The role of the spin connection in quantum Hall effect: A perspective from geometric quantization

TL;DR

This paper explores the interplay between the spin connection and gauge fields in the quantum Hall effect, highlighting geometric quantization and symplectic transformations as key tools for understanding topological aspects across dimensions.

Contribution

It introduces a novel perspective linking the Abelian potential and spin connection through geometric quantization and symplectic algebra in quantum Hall systems.

Findings

The Abelian potential combines with the spin connection in topological terms.

This relationship holds for quantum Hall effects on complex projective spaces of any dimension.

The interpretation involves symplectic transformations and the metaplectic correction.

Abstract

The topological terms of the bulk effective action for the integer quantum Hall effect, capturing the dynamics of gauge and gravitational fluctuations, reveal a curiosity, namely, the Abelian potential for the magnetic field appears in a particular combination with the Abelian spin connection. This seems to hold for quantum Hall effect on complex projective spaces of arbitrary dimensions. An interpretation of this in terms of the algebra of symplectic transformations is given. This can also be viewed in terms of the metaplectic correction in geometric quantization.