  Strominger and collaborators recently proposed that soft theorems for gauge
and gravity amplitudes can be interpreted as Ward identities of a 2d CFT at
null infinity. In this paper, we will consider a specific realization of this
CFT known as ambitwistor string theory, which describes 4d Yang-Mills and
gravity with any amount of supersymmetry. Using 4d ambitwistor string theory,
we derive soft theorems in the form of an infinite series in the soft momentum
which are valid to subleading order in gauge theory and sub-subleading order in
gravity. Furthermore, we describe how the algebra of soft limits can be encoded
in the braiding of soft vertex operators on the worldsheet and point out a
simple relation between soft gluon and soft graviton vertex operators which
suggests an interesting connection to color-kinematics duality. Finally, by
considering ambitwistor string theory on a genus one worldsheet, we compute the
1-loop correction to the subleading soft graviton theorem due to infrared
divergences.
