# On the full asymptotic of analytic torsion

## TL;DR

This paper investigates the detailed asymptotic expansion of Ray-Singer analytic torsion for high tensor powers of line bundles, providing explicit coefficients in the Kahler case and extending results to orbifolds, with applications to quantum Hall effect.

## Contribution

It derives the full asymptotic expansion of analytic torsion, including explicit coefficients in the Kahler case and extends the analysis to orbifolds, answering recent open questions.

## Key findings

- Asymptotic expansion contains only specific powers and logs of p.
- Explicit calculation of coefficients for leading terms in Kahler case.
- Extension of asymptotic analysis to orbifolds.

## Abstract

The purpose of this article is to study the asymptotic expansion of Ray-Singer analytic tosion associated with increasing powers p of a given positive line bundle. Here we prove that the asymptotic expansion associated to a manifold contains only the terms of the form $p^{n-i} \log p, p^{n-i}$ for $i$-natural. For the two leading terms it was proved by Bismut and Vasserot in 1989. We will calculate the coefficients of the terms $p^{n-1} \log p, p^{n-1}$ in the Kahler case and thus answer the question posed in the recent work of Klevtsov, Ma, Marinescu and Wiegmann about quantuum Hall effect. Our second result concerns the general asymptotic expansion of Ray-Singer analytic torsion for an orbifold.