  A finite-temperature effective free energy of the boundary of a quantized
thermal Hall system is derived microscopically from the bulk two-dimensional
Dirac fermion coupled with a gravitational field. In two spatial dimensions,
the thermal Hall conductivity of fully gapped insulators and superconductors is
quantized and given by the bulk Chern number, in analogy to the quantized
electric Hall conductivity in quantum Hall systems. From the perspective of
effective action functionals, two distinct types of the field theory have been
proposed to describe the quantized thermal Hall effect. One of these, known as
the gravitational Chern-Simons action, is a kind of topological field theory,
and the other is a phenomenological theory relevant to the St\v{r}eda formula.
In order to solve this problem, we derive microscopically an effective theory
that accounts for the quantized thermal Hall effect. In this paper, the
two-dimensional Dirac fermion under a static background gravitational field is
considered in equilibrium at a finite temperature, from which an effective
boundary free energy functional of the gravitational field is derived. This
boundary theory is shown to explain the quantized thermal Hall conductivity and
thermal Hall current in the bulk by assuming the Lorentz symmetry. The bulk
effective theory is consistently determined via the boundary effective theory
