2D Kac-Moody Symmetry of 4D Yang-Mills Theory

TL;DR

The paper shows that four-dimensional Yang-Mills scattering amplitudes can be recast as two-dimensional correlators on the celestial sphere at null infinity, with the positive-helicity soft gluon theorem identified as the Ward identity of a holomorphic $\mathcal{G}$-Kac-Moody symmetry acting on these correlators. This holomorphic symmetry corresponds to CPT-invariant large-gauge transformations at ${\mathscr I}$, and soft gluons emerge as Goldstone modes of spontaneously broken asymptotic color symmetry; the work also clarifies the role of Wilson lines and the flat connection on ${\mathscr I}$ in organizing radiative versus soft data. A key subtlety is the double-soft limit, which is order-dependent in the nonabelian case and leads to a prescription (positive helicity soft first) to realize a consistent holomorphic Kac-Moody structure; the antiholomorphic sector does not yield a second symmetry. The results provide a concrete 2D current-algebra framework for YM infrared physics, with potential implications for holography, infrared structure, and loop-level extensions via soft-sector factorization and Wilson-line operators.

Abstract

Scattering amplitudes of any four-dimensional theory with nonabelian gauge group $\mathcal G$ may be recast as two-dimensional correlation functions on the asymptotic two-sphere at null infinity. The soft gluon theorem is shown, for massless theories at the semiclassical level, to be 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. The Kac-Moody transformations are a $CPT$ invariant subgroup of gauge transformations which act nontrivially at null infinity and comprise the four-dimensional asymptotic symmetry group.