A BMS-invariant free scalar model

TL;DR

The paper provides the first explicit free scalar model with BMS$_3$ symmetry, realized as the ultra-relativistic limit of a CFT$_2$, and shows $c_L=2$, $c_M=0$. It uncovers novel representation theory for BMSFTs at $c_M=0$, including primary multiplets and an enlarged staggered module built from an extra quasi-primary $K$, with explicit operator content and correlation functions. The authors compute the torus partition function and prove modular invariance, reinforcing the consistency of BMS invariant holography ideas in flat space. They also connect the intrinsic BMSFT construction to the UR limit of a relativistic CFT, clarifying plane/cylinder subtleties and providing a detailed bridge between CFT$_2$ and BMS$_3$ quantum field theories. The results offer a concrete, nontrivial testbed for flatspace holography and illuminate new logarithmic-like structures (staggered modules) in BMS representation theory.

Abstract

The BMS (Bondi-van der Burg-Metzner-Sachs) symmetry arises as the asymptotic symmetry of flat spacetime at null infinity. In particular, the BMS algebra for three dimensional flat spacetime (BMS$_3$) is generated by the super-rotation generators which form a Virasoro sub-algebra with central charge $c_L$, together with mutually-commuting super-translation generators. The super-rotation and super-translation generators have non-trivial commutation relations with another central charge $c_M$. In this paper, we study a free scalar theory in two dimensions exhibiting BMS$_3$ symmetry, which can also be understood as the ultra-relativistic limit of a free scalar CFT$_2$. Upon canonical quantization on the highest weight vacuum, the central charges are found to be $c_L=2$ and $c_M=0$. Because of the vanishing central charge $c_M=0$, the theory features novel properties: there exist primary states which form a multiplet, and the Hilbert space can be organized by an enlarged version of BMS modules dubbed the staggered modules. We further calculate correlation functions and the torus partition function, the later of which is also shown explicitly to be modular invariant.

Figures (5)

• Figure 1: Staggered module
• Figure 2: The vacuum module: the vacuum state with $\Delta=0$; states with $\Delta=2$ form a quasi-primary multiplet, from left to right: $|M\rangle,\, |T\rangle, \, |K\rangle$; states with $\Delta=3$ from left to right: $L_{-1}|M\rangle,\, L_{-1}|T\rangle,\, M_{-1}|K\rangle, \,L_{-1} |K\rangle$. The four states at level 3 split into two multiplets: a singlet $L_{-1}|T\rangle-3M_{-1}|K\rangle$, and a triplet consisting of $L_{-1}|M\rangle$, $L_{-1}|T\rangle+M_{-1}|K\rangle$ and $L_{-1} |K\rangle$.
• Figure 3: $\boldsymbol{O}$ module, part I: states with $\Delta=1$: $|O_0\rangle,\, |O_1\rangle$; states with $\Delta=2$: $L_{-1}|O_0\rangle,\, L_{-1}|O_1\rangle$; states with $\Delta=3$: $L^2_{-1}|O_0\rangle,\, L^2_{-1}|O_1\rangle$.
• Figure 4: $\boldsymbol{O}$ module, part II : states with $\Delta=1$: $|O_0\rangle,\, |O_1\rangle$; states with $\Delta=3$: $M_{-2}|O_0\rangle,\,L_{-2}|O_0\rangle,\, L_{-2}|O_1\rangle,\, |KO_1\rangle$.
• Figure 5: States up to $\Delta=3$