Asymptotics of zeta determinants of Laplacians on large degree abelian 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 $N\to +\infty$, 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 $N\to +\infty$ 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 $M$.