Asymptotics of zeta determinants of Laplacians on large degree abelian covers

TL;DR

The paper investigates the asymptotics of zeta determinants of Laplacians on towers of large-degree abelian covers and establishes the existence of a density limit for log det_ζ(Δ_N) normalized by the volume. It develops a robust strategy based on heat-kernel identities, Li–Yau bounds, and a decomposition into twisted Laplacians via a fiberwise Fourier transform, then passes to a periodic infinite cover M_∞ to express the limit in terms of the heat kernel on the limit space. The main contributions include a precise limit formula, uniform spectral-gap control via Kato perturbation theory, and the extension to twisted Bochner Laplacians on flat vector bundles, with concrete examples illustrating nonzero limits in the torus and hyperbolic settings. These results connect spectral invariants of large geometric structures to the local geometry of the periodic limit, providing insights relevant to quantum field theory and spectral geometry on noncompact covers.

Abstract

Let $(M,g)$ be some smooth, closed, compact Riemannian manifold and $(M_N\mapsto M)_N$ be an increasing sequence of large degree cyclic covers of $M$ that converges when $N\rightarrow +\infty$, in a suitable sense, to some limit $\mathbb{Z}^p$ cover $M_\infty$ over $M$. Motivated by recent works on zeta determinants on random surfaces and some natural questions in Euclidean quantum field theory, we show the convergence of the sequence $ \frac{\log\det_ζ(Δ_{N})}{\text{Vol}(M_N)} $ when $N\rightarrow +\infty$ where $Δ_N$ is the Laplace-Beltrami operator on $M_N$. We also generalize our results to the case of twisted Laplacians coming from certain flat unitary vector bundles over $M$.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2: Kirchhoff, Sarnak-moser, Kirchhoff
• Lemma 2.1: Milnor-Schwarz type Lemma
• Theorem 2.1: Li-Yau heat kernel bounds
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7