# The BMS group In addition to the Poincare group, the BMS group consists of supertranslation. More precisely, the BMS group is the semi-product of Lorentz group with supertranslations (STs). There are several ways in which the BMS group has been generalised: - *generalised* BMS: Diff$(S^2)\ltimes$ST - *extended* BMS: (Witt$\oplus$Witt)$\ltimes$ST - Weyl BMS: Diff$(S^2)\ltimes$Weyl$\ltimes$ST ## Refs - original BMS: - [[1962#Bondi, van der Burg, Metzner]] - [[1962#Sachs]] - algebra etc in [[2020#Banerjee, Ghosh, Gonzo]] - importance in flat holography: [[2019#Fotopoulos, Stieberger, Taylor, Zhu]] - BMS for all five infinities: [[2023#Compere, Gralla, Wei]] - non-linear Weyl BMS: [[2023#Flanagan, Nichols]] ## Unextended BMS - semi-product of the $SL(2,\mathbb{C})$ subgroup of global conformal transformations of the [[0022 Celestial sphere|celestial sphere]] (isomorphic to the Lorentz group) and the Abelian subgroup of supertranslations ## Transformations (infinitesimal) - superrotation - holomorphic, ==$L_n$== - $z \rightarrow z+\epsilon z^{n+1}, \quad \bar{z} \rightarrow \bar{z}, \quad u \rightarrow u+\frac{1}{2} \epsilon z^{n+1}, \quad n \in \mathbb{Z}$ - antiholomorphic, ==$\bar{L}_n$== - $z \rightarrow z, \quad \bar{z} \rightarrow \bar{z}+\bar{\epsilon} \bar{z}^{n+1}, \quad u \rightarrow u+\frac{1}{2} \bar{\epsilon} \bar{z}^{n+1}, \quad n \in \mathbb{Z}$ - supertranslation, ==$P_{a,b}$== - $z \rightarrow z, \quad \bar{z} \rightarrow \bar{z}, \quad u \rightarrow u+\epsilon z^{a+1} \bar{z}^{b+1}, \quad a, b \in \mathbb{Z}$ - global part $a,b=-1,0$ ## Algebra - $\left[L_{m}, L_{n}\right]=(m-n) L_{m+n}$ - = centre-less Virasoro - $\left[\bar{L}_{m}, \bar{L}_{n}\right]=(m-n) \bar{L}_{m+n}$ - = another copy of centre-less Virasoro - $\left[P_{a, b}, P_{a^{\prime}, b^{\prime}}\right]=0$ - $\left[L_{n}, P_{a, b}\right]=\left(\frac{n-1}{2}-a\right) P_{a+n, b}$ - $\left[\bar{L}_{n}, P_{a, b}\right]=\left(\frac{n-1}{2}-b\right) P_{a, b+n}$ ## Non-factorisation ([[2020#Banerjee, Ghosh, Gonzo]]) - BMS group is not a direct product of holomorphic and anti-holo. transformation - <- because supertranslations have weights of both - => different from usual CFT ## Transformation of fields - holomorphic superrotation - $\delta \phi_{h, \bar{h}}(u, z, \bar{z})=\epsilon\left[z^{n+1} \partial+(n+1)\left(h+\frac{1}{2} u \partial_{u}\right) z^{n}\right] \phi_{h, \bar{h}}(u, z, \bar{z})$ - antiholom. superrot. - $\delta \phi_{h, \bar{h}}(u, z, \bar{z})=\epsilon\left[\bar{z}^{n+1} \bar{\partial}+(n+1)\left(\bar{h}+\frac{1}{2} u \partial_{u}\right) \bar{z}^{n}\right] \phi_{h, \bar{h}}(u, z, \bar{z})$ - supertranslation - $\delta \phi_{h, \bar{h}}(u, z, \bar{z})=\epsilon z^{a+1} \bar{z}^{b+1} \partial_{u} \phi_{h, \bar{h}}(u, z, \bar{z})$ ## Extensions - [[Prabhu202112]][](https://arxiv.org/pdf/2112.07186.pdf) - understanding from [[0103 Two-point functions|Green's function]]: [[BatlleCampelloGomis2022]][](https://arxiv.org/pdf/2207.12299.pdf) - on effective action of superrotation modes [[NguyenSalzer2020]] [arXiv](https://arxiv.org/pdf/2008.03321.pdf) - quantum [[2020#Donnay, Giribet, Rosso]] ## History - see [[0063 Symmetry of CCFT]] ![[0064_sum.png]] ## Novel descriptions - Wald-Zoupas prescription [[GrantPrabhuShehzad2021]][](https://arxiv.org/pdf/2105.05919.pdf) - [[0330 Twistor theory|twistorial]] description: [[Prabhu202111]][](https://arxiv.org/pdf/2111.00478.pdf)