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Shanzhong Sun

Publications and source records attributed to Shanzhong Sun.

At least 19 recordsLinked to original sources
arXiv2026

Two Regularized Determinants of Laplacian through Resurgence theory

We study two types of regularizations of the determinant of Laplacian on Riemann manifold from the viewpoint of resurgence theory. One is the formal logarithmic derivative of the determinant, and the other is its exponential deformation. Under appropriate conditions, the close formulas for both regularized determinant are established through Borel-Laplace resummation which takes into account the contribution of the singularities along the analytic continuation of Theta series $\hat{\Theta}_{D_X}$. The series resembles the trace of the heat kernel, but is defined via the spectrum of the square-root of the Laplacian. As applications, we revisit the well known formal logarithmic derivative of determinant on $S^1$ and compact Riemann surface with higher genus ($\geq2$) corresponding to the Poisson summation formula and Selberg trace formula respectively. Furthermore, the 1-Gevrey asymptotic behavior of the exponential deformation regularization at infinity is considered whose coefficients are determined by the trace of the heat kernel. In the end, we establish the relationship between the two regularized determinants. In fact, they have the same derivatives when the deformation parameter tends to $0$ in exponentially deformed regularization.

math-ph↗
arXiv2025

Degeneracy of Planar Central Configurations in the $N$-Body Problem

The degeneracy of central configurations in the planar $N$-body problem makes their enumeration problem hard and the related dynamics appealing. To truly understand the bifurcations of central configurations, we should work in the FULL configuration space which also facilitates the computer-aided methods. The degeneracy is always intertwined with the symmetry of the system of central configurations which makes the problem subtle. By analyzing the Jacobian matrix of the system, we systematically explore the direct method to single out trivial zero eigenvalues associated with translational, rotational and scaling symmetries, thereby isolating the non-trivial part of the Jacobian to study the degeneracy. Four distinct formulations of degeneracy are presented, each tailored to handle different forms of the system appeared in the literature. The method is applied to such well-known examples as Lagrange's equilateral triangle solutions for arbitrary masses, the square configuration for four equal masses and the equilateral triangle with a central mass revealing specific mass values for which degeneracy occurs. Combining with the interval algorithm, the nondegeneracy of rhombus central configurations for arbitrary mass is also established.

math.DS↗
arXiv2025

A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres

Let $X$ be a general Seifert fibered integral homology $3$-sphere with $r\ge3$ exceptional fibers. For every root of unity $ζ\not=1$, we show that the SU(2) WRT invariant of $X$ evaluated at $ζ$ is (up to an elementary factor) the non-tangential limit at $ζ$ of the GPPV invariant of $X$, thereby generalizing a result from [Andersen-Mistegard 2022]. Based on this result, we apply the quantum modularity results developed in [Han-Li-Sauzin-Sun 2023] to the GPPV invariant of $X$ to prove Witten's asymptotic expansion conjecture [Witten 1989] for the WRT invariant of $X$. We also prove that the GPPV invariant of $X$ induces a higher depth strong quantum modular form. Moreover, when suitably normalized, the GPPV invariant provides an ``analytic incarnation'' of the Habiro invariant.

math.CV↗
arXiv2025

Resurgence in the Universal Structures in B-model Topological String Theory

We propose a systematic analysis of Alim-Yau-Zhou's double scaling limit and Couso-Santamaría's large radius limit for the perturbative free energies in B-model topological string theory based on Écalle's Resurgence Theory. Taking advantage of the known resurgent properties of the formal solutions to the Airy equation and of the stability of resurgent series under exponential/logarithm and nonlinear changes of variable, we show how to rigorously derive the non-perturbative information from the perturbative one by means of alien calculus in this context, spelling out the notions of formal integral and Bridge Equation, typical of the resurgent approach to ordinary differential equations. We also discuss the Borel-Laplace summation of the obtained resurgent transseries, including a study of real analyticity based on the connection formulas stemming from the resummation of the Bridge Equation.

math-ph↗
arXiv2023

On the Uniqueness of Convex Central Configurations in the Planar $4$-Body Problem

In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem. Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its surrounding neighborhood.

math.DS↗
arXiv2022

A Weak $\infty$-Functor in Morse Theory

In the spirit of Morse homology initiated by Witten and Floer, we construct two $\infty$-categories $\mathcal{A}$ and $\mathcal{B}$. The weak one $\mathcal{A}$ comes out of the Morse-Samle pairs and their higher homotopies, and the strict one $\mathcal{B}$ concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak $\infty$-functor $\mathcal{F}: \mathcal{A}\rightarrow \mathcal{B}$. Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.

math.AT↗
arXiv2022

Resurgence and Partial Theta Series

We consider partial theta series associated with periodic sequences of coefficients, of the form $Θ(τ) := \sum_{n>0} n^νf(n) e^{iπn^2τ/M}$, with $ν$ non-negative integer and an $M$-periodic function $f : \mathbb{Z} \rightarrow \mathbb{C}$. Such a function is analytic in the half-plane $\{Im(τ)>0\}$ and as $τ$ tends non-tangentially to any $α\in\mathbb{Q}$, a formal power series appears in the asymptotic behaviour of $Θ(τ)$, depending on the parity of $ν$ and $f$. We discuss the summability and resurgence properties of these series by means of explicit formulas for their formal Borel transforms, and the consequences for the modularity properties of $Θ$, or its ``quantum modularity'' properties in the sense of Zagier's recent theory. The Discrete Fourier Transform of $f$ plays an unexpected role and leads to a number-theoretic analogue of Écalle's ``Bridge Equations''. The motto is: (quantum) modularity = Stokes phenomenon + Discrete Fourier Transform.

math.CV↗
arXiv2021

Elliptic fixed points with an invariant foliation: Some facts and more questions

We address the following question: let F:(R^2,0)->(R^2,0) be an analytic local diffeomorphism defined in the neighborhood of the non resonant elliptic fixed point 0 and let Φbe a formal conjugacy to a normal form N. Supposing F leaves invariant the foliation by circles centered at 0, what is the analytic nature of Φand N?

math.DS↗
arXiv2020

On the Moyal Star Product of Resurgent Series

We analyze the Moyal star product in deformation quantization from the resurgence theory perspective. By putting algebraic conditions on Borel transforms, one can define the space of ``algebro-resurgent series'' (a subspace of $1$-Gevrey formal series in $i\hbar/2$ with coefficients in $C\{q,p\}$), which we show is stable under Moyal star product.

math-ph↗
arXiv2020

Hadamard Product and Resurgence Theory

We discuss the analytic continuation of the Hadamard product of two holomorphic functions under assumptions pertaining to Ecalle's Resurgence Theory, proving that if both factors are endlessly continuable with prescribed sets of singular points $A$ and $B$, then so is their Hadamard product with respect to the set $\{0\}\cup A \cdot B$. In this generalization of the classical Hadamard Theorem, all the branches of the multivalued analytic continuation of the Hadamard product are considered.

math.CV↗
arXiv2020

Non-Embedding Theorems of Nilpotent Lie groups and Sub-Riemannian Manifolds

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the RCD(0,N), with N > 1. In fact, we can prove that a subRiemannian manifold whose generic degree of nonholonomy is not smaller than 2 can not be biLipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the CD(K,N) with N > 1. We also prove that the subRiemannian manifold is infinitesimally Hilbert space.

math.DG↗
arXiv2018

The Baker-Campbell-Hausdorff formula via mould calculus

The well-known Baker-Campbell-Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product e X e Y can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction t{ó} Ecalle's mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin explicit formula [Dy47] for the logarithm, as well as another formula recently obtained by T. Kimura [Ki17] for the product of exponentials itself. We also analyse the relation between the two formulas and indicate their mould calculus generalization to a product of more exponentials.

math.RA↗
arXiv2016

Gutzwiller's Semiclassical Trace Formula and Maslov-Type Index Theory for Symplectic Paths

Gutzwiller's famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with an view towards the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley-Zhender index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley-Zehnder's index theory which we will recall all essential ingredients is the Maslov-type index theory for symplectic paths developed by Long and his collaborators which would shed new light on the computations and understandings on the semiclassical trace formula. The insights in Gutzwiller's work also seems plausible to the studies on Hamiltonian systems.

math.DS↗
arXiv2015

A Double Poisson Algebra Structure on Fukaya Categories

Let $M$ be an exact symplectic manifold with $c_1(M)=0$. Denote by $\mathrm{Fuk}(M)$ the Fukaya category of $M$. We show that the dual space of the bar construction of $\mathrm{Fuk}(M)$ has a differential graded noncommutative Poisson structure. As a corollary we get a Lie algebra structure on the cyclic cohomology $\mathrm{HC}^\bullet(\mathrm{Fuk}(M))$, which is analogous to the ones discovered by Kontsevich in noncommutative symplectic geometry and by Chas and Sullivan in string topology.

math.SG↗
arXiv2014

Linear stability of elliptic Lagrangian solutions of the planar three-body problem via index theory

It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter $\bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2\in [0,9]$ and the eccentricity $e\in [0,1)$. We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle $[0,9]\times [0,1)$, besides perturbation methods for $e>0$ small enough, blow-up techniques for $e$ sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full $(\bb,e)$ range $[0,9]\times [0,1)$ via the $\om$-index theory of symplectic paths for $\om$ belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the $\om$-index decreasing property of the solutions in $\bb$ for fixed $e\in [0,1)$, we prove the existence of three curves located from left to right in the rectangle $[0,9]\times [0,1)$, among which two are -1 degeneracy curves and the third one is the right envelop curve of the $\om$-degeneracy curves for $\om\not=1$, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter $(\bb,e)$ passes through each of these three curves. Interesting symmetries of these curves are also observed. The singular case when the eccentricity $e$ approaches to 1 is also analyzed in details concerning the linear stability.

math.DS↗
arXiv2012

Cyclic Homology of Fukaya Categories and the Linearized Contact Homology

Let $M$ be an exact symplectic manifold with contact type boundary such that $c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya category of $M$ has the structure of an involutive Lie bialgebra. Inspired by a work of Cieliebak-Latschev we show that there is a Lie bialgebra homomorphism from the linearized contact homology of $M$ to the cyclic cohomology of the Fukaya category. Our study is also motivated by string topology and 2-dimensional topological conformal field theory.

math.SG↗