A 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 sub${}^n$-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 explicit 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.