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Algebraic geometry of the multilayer model of the fractional quantum Hall effect on a torus

Igor Burban, Semyon Klevtsov

TL;DR

This work builds a rigorous algebro-geometric framework for Keski-Vakkuri–Wen multilayer fractional quantum Hall states on a torus. It constructs a holomorphic magnetic vector bundle 𝔅 on the g-fold torus A whose fibers recover the KV–Wen ground-state spaces, and proves 𝔅 is simple and semi-homogeneous with a first Chern class computable from the adjunct K^♯, yielding a total Chern class c(𝔅) = (1 + c1(𝔅)/δ)^δ. Two natural hermitian metrics induce Bott–Chern connections, which are projectively flat in the canonical cases, linking the center-of-mass dynamics and the many-body Hilbert space to the differential-geometric structure. Using Fourier–Mukai transforms on abelian varieties, the paper identifies 𝔅 as the FM image of a line bundle and analyzes its dual and restricted forms, deriving explicit expressions for c1 and slope. The restricted bundle 𝕌 on the diagonal E has slope −ρ/δ and is stable precisely when K is primary, with Jain fractions arising as slopes in key examples, thereby connecting algebro-geometric invariants to fractional quantum Hall data.

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 positive integral coefficients. The space of the corresponding wave functions turns out to be $δ$-dimensional, where $δ$ is the determinant of $K$. We construct a hermitian holomorphic bundle of rank $δ$ 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. This bundle turns out to be simple and semi-homogeneous. This bundle 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.

Algebraic geometry of the multilayer model of the fractional quantum Hall effect on a torus

TL;DR

This work builds a rigorous algebro-geometric framework for Keski-Vakkuri–Wen multilayer fractional quantum Hall states on a torus. It constructs a holomorphic magnetic vector bundle 𝔅 on the g-fold torus A whose fibers recover the KV–Wen ground-state spaces, and proves 𝔅 is simple and semi-homogeneous with a first Chern class computable from the adjunct K^♯, yielding a total Chern class c(𝔅) = (1 + c1(𝔅)/δ)^δ. Two natural hermitian metrics induce Bott–Chern connections, which are projectively flat in the canonical cases, linking the center-of-mass dynamics and the many-body Hilbert space to the differential-geometric structure. Using Fourier–Mukai transforms on abelian varieties, the paper identifies 𝔅 as the FM image of a line bundle and analyzes its dual and restricted forms, deriving explicit expressions for c1 and slope. The restricted bundle 𝕌 on the diagonal E has slope −ρ/δ and is stable precisely when K is primary, with Jain fractions arising as slopes in key examples, thereby connecting algebro-geometric invariants to fractional quantum Hall data.

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 and a symmetric positively definite matrix of size with positive integral coefficients. The space of the corresponding wave functions turns out to be -dimensional, where is the determinant of . We construct a hermitian holomorphic bundle of rank on the abelian variety (which is the -fold product of the torus 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. This bundle turns out to be simple and semi-homogeneous. This bundle 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.
Paper Structure (11 sections, 33 theorems, 147 equations, 1 figure)

This paper contains 11 sections, 33 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Holomorphic line bundles on complex tori and theta-functions
  3. Haldane--Rezayi wave functions on a torus
  4. Multivariate theta functions and generalized Heisenberg groups
  5. Multi-layer model of FQHE on a torus
  6. Vector bundles and Fourier--Mukai transforms on abelian varieties
  7. Dual complex torus
  8. Line bundles on complex tori and Fourier--Mukai transforms
  9. Poincaré line bundle revisited
  10. Magnetic vector bundle of the multilayer torus model of FQHE
  11. Restricted magnetic vector bundle

Key Result

Proposition 2.1

As a function of $z$, the theta-series (E:ThetaWithCharacteristics) converges absolutely and uniformly on compact subsets of $\mathbb C$ and satisfies the following quasi-periodic conditions: Modulo the lattice $\Lambda$, the function $\vartheta[a, b](z | \tau)$ has a unique zero at the point Moreover, this zero is simple. In particular, the function has a unique simple zero at $z = 0$ modulo $

Figures (1)

  • Figure :

Theorems & Definitions (88)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • ...and 78 more