A General Hierarchy of Charges at Null Infinity via the Todd Polynomials

TL;DR

This paper develops a universal, gauge-independent method to construct an extended phase space at null infinity for Yang-Mills theory, enabling systematic derivation of charges linked to all orders of soft theorems, with a hierarchy governed by Todd genus Bernoulli numbers.

Contribution

It introduces a general, coordinate-independent procedure for extended phase space construction at null infinity, revealing a hierarchy controlled by Todd genus Bernoulli numbers and recursion relations at all orders.

Findings

Hierarchy of charges governed by Bernoulli numbers

Explicit tree-level example in radial gauge

Recursion relations for equations of motion and charges

Abstract

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.