Chern classes of Laughlin bundles on the quasihole moduli space

TL;DR

This paper investigates the geometric and topological properties of Laughlin quasihole states on Riemann surfaces, deriving Chern classes and connecting them to Berry phases and fractional statistics.

Contribution

It constructs vector bundles for quasihole states on Riemann surfaces, computes their Chern characters, and relates these to Berry phases and fractional statistics, including explicit wave-functions in genus zero and one.

Findings

Chern character of quasihole bundles determined via Grothendieck-Riemann-Roch.

Vector bundle compatibility with projective flatness in maximal filling state.

Explicit wave-functions constructed for genus zero and one, matching predicted Chern classes.

Abstract

We study fractional quantum Hall states with quasihole excitations, on Riemann surfaces of arbitrary genus. For configurations with $m$ quasiholes we construct a vector bundle above the $m$-th symmetric power of the curve so that the fiber at a point $\lbrace w_1,\dots,w_m \rbrace$ corresponds to the state with quasiholes localized at these positions. We determine the Chern character of this bundle via the Grothendieck-Riemann-Roch theorem and show that in the completely filled state, i.e. when the number of particles is maximal, the vector bundle is compatible with the condition of projective flatness. Furthermore, we obtain a generalization of this result to the case of multiple layers and multiple quasihole types. In genus zero and one, we construct explicit wave-functions and verify that the curvature of the associated Chern connection reproduces the predicted Chern classes. The Chern classes obtained match, term by term, the predicted decomposition of the Berry phase under quasihole exchange, into an extensive Aharonov--Bohm contribution and a fractional statistical contribution.