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

TL;DR

This paper demonstrates that two different theoretical proposals for conductivity flow diagrams in the quantum Hall effect produce identical flow patterns despite differences in their beta-functions and traversal rates, highlighting a fundamental equivalence.

Contribution

It reveals that two seemingly distinct modular symmetry-based models for quantum Hall conductivity flows are actually equivalent in their flow diagrams.

Findings

Flow diagrams are identical despite different beta-functions.

Flow line tangents are the same in both models.

Flow traversal rates differ between models.

Abstract

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.