Asymptotic Symmetries and Celestial CFT

TL;DR

This work unifies the celestial CFT framework with the asymptotic symmetry program by showing that conformal primary wavefunctions built from the principal continuous series ( elta=1+ilambda) form a complete finite-energy basis, while conformally soft modes with analytically continued elta are contour integrals over this basis and do not enlarge it. It elucidates how spin-1 and spin-2 conformal primaries and their shadows generate large U(1) and BMS symmetries, including shadow superrotations, thereby tying the 2D celestial stress tensor and its shadow to soft charges. The analysis introduces a rigorous treatment of analytically continued quantum modes via generalized delta distributions, contour deformations, and renormalized symplectic structures, enabling finite soft charges for Diff(S^2) and Virasoro-related symmetries. The results place Virasoro and Diff(S^2) on equal footing within a shadow-extended asymptotic-symmetry group for Einstein gravity at null infinity, with direct implications for celestial CFT operator identities and memory effects.

Abstract

We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension $Δ$. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series $Δ\in 1+i\mathbb{R}$ and form a complete basis. We show that conformal primaries with analytically continued conformal dimension can be understood as certain contour integrals on the principal series. This clarifies how conformally soft Goldstone modes fit in but do not augment this basis. Conformally soft gravitons of dimension two and zero which are related by a shadow transform are shown to generate superrotations and non-meromorphic diffeomorphisms of the celestial sphere which we refer to as shadow superrotations. This dovetails the Virasoro and Diff(S$^2$) asymptotic symmetry proposals and puts on equal footing the discussion of their associated soft charges, which correspond to the stress tensor and its shadow in the two-dimensional celestial CFT.

Figures (3)

• Figure 1: Superrotations imply the subleading soft graviton theorem so long as the asymptotic Virasoro symmetry group is enhanced to include shadow superrotations.
• Figure 2: Conformal primaries with general conformal dimension $\Delta$ can be expressed as integrals along the principal series contour $z=1+i\lambda$ with $\lambda \in \mathbb{R}$. Depending on whether ${\rm Re}(\Delta) \gtrless 1$ the contour is deformed to the right (\ref{['fig:right']}) or left (\ref{['fig:left']}). For illustration we have drawn the pole positions corresponding to $\Delta=2$ and $\Delta=0$ in (\ref{['fig:right']}) and (\ref{['fig:left']}), respectively. Using \ref{['SHADelta']} and \ref{['SHhDelta']}, we see the same contours are relevant for the shadow modes with $\tilde{\Delta}=2$ and $\tilde{\Delta}=0$, respectively.
• Figure 3: The shadow Dolan:2011dv and renormalization Compere:2018ylh procedures commute. The renormalized quantities $\{T_{ww},\widetilde{T}_{{\bar{w}}{\bar{w}}}\}$ correspond to soft charges appearing in Kapec:2014opa and Campiglia:2014yka, respectively.