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arXiv · 2505.01586

Asymptotics of zeta determinants of Laplacians on large degree abelian covers

Nguyen Viet Dang·Jiasheng Lin·Frédéric Naud
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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 $M$.

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Nguyen Viet Dang, Jiasheng Lin, Frédéric Naud. 2026-01-12. Asymptotics of zeta determinants of Laplacians on large degree abelian covers. https://arxiv.org/abs/2505.01586

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