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Geometric Zabrodin-Wiegmann conjecture for integer Quantum Hall states

Shu Shen, Jianqing Yu

TL;DR

This work establishes a geometric version of the Zabrodin–Wiegmann conjecture for integer quantum Hall states on a compact Riemann surface by constructing a canonical section $s_p$ in the determinant line $oldsymbol ext{det}(H^ullet(X,L^poxtimes E))$ from a reduced divisor $D$ and initial data. The partition function $Z_p$, defined as the squared $L^2$-norm of $s_p$, admits a precise large-$p$ asymptotic expansion with leading quadratic and logarithmic corrections; its constant term encodes holomorphic torsion, a global spectral invariant. The proof blends determinant-line technology, the Quillen metric, and analytic torsion via Bismut–Lebeau’s embedding formula and Bismut–Vasserot–Finski asymptotics, yielding explicit expressions for the expansion coefficients including a torsion-dependent $a_0$. The approach generalizes and recovers known cases for genus 0 and 1, clarifies the role of normalized divisors, and clarifies the limitations to $eta=1$ and compact settings, providing a robust bridge between quantum Hall physics and complex-analytic index theory.

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.

Geometric Zabrodin-Wiegmann conjecture for integer Quantum Hall states

TL;DR

This work establishes a geometric version of the Zabrodin–Wiegmann conjecture for integer quantum Hall states on a compact Riemann surface by constructing a canonical section in the determinant line from a reduced divisor and initial data. The partition function , defined as the squared -norm of , admits a precise large- asymptotic expansion with leading quadratic and logarithmic corrections; its constant term encodes holomorphic torsion, a global spectral invariant. The proof blends determinant-line technology, the Quillen metric, and analytic torsion via Bismut–Lebeau’s embedding formula and Bismut–Vasserot–Finski asymptotics, yielding explicit expressions for the expansion coefficients including a torsion-dependent . The approach generalizes and recovers known cases for genus 0 and 1, clarifies the role of normalized divisors, and clarifies the limitations to and compact settings, providing a robust bridge between quantum Hall physics and complex-analytic index theory.

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 -tensor power of the line bundle. When endowed with proper metric data, the square of the -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 . 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.
Paper Structure (31 sections, 17 theorems, 170 equations)

This paper contains 31 sections, 17 theorems, 170 equations.

Table of Contents

  1. Physical background
  2. Mathematical background
  3. Main results
  4. Constructions of $s_{p}\in \lambda_{p}$
  5. Proofs of (\ref{['eq:zjj1']}) and (\ref{['eq:aaaz']})
  6. Remarks on our assumption on the divisor $D$
  7. Comparison with related works
  8. Organisation of the paper
  9. Acknowledgement
  10. Preliminary
  11. Determinant functors
  12. Holomorphic torsion and Quillen metric
  13. Slater determinant and partition function
  14. The canonical section in the determinant line
  15. A positive line bundle
  16. ...and 16 more sections

Key Result

Theorem 1

Given an initial data $s_{0 }\in \lambda_{0}\left(E\right)$ and some non-zero local normalisation data $s^{L}_{D}\in \lambda \left(L_{|D} \right)$, $s^{E}_{D}\in \lambda \left(E_{|D}\right)$, one can use $s_{D}$ to canonically associate, for each $p\in \mathbf{N}$, a non-zero element $s_{p}$ in $\la All the constants above are explicitly determined in eq:koff and eq:dbkb. In particular, where $\t

Theorems & Definitions (54)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Example 1.5
  • Definition 1.6
  • Proposition 1.7
  • ...and 44 more