BMS-invariant free scalar model

The BMS (Bondi--van der Burg--Metzner--Sachs) symmetry arises as the asymptotic of flat spacetime at null infinity. In particular, algebra for three-dimensional (${\mathrm{BMS}}_{3}$) is generated by super-rotation generators that form a Virasoro subalgebra with central charge ${c}_{L}$, together mutually commuting super-translation generators. and supertranslation have nontrivial commutation relations another ${c}_{M}$. this paper, we study free scalar theory in two dimensions exhibiting ${\mathrm{BMS}}_{3}$ symmetry, which can also be understood ultrarelativistic limit ${\mathrm{CFT}}_{2}$ flipped representation. Upon canonical quantization on highest weight vacuum, charges are found to ${c}_{L}=2$ ${c}_{M}=0$. Because vanishing ${c}_{M}=0$, features novel properties: there exist primary states multiplet, Hilbert space organized an enlarged version modules dubbed staggered modules. We further calculate correlation functions torus partition function, latter shown explicitly modular invariant. Is it interesting note our model has ${c}_{M}$, feature shared so-called chiral gravity Bagchi et al. [Flat-Space Chiral Gravity, Phys. Rev. Lett. 109, 151301 (2012)].