Supersymmetric AdS vacua and separation of scales

Abstract

The moduli space of the supersymmetric massive IIA AdS 4 × S 2 (B 4 ) vacua, where S 2 (B 4 ) is a two-sphere bundle over a four-dimensional Kähler-Einstein base B 4 , includes three independent parameters which can be thought of as corresponding to the sizes of AdS 4 , B 4 and the S 2 fiber. It might therefore be expected that these vacua do not suffer from the absence of scale separation. We show that the independence of the geometric moduli survives flux quantization. However, we uncover an attractor behavior whereby all sizes flow to equality in some neighborhood of spacetime independently of the initial conditions set by the parameters of the solution. This is further confirmed by the study of the ratio of internal to external scalar curvatures. We also show that the asymptotic Kaluza-Klein spectrum of a ten-dimensional massive scalar is governed by a scale of the order of the AdS 4 radius. Furthermore we point out that the curvature ratio in supersymmetric IIA AdS 4 vacua with rigid SU(3) structure is of order one, indicating the absence of scale separation in this large class of vacua.

On the spin-2 Kaluza-Klein spectrum of AdS 4 × S 2 ℬ 4 $$ {\mathrm{AdS}}_4\times {S}^2\left({\mathrm{\mathcal{B}}}_4\right) $$

“…Moreover it was shown that the asymptotic Kaluza-Klein (KK) spectrum of a ten-dimensional scalar in the background of [1] is governed by a scale of the order of the AdS 4 radius. Nevertheless the results of [2] were inconclusive as to the existence or absence of scale separation in the vacua of [1]. Given the potential phenomenological interest of AdS vacua with scale separation it is important to study this problem further.…”

“…These solutions naively seem to have enough parameters to independently control the sizes of the external AdS 4 and the internal S 2 (B 4 ) space. However [2] observed an 'atractor' behavior: although the warp factors of the solution that control the relative scales of the external and internal spaces can be chosen to have an arbitrarily large ratio at the north pole of the S 2 , that 1 For the purposes of the present paper we may think of the (radius)…”

New examples of flux vacua

“…We will later split these coordinates further into, 9) and restrict w to dependence only on {x 2 , y 2 , z 2 }, but there is no reason do that yet. The 10) and Ricci tensor,…”

“…In the main text, we will use a specific choice for α i stated below. For example, we take the convention that the epsilon tensors (not symbols) satisfy: 01···9 = α 1 −g (10) , 0123 = α 1 −g (4) . (A.1)…”

“…The possibility of classical IIA solutions with scale separation was discussed in [10], but no examples are currently known. The status of various attempts to build type IIA and massive IIA flux vacua is summarized in [11].…”