Lowest Landau level on a cone and zeta determinants

TL;DR

This paper investigates the impact of gravitational anomalies on quantum Hall states on conical Riemann surfaces, proposing a formula linking wave function normalization to zeta determinants and validating it on specific geometries.

Contribution

It introduces a novel formula connecting wave function normalization to zeta determinants on conical surfaces, enhancing understanding of gravitational anomalies in quantum Hall states.

Findings

Derived a formula relating normalization factor to zeta determinants.

Validated the formula on specific conical geometries.

Discussed potential extensions to fractional quantum Hall states.

Abstract

We consider the integer QH state on Riemann surfaces with conical singularities, with the main objective of detecting the effect of the gravitational anomaly directly from the form of the wave function on a singular geometry. We suggest the formula expressing the normalisation factor of the holomorphic state in terms of the regularized zeta determinant on conical surfaces and check this relation for some model geometries. We also comment on possible extensions of this result to the fractional QH states.