Topology change and the Gauss Bonnet term in causal set theory: the sign of the cosmological term and the axion decay constant

Abstract

Causal set theory permits changes of spatial topology without the metric degeneracy that forbids them in continuum Lorentzian gravity. In four dimensions such changes enter the gravitational action only through the Gauss--Bonnet term, which is the Euler density. We show that the Euler density of a discrete space is exact, local and combinatorial: it sums to the Euler characteristic at any resolution and is supported on topological defects. The Gauss--Bonnet term of a causal set is therefore fixed rather than postulated. Because it is linear in the number of elements, which is the four-volume, it acts as a cosmological constant. Its coefficient follows without forming any functional derivative, so the known obstruction in the continuum does not arise. The everpresent cosmological constant of causal set theory assumes that the mean topological charge vanishes. We show that it does not. The species of opposite sign solves the field equations only if a positive cosmological constant is already present, and, being simply connected, it cannot carry the axion charge that supports the other species. Only one sign survives, and the sign of the cosmological term is derived rather than assumed. We recover the surviving transition from the causal order alone in every simulated sprinkling. The action of the corresponding axion wormhole then fixes the magnitude. The observed value requires an axion decay constant near the grand unification scale, and this result is insensitive to the one-loop prefactor and to the Gauss--Bonnet coupling.