Heat transport as torsional responses and Keldysh formalism in a curved spacetime

We revisit a theory of heat transport in the light of a gauge theory of gravity and find the proper heat current with a corresponding gauge field, which yields the natural definitions of the heat magnetization and the Kubo-formula contribution to the thermal conductivity as torsional responses. We also develop a general framework for calculating gravitational responses by combining the Keldysh and Cartan formalisms. By using this framework, we explicitly calculate these two quantities and reproduce the Wiedemann-Franz law for the thermal Hall conductivity in the clean and non-interacting case. Finally, we discuss an effective action for the quantized thermal Hall effect in (2 + 1)-D topological superconductors.