  We present a detailed study of thermal transport in the disordered Fermi
liquid with short-range interactions. At temperatures smaller than the impurity
scattering rate, i.e., in the diffusive regime, thermal conductivity acquires
non-analytic quantum corrections. When these quantum corrections become large
at low temperatures, the calculation of thermal conductivity demands a
theoretical approach that treats disorder and interactions on an equal footing.
In this paper, we develop such an approach by merging Luttinger's idea of using
gravitational potentials for the analysis of thermal phenomena with a
renormalization group calculation based on the Keldysh nonlinear sigma model.
The gravitational potentials are introduced in the action as auxiliary sources
that couple to the heat density. These sources are a convenient tool for
generating expressions for the heat density and its correlation function from
the partition function. Already in the absence of the gravitational potentials,
the nonlinear sigma model contains several temperature-dependent
renormalization group charges. When the gravitational potentials are introduced
into the model, they acquire an independent renormalization group flow. We show
that this flow preserves the phenomenological form of the correlation function,
reflecting its relation to the specific heat and the constraints imposed by
energy conservation. The main result of our analysis is that the
Wiedemann-Franz law holds down to the lowest temperatures even in the presence
of disorder and interactions and despite the quantum corrections that arise for
both the electric and thermal conductivities.
