Asymptotic Symmetries in Four-Dimensional Gauge and Gravity Theories

Abstract

Recent developments have uncovered a deep relationship between soft theorems in quantum field theories and asymptotic symmetries. We investigate five explicit examples wherein these connections are studied and verified.

First, we show that the Weinberg's soft-photon theorem may be recast as the Ward identity for $CPT$-invariant large $U(1)$ gauge transformations that asymptotically approach an arbitrary function $\varepsilon$ of the conformal sphere at null infinity, but are independent of retarded time. The symmetries for which $\varepsilon\neq$ constant are spontaneously broken in the perturbative quantum field theory vacuum and the associated Goldstone modes are the zero-momentum photons. These comprise a $U(1)$ boson living on the conformal sphere.

Second, we generalize the construction to non-abelian gauge theories with gauge group ${\mathcal G}$ and show that the massless tree-level soft-gluon theorem is the Ward identity of a holomorphic two-dimensional ${\mathcal G}$-Kac-Moody symmetry acting on these correlation functions. Holomorphic Kac-Moody current insertions are positive helicity soft-gluon insertions. These symmetries are also spontaneously broken and the soft-gluons are the Goldstone modes.

Third, we generalize to supersymmetric ${\mathcal N}=1$ abelian gauge theories with massless charged matter and establish the existence of infinitely many fermionic asymptotic symmetries at null infinity, parametrized by a function on~$S^2$, whose Ward identities give rise to the soft photino theorem. Unlike large gauge transformations, these symmetries are not manifest at the level of the Lagrangian. They are spontaneously broken, and the soft photinos are the associated Goldstone fermions. Unbroken global supersymmetry relates this fermionic charge to the $U(1)$ large gauge charge.

Fourth, we consider gravitational theories and show that Weinberg's soft-graviton theorem is the Ward identity corresponding to a certain infinite-dimensional ``diagonal" subgroup of BMS supertranslations acting on past and future null infinity (${\mathcal I}^-$ and ${\mathcal I}^+$). The soft-gravitons are the Goldstone bosons of spontaneously broken supertranslation invariance.

Finally, we use the sub-leading soft-graviton theorem to construct an operator $T_{zz}$ whose insertion in the four-dimensional tree-level quantum gravity $\mathcal{S}$-matrix obeys the Virasoro-Ward identities of the energy-momentum tensor of a two-dimensional conformal field theory (CFT$_2$). The celestial sphere at Minkowskian null infinity plays the role of the Euclidean sphere of the CFT$_2$, with the Lorentz group acting as the unbroken $SL(2,\mathbb{C})$ subgroup.