The massive BMS character in 3D quantum gravity

We derive the one-loop partition function for three-dimensional quantum gravity in a finite-radius thermal twisted flat space with a conical defect, reproducing the massive BMS$_3$ character. We perform the computation in both discrete and continuum geometry formulations, showing consistency between them. In the discrete case, we integrate out bulk degrees of freedom in a Regge gravity framework, while in the continuum, we construct a dual non-local boundary field theory encoding geodesic length fluctuations. Our study shows that the additional modes of the massive character, compared to the vacuum case, originate from the explicit breaking of radial diffeomorphism symmetry by the defect. This provides a concrete geometric mechanism in Regge gravity, tracing the appearance of massive BMS$_3$ particles to diffeomorphism breaking by conical defects, and highlights the broader relevance of discrete geometry approaches to quantum gravity with matter.