arXiv subjects

Siarhei Finski

Publications and source records attributed to Siarhei Finski.

24 records · Page 2Linked to original sources

arXiv2020

Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion

We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a half-translation surface endowed with a flat unitary vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the sum of cycle-rooted spanning forests weighted by the monodromy of the connection of the unitary vector bundle, to the corresponding zeta-regularized determinants. As one application, by combining our result with a recent work of Kassel-Kenyon, modulo some universal topological constants, we give an explicit formula for the limit of the probability that a cycle-rooted spanning forest with non-contractible loops, sampled uniformly on discretizations approaching a given surface, induces the given lamination by its cycles. We also calculate an explicit value for the limit of certain topological observables on the associated loop measures.

math.PR↗

arXiv2020

Finite difference method on flat surfaces with a flat unitary vector bundle

We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with von Neumann boundary conditions. As an interesting byproduct of our study, we obtain Harnack-type estimates on "almost harmonic" discrete functions, defined on the graphs, which approximate a given surface. The results of this paper will be later used to relate the asymptotic expansion of the number of spanning trees, spanning forests and weighted cycle-rooted spanning forests on the discretizations to the corresponding zeta-regularized determinants.

math.DG↗

arXiv2019

Quillen metric for singular families of Riemann surfaces with cusps and compact perturbation theorem

We study the behavior of the Quillen metric for the family of Riemann surfaces with cusps when the additional cusps are created by degeneration. More precisely, in our previous paper, we've seen that the renormalization of the Quillen metric associated with a family of Riemann surfaces with cusps extends continuously over the locus of singular curves. The main result of this article shows that, modulo some explicit universal constant, this continuous extension coincides with the Quillen metric of the normalization of singular curves. This result shows that the Quillen metric is compatible with the adjunction of cusps. When this theorem is applied directly to the moduli space of curves, we obtain the compatibility of the Quillen metric with clutching morphisms in the moduli space of pointed stable curves. As one application, we obtain the compatibility between our definition of the analytic torsion and the definition of Takhtajan-Zograf using lengths of closed geodesics. As a consequence of the proof of the main theorem, we get an explicit relation in terms of Bott-Chern forms between the Quillen metric associated with a cusped metric and the Quillen metric associated with a metric on the compactified Riemann surface. This refines relative compact perturbation theorem we obtained before by pinning down the universal constant.

math.DG↗

arXiv2018

Analytic torsion for surfaces with cusps II. Regularity, asymptotics and curvature theorem

In this article we study the Quillen norm on the determinant line bundle associated with a family of complex curves with cusps, which admit singular fibers. More precisely, we fix a family of complex curves $π: X \to S$, which admit at most double-point singularities. Let $(ξ, h^ξ)$ be a holomorphic Hermitian vector bundle over $X$. Let $σ_1, \ldots, σ_m : S \to X$ be disjoint holomorphic sections. We denote the divisor $D_{X/S} := \rm{Im}(σ_1) + \cdots + \rm{Im}(σ_m)$, and endow the relative canonical line bundle $ω_{X/S}$ with a Hermitian norm such that its restriction at each fiber of $π$ induces Kähler metric with hyperbolic cusps. This Hermitian norm induces the Hermitian norm on the twisted relative canonical line bundle $ω_{X/S}(D) := ω_{X/S} \otimes \mathscr{O}_{X}(D_{X/S})$. For $n \leq 0$, we study the determinant line bundle $λ(j^*(ξ\otimes ω_{X/S}(D)^n))$. We endow it with the Quillen norm by using the analytic torsion from the first paper of this series. Then we study the regularity of this Quillen norm and its asymptotics near the locus of singular curves. The singular terms of the asymptotics turn out to be reasonable enough, so that the curvature of $λ(j^*(ξ\otimes ω_{X/S}(D)^n))$ is well-defined as a current over $S$. We derive the explicit formula for this current, which gives a refinement of Riemann-Roch-Grothendieck theorem at the level of currents. This generalizes the curvature formulas of Takhtajan-Zograf and Bismut-Bost. As a consequence of our study, we also get some regularity results on the Weil-Petersson form over the moduli space of pointed curves, which are enough to conclude the well-known fact, originally due to Wolpert, that the Weil-Petersson volume of the moduli space of pointed stable curves is a rational multiple of a power of $π$.

math.DG↗

arXiv2018

Analytic torsion for surfaces with cusps I. Compact perturbation theorem and anomaly formula

Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincaré metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted canonical line bundle $ω_M(D)$ with the induced Hermitian norm $\|\cdot\|_M$ over $\overline{M} \setminus D_M$. Let $(ξ, h^ξ)$ be a holomorphic Hermitian vector bundle over $\overline{M}$. In this article we define the analytic torsion $T(g^{TM}, h^ξ \otimes \|\cdot\|_M^{2n})$ associated with $(M, g^{TM})$ and $(ξ\otimes ω_M(D)^n, h^ξ \otimes \|\cdot\|_M^{2n})$ for $n \leq 0$. We prove that $T(g^{TM}, h^ξ \otimes \|\cdot\|_M^{2n})$ is related to the analytic torsion of non-cusped surfaces. Then we prove the anomaly formula for the associated Quillen norm. The results of this paper will be used in the sequel to study the regularity of the Quillen norm and its asymptotics in a degenerating family of Riemann surfaces with cusps and to prove the curvature theorem. We also prove that our definition of the analytic torsion for hyperbolic surfaces is compatible with the one obtained through Selberg trace formula by Takhtajan-Zograf.

math.DG↗

arXiv2018

On the full asymptotic of analytic torsion

The purpose of this article is to study the asymptotic expansion of Ray-Singer analytic tosion associated with increasing powers p of a given positive line bundle. Here we prove that the asymptotic expansion associated to a manifold contains only the terms of the form $p^{n-i} \log p, p^{n-i}$ for $i$-natural. For the two leading terms it was proved by Bismut and Vasserot in 1989. We will calculate the coefficients of the terms $p^{n-1} \log p, p^{n-1}$ in the Kahler case and thus answer the question posed in the recent work of Klevtsov, Ma, Marinescu and Wiegmann about quantuum Hall effect. Our second result concerns the general asymptotic expansion of Ray-Singer analytic torsion for an orbifold.

math.DG↗