  A duality transformation in quantum field theory is usually established first
through partition functions. It is always important to explore the dual
relations between various correlation functions in the transformation. Here, we
explore such a dual relation to study quantum phases and phase transitions in
an extended boson Hubbard model at 1/3 (2/3) filling on a triangular lattice.
  We develop systematically a simple and effective way to use the vortex degree
of freedoms on dual lattices to characterize both the density wave and valence
bond symmetry breaking patterns of the boson insulating states in the direct
lattices.
  In addition to a checkerboard charge density wave (X-CDW) and a stripe CDW,
we find a novel CDW-VBS phase which has both local CDW and local valence bond
solid (VBS) orders. Implications on QMC simulations are addressed.
  The possible experimental realizations of cold atoms loaded on optical
lattices are discussed.
