A holographic connection between strings and causal diamonds

Abstract

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 ${\mathbb K}$, 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 ${\mathbb R}^{d,2}$ 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 ${\mathbb K}$ 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 $\partial_\pm X$ 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 $SO(1,1)\times SO(1,d-1)$ gauge structure incorporated into a Grassmannian $\sigma$-model targeted in ${\mathbb K}$. 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 ${\mathbb K}$ (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.