the Heun equation in question, some harmonics were found and identified with their dual mesonic states in [9]. Here we extend the available results by finding the shortened KK spectrum of IIB supergravity compactified on Y p,q. We conjecture that this spectrum is complete and perform two checks involving AdS/CFT. This shortened spectrum is the main result of this work and can be found in tables 3 and 4. The shortened KK spectrum enables us to see the spectrum of all the protected singletrace operators from the gravity side. It also allows us to compute the superconformal index from supergravity and find matching with the earlier field theoretical computation of [7]. Another use of the shortened spectrum is in holographically reproducing the difference of the gauge theory central charges c a [11]. This is a rather non-trivial test of AdS/CFT beyond large-N . Our success in reproducing the field theory value, c a = p/8, from the shortened spectrum on the gravity side helps to address some of the issues raised in [11]. This paper is organized as follows. Section 2 reviews the KK spectrum of IIB supergravity compactified on a generic Sasaki-Einstein 5-manifold and the possible multiplet shortening patterns. In section 3 we present the shortened KK spectrum of IIB supergravity on Y p,q. Section 4 illustrates how the mesonic chiral ring of the dual field theories is mapped to the supergravity states. In section 5 we demonstrate that the untwisted shortened spectrum on S5/Z2p and T 1,1/Zp may be obtained from the shortened spectrum presented here upon setting p = q or q = 0, respectively. The superconformal index is computed in section 6, and the holographic c a is computed in section 7. The closing section includes comments on how the results of the present paper shed light on the issues concerning the 1/N 2 corrections to the holographic Weyl anomaly raised in [11]. The KK spectrum of IIB supergravity on SE5 A generic compactification of IIB supergravity on AdS5 SE5 yields N = 2 gauged supergravity coupled to a KK tower that can be arranged into N = 2 representations of the SU(2, 2|1) supergroup. The compactification on S5 preserves N = 8 supersymmetry, and the KK spectrum was obtained in [12, 13]. The result is particularly simple when given in terms of shortened representations of SU(2, 2|4); at level p (p 2), the states transform under the representation D(p, 0, 0; 0, p, 0), where we have used the notation D(E0, s1, s2; l1, l2, l3) where (l1, l2, l3) are the Dynkin labels of the SU(4)R representation. Subsequently the KK spectroscopy for T 1,1 was investigated in [14, 15]. The resulting spectrum was given in terms of nine generic KK multiplets Graviton, Gravitinos I through IV, and Vectors I through IV along with a Betti vector and Betti hypermultiplet. It was then shown in [16] that this decomposition in terms of nine generic multiplets persists for general N = 2 compactifications. The full spectrum consists of these generic multiplets along with the possible addition of special KK multiplets and Betti multiplets. In fact, the analysis of [1416] demonstrates that the generic KK tower can be obtained solely from knowledge of the eigenvalues of the scalar Laplacian on SE5. Essentially, the vector and tensor harmonics needed in the decomposition of IIB fields on SE5 may be related to a combination of scalar harmonics and invariant tensors related to the structure of the manifold. Hence information from the scalar harmonics is sufficient. It is convenient to define the eigenvalues of the scalar Laplacian on SE5 according to Y = e0(e0 + 4)Y, Supermultiplet D e0 + 32 , 12 , 0; r + 1 + D e0 + 23 , 0, 21 ; r 1 D e0 + 92 , 12 , 0; r 1 + D e0 + 29 , 0, 21 ; r + 1 e0 0 e0 0 e0 0 e0 0 where we take e0 0. Note that the eigenvalues e0 will depend on the R-charge as well as other global quantum numbers on SE5. Moreover, it was shown in [16] that e0 satisfies the bound |r| 2 min(j, ). As an example, for T 1,1, we have e0(e0 + 4) = 6[j(j + 1) + ( + 1) r2/8], where (j, l, r) labels the representation under the isometry group SU(2)j SU(2) U(1)r of T 1,1, and the R-charge satisfies the bound The e0 23 r bound is saturated when j = = |r|/2. In general, the isometry group may be different. However, the conserved U(1)r will always be present, as demanded by N = 2 supersymmetry. Thus the KK spectrum can be arranged into representations of SU(2, 2|1) based on the e0 and r eigenvalues of the scalar Laplacian. The generic KK spectrum is given in table 1. In addition to the generic spectrum, there may be KK towers of special multiplets as (...truncated)