  The soft factorization theorem for 4D abelian gauge theory states that the
$\mathcal{S}$-matrix factorizes into soft and hard parts, with the universal
soft part containing all soft and collinear poles. Similarly, correlation
functions on the sphere in a 2D CFT with a $U(1)$ Kac-Moody current algebra
factorize into current algebra and non-current algebra factors, with the
current algebra factor fully determined by its pole structure. In this paper,
we show that these 4D and 2D factorizations are mathematically the same
phenomena. The soft `tHooft-Wilson lines and soft photons are realized as a
complexified 2D current algebra on the celestial sphere at null infinity. The
current algebra level is determined by the cusp anomalous dimension. The
associated complex $U(1)$ boson lives on a torus whose modular parameter is
$\tau =\frac {2\pi i }{e^2}+\frac{\theta}{ 2 \pi}$. The correlators of this 2D
current algebra fully reproduce the known soft part of the 4D
$\mathcal{S}$-matrix, as well as a conjectured generalization involving
magnetic charges.
