Citation: Kapustin, Anton, and Lev Spodyneiko. "Thermal Hall conductance and a relative topological invariant of gapped two-dimensional systems." Physical Review B 101.4 (2020): 045137.
Web: https://arxiv.org/abs/1905.06488
Tags: Quantum-hall-effect, Conformal-field-theory, No-go
This paper discusses the chiral central charge. It recovers the results of Appendix D of
> Kitaev, Alexei. "Anyons in an exactly solved model and beyond." Annals of Physics 321.1 (2006): 2-111.
in a way which generalizes to non-frustration-free models, and with more detail than Alexei. It's a good place to supplement the ideas of Kitaev if you're curious about the chiral central charge in terms of bulk currents.
A point that the authors emphasize is that it is hard to define the chiral central charge in the bulk. Naturally one is led to a response formula which gives the derivative of chiral central charge, and not chiral central charge itself. The point is then to connect the given topological state to the trivial state by a continuous path through parameter space, and then integrate along that path to define the chiral central charge. This approach fails if you pass through a thermal phase transition, since at those points the response formula breaks down. So, one needs to be able to connect every topological phase to a maximally mixed state without a thermal phase transition. Constructing such a path is currently an open problem.
Note importantly: In the Kubo formula they give, the new Kubo-like term vanishes in the frustration-free case, reducing to the formula that Alexei had in his appendix.