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

TL;DR

This paper explores the algebraic geometry underlying a multilayer fractional quantum Hall effect model on a torus, constructing a hermitian holomorphic bundle and analyzing its properties using Fourier-Mukai transforms.

Contribution

It provides a rigorous geometric construction of the wave function space as a hermitian holomorphic bundle on an abelian variety, linking quantum Hall physics with algebraic geometry techniques.

Findings

Constructed a hermitian holomorphic bundle of wave functions on an abelian variety.

Proved the bundle is simple and semi-homogeneous.

Analyzed the projective flatness of the Bott-Chern connection.

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 $\delta$-dimensional, where $\delta$ is the determinant of $K$. We construct a hermitian holomorphic bundle of rank $\delta$ 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.