$N$-jettiness soft function at next-to-next-to-leading order in perturbative QCD

We derive a compact representation of the renormalized $N$-jettiness soft function that is free of infrared and collinear divergences through next-to-next-to-leading order in perturbative QCD. The number of hard partons $N$ is a parameter in the formula for the finite remainder. Cancellation of all infrared and collinear singularities between the bare soft function and its renormalization matrix in color space is demonstrated analytically.