---
_id: '66291'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n          <jats:p>In 1993 Keski-Vakkuri
    and Wen introduced a model for the fractional quantum Hall effect based on multilayer
    two-dimensional electron systems satisfying quasi-periodic boundary conditions.
    Such a model is essentially specified by a choice of a complex torus <jats:italic>E</jats:italic>
    and a symmetric positively definite matrix <jats:italic>K</jats:italic> of size
    <jats:italic>g</jats:italic> with non-negative integral coefficients, satisfying
    some further constraints. The space of the corresponding wave functions turns
    out to be <jats:inline-formula>\r\n              <jats:alternatives>\r\n                <jats:tex-math>$$\\delta
    $$</jats:tex-math>\r\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>δ</mml:mi>\r\n                </mml:math>\r\n              </jats:alternatives>\r\n
    \           </jats:inline-formula>-dimensional, where <jats:inline-formula>\r\n
    \             <jats:alternatives>\r\n                <jats:tex-math>$$\\delta
    $$</jats:tex-math>\r\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>δ</mml:mi>\r\n                </mml:math>\r\n              </jats:alternatives>\r\n
    \           </jats:inline-formula> is the determinant of <jats:italic>K</jats:italic>.
    We construct a hermitian holomorphic bundle of rank <jats:inline-formula>\r\n
    \             <jats:alternatives>\r\n                <jats:tex-math>$$\\delta
    $$</jats:tex-math>\r\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>δ</mml:mi>\r\n                </mml:math>\r\n              </jats:alternatives>\r\n
    \           </jats:inline-formula> on the abelian variety <jats:italic>A</jats:italic>
    (which is the <jats:italic>g</jats:italic>-fold product of the torus <jats:italic>E</jats:italic>
    with itself), whose fibres can be identified with the space of wave function of
    Keski-Vakkuri and Wen. A rigorous construction of this “magnetic bundle” involves
    the technique of Fourier–Mukai transforms on abelian varieties. The constructed
    bundle turns out to be simple and semi-homogeneous and it can be equipped with
    two different (and natural) hermitian metrics: the one coming from the center-of-mass
    dynamics and the one coming from the Hilbert space of the underlying many-body
    system. We prove that the canonical Bott–Chern connection of the first hermitian
    metric is always projectively flat and give sufficient conditions for this property
    for the second hermitian metric.</jats:p>"
article_number: '97'
author:
- first_name: Igor
  full_name: Burban, Igor
  id: '72064'
  last_name: Burban
- first_name: Semyon
  full_name: Klevtsov, Semyon
  last_name: Klevtsov
citation:
  ama: Burban I, Klevtsov S. Algebraic Geometry of the Multilayer Model of the Fractional
    Quantum Hall Effect on a Torus. <i>Communications in Mathematical Physics</i>.
    2025;406(5). doi:<a href="https://doi.org/10.1007/s00220-025-05267-9">10.1007/s00220-025-05267-9</a>
  apa: Burban, I., &#38; Klevtsov, S. (2025). Algebraic Geometry of the Multilayer
    Model of the Fractional Quantum Hall Effect on a Torus. <i>Communications in Mathematical
    Physics</i>, <i>406</i>(5), Article 97. <a href="https://doi.org/10.1007/s00220-025-05267-9">https://doi.org/10.1007/s00220-025-05267-9</a>
  bibtex: '@article{Burban_Klevtsov_2025, title={Algebraic Geometry of the Multilayer
    Model of the Fractional Quantum Hall Effect on a Torus}, volume={406}, DOI={<a
    href="https://doi.org/10.1007/s00220-025-05267-9">10.1007/s00220-025-05267-9</a>},
    number={597}, journal={Communications in Mathematical Physics}, publisher={Springer
    Science and Business Media LLC}, author={Burban, Igor and Klevtsov, Semyon}, year={2025}
    }'
  chicago: Burban, Igor, and Semyon Klevtsov. “Algebraic Geometry of the Multilayer
    Model of the Fractional Quantum Hall Effect on a Torus.” <i>Communications in
    Mathematical Physics</i> 406, no. 5 (2025). <a href="https://doi.org/10.1007/s00220-025-05267-9">https://doi.org/10.1007/s00220-025-05267-9</a>.
  ieee: 'I. Burban and S. Klevtsov, “Algebraic Geometry of the Multilayer Model of
    the Fractional Quantum Hall Effect on a Torus,” <i>Communications in Mathematical
    Physics</i>, vol. 406, no. 5, Art. no. 97, 2025, doi: <a href="https://doi.org/10.1007/s00220-025-05267-9">10.1007/s00220-025-05267-9</a>.'
  mla: Burban, Igor, and Semyon Klevtsov. “Algebraic Geometry of the Multilayer Model
    of the Fractional Quantum Hall Effect on a Torus.” <i>Communications in Mathematical
    Physics</i>, vol. 406, no. 5, 97, Springer Science and Business Media LLC, 2025,
    doi:<a href="https://doi.org/10.1007/s00220-025-05267-9">10.1007/s00220-025-05267-9</a>.
  short: I. Burban, S. Klevtsov, Communications in Mathematical Physics 406 (2025).
date_created: 2026-07-07T06:18:00Z
date_updated: 2026-07-07T06:18:58Z
doi: 10.1007/s00220-025-05267-9
intvolume: '       406'
issue: '5'
language:
- iso: eng
publication: Communications in Mathematical Physics
publication_identifier:
  issn:
  - 0010-3616
  - 1432-0916
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect
  on a Torus
type: journal_article
user_id: '72064'
volume: 406
year: '2025'
...