The gravity side of the $p$-adic AdS${}_2$/CFT${}_1$

Abstract

By means of a higher-dimensional embedding approach, together with the study of distance, measure, and curvature, we construct a gravitational theory on the two-dimensional space over $p$-adic numbers. Before constructing this gravitational theory, we provide supplementary calculations based on existing literature and justify the validity of the gravitational theory by comparison with the real-number case, for instance, the curvatures of de Sitter(dS), anti-de Sitter(AdS), and Euclidean space should be positive, negative, and zero, respectively. Finally, as an example on the gravity side of the $p$-adic AdS${}_2$/CFT${}_1$, we present the action for a scalar field coupled to gravity.

10 pages, 7 figures, 2 tables. Dimensional analysis has been rigorously treated, the one-dimensional examples have been removed, a more general definition of curvature has been introduced, the coupling between gravity and a scalar field has been incorporated, two conjectural statements have been deleted, minor errors and typos have been corrected