S-algebra in gauge theory: twistor, spacetime and holographic perspectives

The celestial S-algebra arose from a reinterpretation of collinear limits of the Yang–Mills S-matrix as OPEs in celestial holography. It was subsequently represented via asymptotic charge aspects defined in the Yang–Mills radiative phase space defined at null infinity on the one hand, and via a twisted holography vertex algebra construction in twistor space on the other. Here we first identify it with the traditional symmetry algebra of self-dual Yang–Mills theory as an integrable system via its hierarchies of conserved quantities and associated flows; the self-dual phase space can be canonically identified with that of full Yang–Mills at null infinity . We derive the associated canonical generators from the twistor space action, identifying two infinite towers of charges corresponding to the two gluon helicities. These expressions are translated into spacetime data at null infinity using twistor integral formulae. Examining the charge algebra at spacelike infinity reveals the vertex algebras studied in the context of twisted holography. Our discussion extends directly to the celestial symmetries of self-dual gravity. This analysis provides a unified framework for celestial symmetries, connecting twistor, spacetime, and holographic approaches and culminating in a nonlinear extrapolate dictionary for self-dual gauge theory.