A symmetry-based analysis (projective symmetry group) is used to study spin-liquid phases on the triangular and Kagom\'e lattices in the Schwinger boson framework. A maximum of eight distinct
Z2 spin-liquid states are found for each lattice, which preserve all symmetries. Out of these only a few have nonvanishing nearest-neighbor amplitudes, which are studied in greater detail. On the triangular lattice, only two such states are present---the first (zero-flux state) is the well-known state introduced by Sachdev, which on condensation of spinons leads to the 120\ifmmode^\circ\else\textdegree\fi{} ordered state. The other solution, which we call the $\ensuremath{\pi}$-flux state has not previously been discussed. Spinon condensation leads to an ordering wave vector at the Brillouin zone edge centers, in contrast to the 120\ifmmode^\circ\else\textdegree\fi{} state. While the zero-flux state is more stable with just nearest-neighbor exchange, we find that the introduction of either next-neighbor antiferromagnetic exchange or four-spin ring exchange (of the sign obtained from a Hubbard model) tends to favor the $\ensuremath{\pi}$-flux state. On the Kagom\'e lattice four solutions are obtained---two have been previously discussed by Sachdev, which on spinon condensation give rise to the
q=0 and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ spin-ordered states. In addition we find two states with significantly larger values of the quantum parameter at which magnetic ordering occurs. For one of them this even exceeds unity ${\ensuremath{\kappa}}_{c}\ensuremath{\approx}2.0$ in a nearest-neighbor model, indicating that if stabilized, could remain spin disordered for physical values of the spin. This state is also stabilized by ring-exchange interactions with signs as derived from the Hubbard model.
We study incompressible ground states of bosons in a two-dimensional rotating square optical lattice. The system can be described by the Bose-Hubbard model in an effective uniform magnetic field present due to the lattice rotation. To study ground states of the system, we map it to a frustrated spin model, followed by Schwinger boson mean-field theory and projective symmetry group analysis. Using symmetry analysis, we identify bosonic fractional quantum Hall states, predicted for bosonic atoms in rotating optical lattices, with possible stable gapped spin-liquid states within the Schwinger boson formalism. In particular, our results indicate that previously obtained fractional quantum Hall states induced by the lattice potential, and with no counterpart in the continuum [M\"oller and Cooper, Phys. Rev. Lett. 103, 105303 (2009)], correspond to ``$\ensuremath{\pi}$-flux'' spin-liquid states of the frustrated spin model.