Anomalies without an action

Modern on-shell methods allow us to construct both the classical and quantum S-matrix for a large class of theories, without utilizing knowledge of the interacting Lagrangian. It was recently shown that the same applies for chiral gauge theories, where the constraints from anomaly cancelation can be recast into the tension between unitarity and locality, without any reference to gauge symmetry. In this paper, we give a more detailed exploration, for chiral QED and QCD. We study the rational terms that are mandated by locality, and show that the factorization poles of such terms reveal a new particle in the spectrum, the Green-Schwarz two-from. We further extend the analysis to six-dimensional gravity coupled to chiral matter, including self-dual two-forms for which covariant actions generically do not exist. Despite this, the on-shell methods define the correct quantum S-matrix by demonstrating that locality of the one-loop amplitude requires combination of chiral matter that is consistent with that of anomaly cancelation.