Having Fun with Theoretical Physics

# Celestial Symmetries

If one were to follow the bottom-up construction of the holographic dual pair, the initial crucial step involves the identification of (non-trivial or physical) symmetries that govern both facets of this pair. Indeed, the celestial story began with Strominger's insight that Ward identities for asymptotic symmetries (supertranslation and superrotation) could be reinterpreted as the well-known (leading and subleading) soft graviton theorems. To make soft theorems manifest, S-matrix elements in four dimensions are recast as correlation functions of primaries in a 2D CFT via a dimensional reduction of the conformal boundary that puts the scattering states into boost eigenstates. In this new basis, it appears that there exists an entire tower of soft theorems corresponding to poles in the complex conformal weight plane, which yields a tower of soft currents in CCFT. Remarkably, the symmetry algebra of these soft currents was identified as w1+∞ symmetry via 2D CFT treatment from leading holomorphic collinear limits. This revelation of a much richer holographic symmetry algebra for 4D asymptotically flat gravity underscores the power of the Celestial Holography framework. As a next step, we are ready to use these symmetries to either organize the data in phase space or constrain the correlation functions (equivalently scattering amplitudes).

## Symmetries on Phase Space

From the 4D phase space point of view, my recent works with Pasterski [2211.14287] and [2307.16801] showed that we are able to systematically realize the celestial symmetries, both w1+∞ for gravity and its general-spin analog, in the free radiative phase space at future or past null infinity. The generators in phase space are naturally organized into a semi-infinite tower of higher-spin operators and at each definite spin, the generator contains linear, quadratic, and higher-order contributions.

## Symmetries of the S-matrix

When viewed as only acting on the in or out state, the celestial symmetries generated by phase space charges are symmetries of the free theory. Then the underlying question is what symmetries of the free theory survive when we turn on interactions. While we can realize the celestial symmetries in terms of the in or out phase space without talking about the EoMs, we need the EoMs as soon as we want to tie these in and out contributions together and discuss symmetries of the full S-matrix. One way to explore these will be by performing amplitude computations and then comparing them with the phase space results.

## Constraining Amplitudes via Symmetries

One core feature of Celestial Holography is using symmetries and standard CFT techniques to constrain amplitudes. For a generic CFT, null states are important objects since they contain inherent information about the CFT.

Banerjee et. al generalized the null state condition within the CCFT framework, resulting in a set of differential equations for tree-level MHV celestial gluon amplitudes. In recent works, we unraveled the amplitude origin of these differential equations [2106.16111] and further established connections with the asymptotic symmetries [2208.11635]. Exciting future endeavors will include extending these results beyond the MHV sector and leveraging this set of differential equations to systematically constrain correlation functions in CCFTs and their corresponding amplitudes in the bulk.