The spectrum of the bosonic ambitwistor string revisited
We revisit the calculation of the spectrum of the bosonic ambitwistor string, understood as the BRST cohomology or, equivalently, as the semi-infinite cohomology of the $\mathrm{BMS}_3$ Lie algebra relative to the centre with values in a particular module. We work in momentum space, which allows us to work algebraically and interpret the BRST cohomology as inducing representations of the Poincaré group. In agreement with the existing literature, we find that all the cohomology resides in the massless sector, but a careful representation-theoretic analysis of the spectrum reveals, in addition to the usual massless sector of the closed bosonic string (dilaton, metric and Kalb--Ramond field), also a massless vector. We devote a large part of the paper to describing the cohomology at a massless momentum $p$ as a module over the stabiliser $H$ of $p$ in the Lorentz group, a task which is made difficult due to $H$ not acting reducibly when $p\neq 0$. This allows us to conclude that the spectrum is not unitary, forbidding the interpretation of the extra massless vector as a Maxwell field.