The Chern character of the Laughlin vector bundle in the Fractional Quantum Hall Effect

TL;DR

This paper mathematically characterizes the Laughlin vector bundle in the fractional quantum Hall effect, proving key physicist conjectures and deriving its Chern characters using algebraic geometry tools.

Contribution

It formulates and solves the mathematical problem of describing the Laughlin vector bundle, confirming physicist conjectures like the Wen-Niu degeneracy and Wen-Zee shift.

Findings

Proved the Wen-Niu topological degeneracy conjecture.

Derived the first Chern class of the Laughlin vector bundle.

Calculated all Chern characters using Grothendieck-Riemann-Roch.

Abstract

We begin by explaining how a physical problem of studying the quantum Hall effect on a closed surface $C$ leads, via Laughlin's approach, to a mathematical question of describing the rank and the first Chern class of a particular vector bundle on the Picard group ${\rm Pic}^g(C)$. Then we formulate and solve the problem mathematically, proving several important conjectures made by physicists, in particular the Wen-Niu topological degeneracy conjecture and the Wen-Zee shift formula. Let $C$ be a closed Riemann surface of genus~$g$ and $S^NC$ its $N$th symmetric power. The product $C \times {\rm Pic}^d(C)$ carries a universal line bundle. On the product $C^N \times {\rm Pic}^d(C)$ we consider the product of $N$ pull-backs of this universal line bundle and twist it by a power of the diagonal on $C^N$. The resulting line bundle descends onto $S^NC \times {\rm Pic}^d(C)$. Its push-forward (as a sheaf) to ${\rm Pic}^d(C)$ is a vector bundle that we call Laughlin's vector bundle. We determine all the Chern characters of the Laughlin vector bundle via a Grothendieck-Riemann-Roch calculation.