Holomorphic and anti-holomorphic conductivity flows in the quantum Hall effect

It is shown that the flow diagrams for the conductivities in the quantum Hall effect, arising from two ostensibly very different proposals based on modular symmetry, are in fact identical. The beta-functions are different, the rate at which the flow lines are traversed are different, but the tangents to the flow lines are the same in both cases, hence the flow diagrams are same in all aspects.