  We show the three-loop integrability of large N plane-wave matrix theory in a
subsector of states comprised of two complex light scalar fields. This is done
by diagonalizing the theory's Hamiltonian in perturbation theory and taking the
large N limit. At one-loop level the result is known to be equal to the
Heisenberg spin-1/2 chain, which is a well-known integrable system. Here,
integrability implies the existence of hidden conserved charges and results in
a degeneracy of parity pairs in the spectrum. In order to confirm integrability
at higher loops, we show that this degeneracy is not lifted and that
(corrected) conserved charges exist. Plane-wave matrix theory is intricately
connected to N=4 Super Yang-Mills, as it arises as a consistent reduction of
the gauge theory on a three-sphere. We find that after appropriately
renormalizing the mass parameter of the plane-wave matrix theory the effective
Hamiltonian is identical to the dilatation operator of N=4 Super Yang-Mills
theory in the considered subsector. Our results therefore represent a strong
support for the conjectured three-loop integrability of planar N=4 SYM and are
in disagreement with a recent dual string theory finding. Finally, we study the
stability of the large N integrability against nonsupersymmetric deformations
of the model.
