  Neutral excitations in fractional quantum Hall (FQH) fluids define the
incompressibility of topological phases, a species of which can show
graviton-like behaviors and are thus called the graviton modes (GMs). Here, we
develop the microscopic theory for multiple GMs in FQH fluids and show
explicitly that they are associated with the geometric fluctuation of
well-defined conformal Hilbert spaces (CHSs), which are hierarchical subspaces
within a single Landau level, each with emergent conformal symmetry and
continuously parameterized by a unimodular metric. This leads to several
statements about the number and the merging/splitting of GMs, which are
verified numerically with both model and realistic interactions. We also
discuss how the microscopic theory can serve as the basis for the additional
Haldane modes in the effective field theory description and their experimental
relevance to realistic electron-electron interactions.
