Generalized GMP Algebra for Three-Dimensional Quantum Hall Fluids of Extended Objects

TL;DR

This paper develops a geometric framework for 3D quantum Hall fluids of extended objects, revealing a noncommutative algebra governing their low-energy kinematics and connecting it to topological field theory.

Contribution

It introduces a generalized GMP algebra for 3D quantum Hall fluids of extended objects and links it to topological BF+BB theory, advancing understanding of their geometric and topological properties.

Findings

Derived a 3D GMP algebra for extended objects in quantum Hall fluids.

Connected the algebra to the quantization of a topological BF+BB theory.

Clarified the geometric and topological structure of 3D quantum Hall phases.

Abstract

We develop a geometric framework for three-dimensional quantum Hall fluids of extended objects (quasi-strings) in the presence of a strong three-form background field associated with a bundle gerbe. In the strong-field regime, fast internal dynamics is frozen and the low-energy kinematics is governed by generalized guiding-center variables consisting of vectorial and tensorial coordinates. We show that these guiding-center variables obey a noncommutative geometry giving rise to a three-dimensional generalization of the Girvin-MacDonald-Platzman (GMP) algebra for projected density operators. Moreover, we relate this algebra to the canonical quantization of a topological BF+BB theory whose level is identified with the Dixmier-Douady invariant. Our results clarify the structure of incompressible quantum Hall-type phases and their geometric and topological features in three spatial dimensions.