On the continuum limit of Benincasa–Dowker–Glaser causal set action

Abstract

We study the continuum limit of the Benincasa–Dowker–Glaser causal set action on a causally convex compact region. In particular, we compute the action of a causal set randomly sprinkled on a small causal diamond in the presence of arbitrary curvature in various spacetime dimensions. In the continuum limit, we show that the action admits a finite limit. More importantly, the limit is composed by an Einstein–Hilbert bulk term as predicted by the Benincasa–Dowker–Glaser action, and a boundary term exactly proportional to the codimension-two joint volume. Our calculation provides strong evidence in support of the conjecture that the Benincasa–Dowker–Glaser action naturally includes codimension-two boundary terms when evaluated on causally convex regions.