Character of the highest weight module of BMS algebra realized on codimensional-two boundary
Bin Chen, Song He, Pujian Mao, Xin-Cheng Mao
TL;DR
The paper develops the highest-weight representations of BMS$_{3}$ and BMS$_{4}$ on codimension-two boundaries, showing that supertranslations act as conformal weight shifts on $S^{1}$ and $S^{2}$, respectively. It constructs the corresponding modules, analyzes their inner products, and derives explicit character formulas, illuminating the relationship between boundary symmetries and bulk gravity. Notably, the BMS$_3$ vacuum character with suitable central charges reproduces the 1-loop partition function of Minkowski 3D gravity up to a phase, providing evidence for flat holography in three dimensions; the BMS$_4$ case yields a vacuum character whose structure hints at a boundary CFT dual encoding the 4D gravitational dynamics. The work highlights non-unitarity in these HWR constructions due to zero-norm states and outlines connections to Carrollian limits and flat holography concepts.
Abstract
In this paper, we investigate the highest weight representation (HWR) of the three and four-dimensional Bondi-Metzner-Sachs (BMS) algebra realized on the codimension-two boundary of asymptotic flat spacetime (AFS). In such a realization, the action of supertranslation shifts the conformal weight of the highest-weight states. As a result, there is no extra quantum number relating to the supertranslation. We construct the highest-weight BMS modules and compute their characters. We show that the BMS$_3$ highest-weight vacuum character with special value of central charges coincides with the 1-loop partition function of three-dimensional asymptotic flat gravity, up to an overall phase factor ``$i$''. We expect the vacuum character of BMS$_4$ may shed light on the flat holography in four dimensions.