A $d$-Dimensional Stress Tensor for Mink$_{d+2}$ Gravity

TL;DR

This work recasts massless scattering in $(d+2)$-dimensional Minkowski space as celestial correlators on a space-like cut ${\mathcal M}_d$ of the null momentum cone, realizing the Lorentz group as the Euclidean conformal group on the boundary and uncovering universal shadow-transformed operators from soft physics. The leading soft-photon theorem yields a boundary $U(1)$ current $J_a$ via a shadow of the soft operator $S_a$, while the subleading soft-graviton theorem yields a boundary stress tensor $T_{ab}$ through the shadow of $B_{ab}$, with explicit Ward identities mirroring those of a conserved current and stress tensor in Euclidean CFT$_d$. The formalism provides a concrete link between infrared structure in the bulk and boundary conformal data, including nonlocal representations of momentum conservation and a discussion of holographic interpretations. Momentum conservation, encoded as a nonlocal constraint on celestial correlators, highlights subtle aspects of translating bulk symmetries into a boundary CFT framework, motivating future work on OPEs, double-soft limits, and supersymmetric extensions. Overall, the paper advances a holographic-like perspective on flat-space gravity, connecting soft theorems to boundary conformal operators and offering a platform for further exploration of celestial CFTs.

Abstract

We consider the tree-level scattering of massless particles in $(d+2)$-dimensional asymptotically flat spacetimes. The $\mathcal{S}$-matrix elements are recast as correlation functions of local operators living on a space-like cut $\mathcal{M}_d$ of the null momentum cone. The Lorentz group $SO(d+1,1)$ is nonlinearly realized as the Euclidean conformal group on $\mathcal{M}_d$. Operators of non-trivial spin arise from massless particles transforming in non-trivial representations of the little group $SO(d)$, and distinguished operators arise from the soft-insertions of gauge bosons and gravitons. The leading soft-photon operator is the shadow transform of a conserved spin-one primary operator $J_a$, and the subleading soft-graviton operator is the shadow transform of a conserved spin-two symmetric traceless primary operator $T_{ab}$. The universal form of the soft-limits ensures that $J_a$ and $T_{ab}$ obey the Ward identities expected of a conserved current and energy momentum tensor in a Euclidean CFT$_d$, respectively.