General hierarchy of charges at null infinity via the Todd polynomials

We give a general procedure for constructing an extended phase space for Yang-Mills theory at null infinity, capable of handling the asymptotic symmetries and construction of charges responsible for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mrow><a:mi>sub</a:mi></a:mrow><a:mi>n</a:mi></a:msup></a:math>-leading soft theorems at all orders. The procedure is coordinate and gauge-choice independent and can be fed into the calculation of both tree and loop-level soft limits. We find a hierarchy in the extended phase space controlled by the Bernoulli numbers arising in Todd genus computations. We give an example of a calculation at tree level, in radial gauge, where we also uncover recursion relations at all orders for the equations of motion and charges.