  A large N expansion technique, based on symplectic (Sp(N)) symmetry, for
frustrated magnetic systems is studied. The phase diagram of a square lattice,
spin S, quantum antiferromagnet with first, second and third neighbor
antiferromagnetic coupling (the J_1-J_2-J_3 model) is determined in the large-N
limit and consequences of fluctuations at finite N for the quantum disordered
phases are discussed. In addition to phases with long range magnetic order, two
classes of disordered phases are found: (i) states similar to those in
unfrustrated systems with commensurate, collinear spin correlations,
confinement of spinons, and spin-Peierls or valence-bond-solid order controlled
by the value of 2S(mod 4) or 2S(mod 2); (ii) states with incommensurate,
coplanar spin correlations, and unconfined bosonic spin-1/2 spinon excitations.
The occurrence of ``order from disorder'' at large S is discussed. Neither
chirally ordered nor spin nematic states are found. Initial results on
superconductivity in the t-J model at N=infinity and zero temperature are also
presented.
