Tunneling, the Quillen metric and analytic torsion for high powers of a holomorphic line bundle

TL;DR

This paper derives a formula for the distribution of small eigenvalues related to high tensor powers of a line bundle, connecting geometric analysis with quantum tunneling phenomena and providing new insights into the Quillen metric asymptotics.

Contribution

It introduces a novel formula for eigenvalue distribution in the context of high tensor powers of line bundles and offers a new proof for Quillen metric asymptotics.

Findings

Formula for eigenvalue distribution of Dolbeault Laplacians

New proof of Quillen metric asymptotics

Comparison with tunneling effects in physics

Abstract

Let L be a line bundle over a compact complex manifold X and endow L and TX with Hermitian metrics. Our main result provides a formula for the average distribution of the exponentially small eigenvalues of the corresponding Dolbeault Laplacians associated to high tensor powers of L; which in physics terminology is a measure of "tunneling" of the Dolbeault complex. Along the way a new proof of the asymptotics of the induced Quillen metric on the corresponding determinant line is obtained. A brief comparison with the tunneling effect for Witten Laplacians and large deviation principles for fermions is also made.