  Motivated by recent progress in understanding the interplay between lattice
and electronic topological phases, we consider quantum-melting transitions of
{\it weak} quantum liquid crystals, a crystal and a nematic phase, in which
electrons form a quantum Hall state. In certain classes of Chern band
insulators and quantum Hall phases, it has been previously demonstrated that
there are topological Chern-Simons terms such as a Hall viscosity term and a
gravitational Chern-Simons term for local lattice deformations. The
Chern-Simons terms can induce anyonic statistics for the topological lattice
defects and furthermore dress the defects with certain symmetry quantum
numbers. On the other hand, the melting transitions of such liquid-crystalline
orders are driven by the condensation of lattice defects. Based on these
observations, we show how the topological terms can change the nature of the
proximate disordered phases of the quantum liquid crystalline phases. We derive
and study the effective dual field theories for the liquid crystalline phases
with the geometric Chern-Simons terms, and carefully examine the symmetry
quantum numbers and statistics of defects. We show that a crystal may go
through a continuous phase transition into another crystal with the new
discrete translational symmetries because the dislocation, the topological
defect in the crystal, carries non-zero crystal momentum due to the Hall
viscosity term. For the nematic phase, the disclination will condense at the
phase transition to the isotropic phase, and we show that the isotropic phase
may support a deconfined fractionally charged excitation due to the Wen-Zee
term, and thus the isotropic phase and the nematic phase have different
electromagnetic Hall responses.
