A new look at anomaly cancellation in heterotic $M$-theory

This paper considers anomaly cancellation for eleven-dimensional supergravity on a manifold with boundary and theories related to heterotic $M$-theory. The Green-Schwarz mechanism is implemented without introducing distributions. The importance of the supersymmetry anomaly in constructing the low energy action is discussed and it is argued that a recently proposed action for heterotic $M$-theory gives a supersymmetric theory to all orders in the gravitational coupling $\kappa$.