2025/06/06 by Boldis, Bercel, Lévay, Péter
#FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2506.06428
In this paper we explore ideas of holography and strings living in the d+1 dimensional Anti-de Sitter space AdSd+1 in a unified framework borrowed from twistor theory. In our treatise of correspondences between geometric structures of the bulk AdSd+1, its boundary and the moduli space of boundary causal diamonds aka the kinematic space \mathbb 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 \mathbb Rd,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 \mathbb K all find a natural place. Then we establish a correspondence between classical strings in AdSd+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(1,1)× SO(1,d-1) gauge structure incorporated into a Grassmannian σ-model targeted in \mathbb K. The d=2 case is worked out in detail. Surprisingly in this case AdS3 with its strings seems to be a natural object which is living inside projective twistor space. On the other hand \mathbb 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.