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Noncommutative Geometry and GMP-type algebra for three-dimensional quantum Hall fluids of extended objects

Giandomenico Palumbo*

  • *Contact author: giandomenico.palumbo@gmail.com
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Phys. Rev. D 114, 046021 – Published 24 August, 2026

DOI: https://doi.org/10.1103/d76r-nzwm

Abstract

We develop a geometric framework for three-dimensional quantum Hall fluids of extended objects, namely quasistrings, 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. The resulting reduced bracket is generally shape dependent; however, after projection onto an assumed isotropic gapped internal sector, these guiding-center variables obey a universal noncommutative geometry giving rise to a three-dimensional Girvin-MacDonald-Platzman (GMP)-type algebra for projected density operators. Moreover, we identify a compatible correspondence between this projected algebra and the canonical quantization of a topological BF+BB theory at the level of suitably smeared collective operators, with its level taken to be associated with the Dixmier-Douady invariant. Our results provide a kinematic starting point for investigating incompressible quantum Hall-type phases and their geometric and topological features in three spatial dimensions.

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Phys. Rev. D 114, 046021– Published 24 August, 2026
  • Received 2 March 2026
  • Accepted 30 July 2026
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