Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus

article Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus published Igor Burban author 72064 Semyon Klevtsov author Abstract 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 E and a symmetric positively definite matrix K of size g with non-negative integral coefficients, satisfying some further constraints. The space of the corresponding wave functions turns out to be $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> -dimensional, where $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> is the determinant of K. We construct a hermitian holomorphic bundle of rank $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> on the abelian variety A (which is the g-fold product of the torus E 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. Springer Science and Business Media LLC2025 eng Communications in Mathematical Physics 0010-3616 1432-091610.1007/s00220-025-05267-9 4065 Burban, Igor, and Semyon Klevtsov. “Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus.” Communications in Mathematical Physics, 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>. Burban I, Klevtsov S. Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus. Communications in Mathematical Physics. 2025;406(5). doi:<a href="https://doi.org/10.1007/s00220-025-05267-9">10.1007/s00220-025-05267-9</a> @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} } Burban, I., & Klevtsov, S. (2025). Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus. Communications in Mathematical Physics, 406(5), Article 97. <a href="https://doi.org/10.1007/s00220-025-05267-9">https://doi.org/10.1007/s00220-025-05267-9</a> I. Burban and S. Klevtsov, “Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus,” Communications in Mathematical Physics, 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>. I. Burban, S. Klevtsov, Communications in Mathematical Physics 406 (2025). Burban, Igor, and Semyon Klevtsov. “Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus.” Communications in Mathematical Physics 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>. 662912026-07-07T06:18:00Z2026-07-07T06:18:58Z