Towards non-AdS Holography via the Long String Phenomenon

Abstract

The microscopic description of AdS space obeys the holographic principle in the sense that the number of microscopic degrees of freedom is given by the area of the holographic boundary. We assume the same applies to the microscopic holographic theories for non-AdS spacetimes, specifically for Minkowski, de Sitter, and AdS below its curvature radius. By taking general lessons from AdS/CFT we derive the cut-off energy of the holographic theories for these non-AdS geometries. Contrary to AdS/CFT, the excitation energy decreases towards the IR in the bulk, which is related to the negative specific heat of black holes. We construct a conformal mapping between the non-AdS geometries and $AdS_3\!\times\! S^{q}$ spacetimes, and relate the microscopic properties of the holographic theories for non-AdS spaces to those of symmetric product CFTs. We find that the mechanism responsible for the inversion of the energy-distance relation corresponds to the long string phenomenon. This same mechanism naturally explains the negative specific heat for non-AdS black holes and the value of the vacuum energy in (A)dS spacetimes.

Figures (5)

• Figure 1: A large causal diamond in AdS consisting of the domain of dependence of the ball bounded by the holographic screen $\mathcal{S}$. The ball and the screen lie in the $t=0$ time slice, and are centered around the origin. The distance of $\mathcal{S}$ to the AdS boundary can be characterized by the time $t$ at which the outward future lightsheet reaches the AdS boundary.
• Figure 2: A small causal diamond in AdS consisting of the domain of dependence of the ball bounded by the holographic screen $\mathcal{S}$. The ball lies in the $t=0$ time slice, and is centered around the origin. The location of $\mathcal{S}$ with respect to the origin can be parametrized by the time $t$ at which the inward future lightsheet arrives at the origin.
• Figure 3: The family of rescalings of the geometry $\tilde{g}$ contains the geometry $g$ as a 'diagonal' described by the equation $\Omega=\Omega_{\mathcal{S}}$. A horizontal line corresponds to a constant rescaling of $\tilde{g}$: $\Omega_{\mathcal{S}_0} \tilde{g}$. On the holographic screen ${\mathcal{S}}_0$ the two geometries coincide: $g_{|\mathcal{S}_0} = \left(\Omega^2_{\mathcal{S}_0} \tilde{g}\right)_{|\mathcal{S}_0}$. By repeating this process for all values of $\Omega$ a foliation of the geometry $g$ is constructed and represented by the diagonal.
• Figure 4: $\!\!\!$The first figure depicts, from left to right, the gluing of the $\left[0,r_0\right]$ region of Weyl rescaled $AdS_3\times S^{d-2}$ to the $[0,R_0]$ region of $AdS_d\!\times\! S^1$. The second figure illustrates, from left to right, the gluing of the $[0,r_0]$ region of massless (Weyl rescaled) $BTZ \times S^{d-2}$ to the $[0,R_0]$ region of $Mink_d \times S^1$. Finally, the third figure shows the gluing of the $\left[0,r_0\right]$ region of (Weyl rescaled) $BTZ\times S^{d-2}$ to the $[0,R_0]$ region of $dS_d\times S^1$.
• Figure 5: The degrees of freedom on the holographic screen at radius $L$ consist of single 'long strings'. The degrees of freedom on holographic screens at radius $R_0= k\ell$ with $k<N$ consist of 'fractional strings'. The total number of long (or fractional) strings is proportional to the area of the holographic screen at radius $L$ (or $R_0$).