Geometric Zabrodin-Wiegmann conjecture for integer Quantum Hall states

TL;DR

This paper proves a geometric version of the Zabrodin-Wiegmann conjecture for integer Quantum Hall states, linking partition functions to geometric invariants on Riemann surfaces and providing asymptotic expansions.

Contribution

It introduces a geometric framework for the Zabrodin-Wiegmann conjecture, connecting partition functions of Quantum Hall states with holomorphic torsion and advanced asymptotic analysis.

Findings

Established an asymptotic expansion for the partition function as p approaches infinity.

Linked the constant term of the expansion to holomorphic analytic torsion.

Validated the geometric Zabrodin-Wiegmann prediction through rigorous proof.

Abstract

The purpose of this article is to show a geometric version of Zabrodin-Wiegmann conjecture for an integer Quantum Hall state. Given an effective reduced divisor on a compact connected Riemann surface, using the canonical holomorphic section of the associated canonical line bundle as well as certain initial data and local normalisation data, we construct a canonical non-zero element in the determinant line of the cohomology of the $p$-tensor power of the line bundle. When endowed with proper metric data, the square of the $ L^{2} $-norm of our canonical element is the partition function associated to an integer Quantum Hall state. We establish an asymptotic expansion for the logarithm of the partition function when $ p\to +\infty$. The constant term of this expansion includes the holomorphic analytic torsion and matches a geometric version of Zabrodin-Wiegmann's prediction. Our proof relies on Bismut-Lebeau's embedding formula for the Quillen metrics, Bismut-Vasserot and Finski's asymptotic expansion for the analytic torsion associated to the higher tensor product of a positive Hermitian holomorphic line bundle.