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

<jats:title>Abstract</jats:title> <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> <jats:alternatives> <jats:tex-math>$$\delta $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula>-dimensional, where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\delta $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> is the determinant of <jats:italic>K</jats:italic>. We construct a hermitian holomorphic bundle of rank <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\delta $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> </jats:alternatives> </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>