  In anomaly-free quantum field theories the integrand in the bosonic
functional integral--the exponential of the effective action after integrating
out fermions--is often defined only up to a phase without an additional choice.
We term this choice ``setting the quantum integrand''. In the low-energy
approximation to M-theory the E(8)-model for the C-field allows us to set the
quantum integrand using geometric index theory. We derive mathematical results
of independent interest about pfaffians of Dirac operators in 8k+3 dimensions,
both on closed manifolds and manifolds with boundary. These theorems are used
to set the quantum integrand of M-theory for closed manifolds and for compact
manifolds with either temporal (global) or spatial (local) boundary conditions.
In particular, we show that M-theory makes sense on arbitrary 11-manifolds with
spatial boundary, generalizing the construction of heterotic M-theory on
cylinders.
