Celestial CFT, Virasoro Algebra, and BMS
Celestial Amplitudes via the Mellin Transform
Standard scattering amplitudes are functions of four-momenta. In the celestial basis we trade energy$\omega$ for a conformal dimension $h$ via a Mellin transform. For massless particles the on-shell momentum is parametrised by a point $(z,\bar{z})$ on the celestial sphere:
$$p^\mu(\omega,z,\bar{z}) = \frac{\omega}{2}\left(1+z\bar{z},\;z+\bar{z},\;-i(z-\bar{z}),\;1-z\bar{z}\right)$$
The celestial operator $\mathcal{O}_h^+(z,\bar{z})$ is the Mellin transform of the positive-helicity graviton creation operator with respect to $\omega$, yielding a conformal primary of weight $(h,\bar{h})$ on $\mathbb{CP}^1$.
Graviton OPE on the Celestial Sphere
The operator product expansion of two positive-helicity celestial gravitons takes the form:
$$\mathcal{O}^+_{h_1}(z_1)\,\mathcal{O}^+_{h_2}(z_2) \;\sim\; \frac{\kappa}{2}\,\frac{B(h_1-1,\,h_2-1)}{z_{12}}\;\mathcal{O}^+_{h_1+h_2-1}(z_2)$$
Here $\kappa = \sqrt{32\pi G}$ is the gravitational coupling,$B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$ the Euler beta function, and$z_{12} = z_1 - z_2$. The $1/z_{12}$ singularity is the hallmark of a holomorphic OPE in a two-dimensional CFT, confirming that celestial graviton scattering organises itself into the structure of a chiral algebra.
Virasoro Generators from Superrotations
The extended BMS group promotes global Lorentz transformations on $S^2$ to local superrotations generated by arbitrary meromorphic vector fields $Y^z(z)$. Expanding in Laurent modes $L_n = -z^{n+1}\partial_z$, the resulting charge algebra is the Virasoro algebra:
$$[L_m,\,L_n] = (m-n)\,L_{m+n} + \frac{c}{12}(m^3 - m)\,\delta_{m+n,0}$$
Together with its anti-holomorphic copy $\bar{L}_n$, this endows the celestial sphere with the full symmetry apparatus of a two-dimensional CFT, including a stress tensor$T(z) = \sum_n L_n\,z^{-n-2}$ whose Ward identities govern all soft and memory effects.
Spin Memory as a CFT Ward Identity
The spin memory Ward identity — relating the permanent angular-momentum shift at$\mathscr{I}^+$ to the superrotation charge flux — is precisely the standard CFT Ward identity for the stress tensor $T(z)$:
$$\oint \frac{dz}{2\pi i}\;T(z)\;\mathcal{O}(w) = h\,\frac{\mathcal{O}(w)}{(z-w)^2} + \frac{\partial\mathcal{O}(w)}{z-w}$$
The residue of the double pole encodes the conformal weight (angular momentum), while the simple pole generates translations. Spin memory is the integrated version of this identity over the full retarded-time interval of the gravitational wave burst.
Stress Tensor from Perelman via Polyakov
The worldsheet stress tensor of the sigma model is
$$T_{zz} = -\frac{1}{\alpha'}\,G_{\mu\nu}\,\partial X^\mu\,\partial X^\nu$$
As $G_{\mu\nu}$ flows under the RG (equivalently under Ricci flow), the Virasoro central charge acquires an effective flow-time dependence:
$$c_{\rm eff}(t) = c_{\rm UV} - 12\pi\int_0^t \beta^i\,G_{ij}\,\beta^j\,dt'$$
The monotone decrease of $c_{\rm eff}(t)$ is the Zamolodchikov c-theorem. Identifying the sigma-model target with spacetime and applying the Polyakov formalism, one finds that Perelman’s $\mathcal{W}$-entropy is the natural $c$-function for the gravitational RG flow — completing the chain from Ricci flow to celestial CFT.