Geometry and large N limits in Laughlin states

TL;DR

This paper surveys the geometric properties of Laughlin states and quantum Hall wave functions on Riemann surfaces, exploring anomalies, Chern-Simons theory, and asymptotic expansions in the context of large N limits.

Contribution

It provides a comprehensive review of geometric and topological aspects of quantum Hall states, connecting various mathematical frameworks and physical phenomena.

Findings

Derivation of electromagnetic and gravitational anomalies

Analysis of adiabatic transport on moduli spaces

Asymptotic expansion of the Bergman kernel

Abstract

In these notes I survey geometric aspects of the lowest Landau level wave functions, integer quantum Hall state and Laughlin states on compact Riemann surfaces. In particular, I review geometric adiabatic transport on the moduli spaces, derivation of the electromagnetic and gravitational anomalies, Chern-Simons theory and adiabatic phase, and the relation to holomorphic line bundles, Quillen metric, regularized spectral determinants, bosonisation formulas on Riemann surfaces and asymptotic expansion of the Bergman kernel.