Geometric Zabrodin-Wiegmann conjecture for integer Quantum Hall states
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911241674424320 |
|---|---|
| author | Shen, Shu Yu, Jianqing |
| author_facet | Shen, Shu Yu, Jianqing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10530 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Zabrodin-Wiegmann conjecture for integer Quantum Hall states Shen, Shu Yu, Jianqing Differential Geometry Mathematical Physics 58J20, 58J52, 81V70, 14H81 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. |
| title | Geometric Zabrodin-Wiegmann conjecture for integer Quantum Hall states |
| topic | Differential Geometry Mathematical Physics 58J20, 58J52, 81V70, 14H81 |
| url | https://arxiv.org/abs/2501.10530 |