Asymptotics of zeta determinants of Laplacians on large degree abelian covers

TL;DR

This paper investigates the asymptotic behavior of zeta determinants of Laplacians on large degree abelian covers of a manifold, showing convergence results and extending to twisted Laplacians, with implications for quantum field theory.

Contribution

It establishes the convergence of normalized zeta determinants of Laplacians on large abelian covers and generalizes results to twisted Laplacians from flat bundles.

Findings

Normalized zeta determinants converge as the cover degree increases.

Results extend to twisted Laplacians on flat vector bundles.

Provides insights relevant to quantum field theory and random surfaces.

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_\zeta(\Delta_{N})}{\text{Vol}(M_N)} $ when $N\rightarrow +\infty$ where $\Delta_N$ 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 $M$.