A holographic connection between strings and causal diamonds

A bstract In this paper we explore ideas of holography and strings living in the d + 1 dimensional Anti-de Sitter space AdS d +1 in a unified framework borrowed from twistor theory. In our treatise of correspondences between geometric structures of the bulk AdS d +1 , its boundary and the moduli space of boundary causal diamonds aka the kinematic space 𝕂, we adopt a perspective offered by projective geometry. From this viewpoint certain lines in the d + 1 dimensional real projective space, defined by two light-like vectors in ℝ d ,2 play an important role. In these projective geometric elaborations objects like Ryu-Takayanagi surfaces, spacelike geodesics with horospheres providing regularizators for them and the metric on 𝕂 all find a natural place. Then we establish a correspondence between classical strings in AdS d +1 and causal diamonds of its asymptotic boundary. At each point on the worldsheet, the tangent vectors ∂ ± X are projected onto boundary coordinates that identify the past and future tips of a causal diamond. Under this projection, the string equations of motion translate into a dynamics of boundary causal diamonds. A procedure for lifting up a causal diamond to get a proper string world sheet is also developed. In this context we identify an emerging SO(1) × SO(1, d − 1) gauge structure incorporated into a Grassmannian σ -model targeted in 𝕂. The d = 2 case is worked out in detail. Surprisingly in this case AdS 3 with its strings seems to be a natural object which is living inside projective twistor space. On the other hand 𝕂 (comprising two copies of two dimensional de Sitter spaces) is a one which is living inside the Klein quadric, as a real section of a complexified space time.