On KKLT/CFT and LVS/CFT dualities

Published for SISSA by Springer Received: January 15, 2015 Revised: June 14, 2015 Accepted: June 22, 2015 Published: July 8, 2015 On KKLT/CFT and LVS/CFT dualities a UCB 390, Physics Department, University of Colorado, Boulder CO 80309, U.S.A. b ICTP, Strada Costiera 11, 34151 Trieste, Italy c DAMTP, CMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K. d Dipartimento di Fisica dell’Università di Trieste and INFN — Sezione di Trieste, Strada Costiera 11, 34151 Trieste, Italy E-mail: , , , Abstract: We present a general discussion of the properties of three dimensional CFT duals to the AdS string theory vacua coming from type IIB Calabi-Yau flux compactifications. Both KKLT and Large Volume Scenario (LVS) minima are considered. In both cases we identify the large ‘central charge’, find a separation of scales between the radius of AdS and the size of the extra dimensions and show that the dual CFT has only a limited number of operators with small conformal dimension. Differences between the two sets of duals are identified. Besides a different amount of supersymmetry (N = 1 for KKLT and N = 0 for LVS) we find that the LVS CFT dual has only one scalar operator with O(1) conformal dimension, corresponding to the volume modulus, whereas in KKLT the whole set of h1,1 Kähler moduli have this property. Also, the maximal number of degrees of freedom is estimated to be larger in LVS than in KKLT duals. In both cases we explicitly compute the coefficient of the logarithmic contribution to the one-loop vacuum energy which should be invariant under duality and therefore provides a non-trivial prediction for the dual CFT. This coefficient takes a particularly simple form in the KKLT case. Keywords: Flux compactifications, AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 1412.6999 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP07(2015)036 JHEP07(2015)036 Senarath de Alwis,a Rajesh Kumar Gupta,b Fernando Quevedob,c and Roberto Valandrob,d Contents 1 2 AdS backgrounds from flux compactifications 2.1 Basics of AdS5 × S5 /CFT4 duality 2.2 Calabi-Yau flux compactifications 4 4 5 3 Properties of the CFT3 duals 3.1 Central charge and number of degrees of freedom 3.2 Conformal dimensions 3.3 Wrapped branes and their dual 9 10 12 14 4 Effective potential and quantum logarithmic effects 4.1 The limit |W0 | → 0 4.2 Effective potential 4.3 Effective potential Γ(1) about AdS background 15 15 16 18 5 Coefficient of ln |W0 |2 in type IIB flux compactifications 5.1 KKLT vacua 5.2 LVS vacua 19 19 22 6 Discussion 28 A N = 1 supergravity Lagrangian 29 B One loop computation B.1 Scalar field B.2 Vector field B.3 Graviton B.4 Dirac fermion B.5 Gravitino 30 30 30 31 31 33 1 Introduction Flux compactifications of type IIB string theory have given rise to two major developments within string theory: AdS/CFT duality [1, 2] (see [3, 4] for a review) and the string landscape [5–16] of moduli stabilised four dimensional (4D) string vacua. In the simplest cases, these four dimensional minima have a negative cosmological constant and hence are AdS4 vacua. It is then natural to inquire if these Anti de Sitter (AdS) vacua of the string –1– JHEP07(2015)036 1 Introduction • The two scenarios realise the separation of scales that allow the neglect of part of the spectrum in different ways. In KKLT this happens because of the small value of the flux superpotential, while in LVS because of the hierarchically large value of the volume of the compactification manifold. In fact, KKLT relies on the possibility of tuning the flux superpotential Wflux to very small values (of the same order of the non-perturbative superpotential), while LVS is based on a generically order one Wflux . • The KKLT AdS4 vacuum preserves N = 1 supersymmetry, whereas the LVS AdS4 vacuum breaks supersymmetry spontaneously, with the breaking being induced by generic fluxes. 1 AdSd+1 /CFTd duality has also been used in Calabi-Yau flux compactifications in a different context that should not be confused with our target in this article. In those cases, conifold geometries such as the Klebanov-Strassler warped throat are embedded in compact Calabi-Yau manifolds and provide a stringy realisation of the Randall-Sundrum set-up with the tip of the throat providing the IR brane and the compact Calabi-Yau at the beginning of the throat providing the UV Planck brane [24]. In these cases AdSd+1 /CFTd duality is used in the sense that 4D field theories are dual to 5D gravity theories in which locally the five dimensions are the 4D spacetime dimensions plus the direction along the throat, i.e. d = 4. On the other hand, in this paper we are concentrating on three-dimensional field theories dual to four-dimensional gravity theories, i.e. d = 3. –2– JHEP07(2015)036 landscape have Conformal Field Theory (CFT) duals and if so what the properties of these theories are. Identifying CFT duals of the AdS (and dS) vacua of the string landscape would be a way to provide a proper non perturbative description of these vacua and put the string landscape on firmer ground. This is the subject of the present article. For previous discussions of this issue see [17–23].1 By now there are two main scenarios of moduli stabilisation in type IIB string compactifications on Calabi-Yau (CY) manifolds: KKLT [10] and the Large Volume Scenario (LVS) [25, 26]. Contrary to the original AdS5 × S 5 background where the flux was enough to stabilise the geometric modulus of S 5 , in KKLT and LVS scenarios the fluxes fix only part of the geometric moduli (this can be read from the ten dimensional equation of motions [8, 9], like for AdS5 × S 5 ) leaving some flat directions. A key ingredient to stabilise the remaining geometric moduli (in a AdS4 vacuum) is the presence of non-perturbative effects in the 4D effective field theory (EFT) obtained after compactification. This makes a full ten dimensional (10D) analysis of these vacua very difficult and we can only rely on the EFT results. Black-brane solutions that were at the origin of the AdS5 × S 5 /CFT4 duality are not available and therefore there is less control on the potential duality in the KKLT and LVS cases. This explains the relative shortage of efforts to study the CFT duals of these vacua during the past ten years. Another difference with AdS5 × S 5 is that in both KKLT and LVS scenarios there is a hierarchy between the size of the internal dimensions and the AdS radius. This is in contrast to the situation in Freund-Rubin compactifications where one needs to establish on a case by case that there is a consistent truncation to the massless modes of the KK tower (see for example the discussion in section 2.2.5 of [3, 4]) . Even though both KKLT and LVS are based on Calabi-Yau flux compactifications of type IIB string theory down to 4D, they have important differences that should be reflected in the dual CFTs. The fact that the LVS vacuu (...truncated)