  New quantum phenomena are continuously being discovered in 2D systems. In
particular, the charge density wave (CDW) has the aspect of a quantum crystal
with a macroscopic wave function (order parameter), so unlike quantum liquids
(superconductivity, quantum Hall liquids $^3$He, $^4$He), new ground states
such as supersolid and Moir\'e solids can be expected. However, it is difficult
to describe these states because of their quantum aspect, hence there is still
no theory that can explain CDW phases in a unified way. The best way to
describe a quantum crystal seems to be a conformal transformation that allows
local deformation (wave properties) and preserves local angles (crystal
properties).
  Here, we propose a unifying conformal description of 2D CDW phases in the
typical 2D CDW material transition metal dichalcogenides (MX$_2$). We discover
that the discommensurate CDW phases in MX$_2$ can be explained beautifully by a
discrete conformal transformation of CDW wavevectors. This conformality is due
to commensurability of CDW with the MX$_2$ lattice. In other words,
interference of harmonic wavefunction induces conformality.
  Using this new conformal formulation, we explain experimental
nearly-commensurate/stripe/T CDW phases in 1$T$-TaS$_2$
($\sqrt{13}\times\sqrt{13}$ structure), 2$H$-TaSe$_2$ ($\sqrt{9}\times\sqrt{9}$
structure), and explain the origin of a new experimental nearly-commensurate
phase in TaSe$_2$ thin-film ($\sqrt{7}\times\sqrt{7}$ structure).
  This theory is very simple in the sense that it includes only
discommensuration and comprises physics as rich as quantum Hall liquids. This
new description will broaden our perspective of quantum crystals.
