  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.
