arXiv subjects

Xinxing Tang

Publications and source records attributed to Xinxing Tang.

9 recordsLinked to original sources

arXiv2026

Heat Kernel and Resurgence

We study the resurgent structure of short-time heat kernel asymptotics from the viewpoint of Picard-Lefschetz theory. For a real analytic Riemannian manifold, we show the heat kernel admits a 1-Gevrey small-time expansion whose Borel transform detects complex-geometric data beyond the real geodesic sector. We formulate an infinite-dimensional Picard-Lefschetz problem of Morse-Floer type for the holomorphic energy functional on the complexified path space, and propose a heat-kernel analogue of the Picard-Lefschetz/Alien correspondence. In this framework, pointed alien operators acting on the asymptotic expansion associated with the real geodesic are predicted to produce the formal heat-kernel sectors associated with other holomorphic geodesics, with coefficients given by signed counts of connecting trajectories of the Morse flow. We perform a confirming test of this proposal on the hyperbolic plane $H^2$.

math-ph↗

arXiv2026

Picard-Lefschetz theory and alien calculus: a case study

We compare Picard--Lefschetz theory and resurgence in three basic one-dimensional exponential integrals: the Airy model, the Bessel model, and the Gamma model. On the Picard--Lefschetz side, we describe the Lefschetz thimbles and compute the connecting trajectories between critical points appearing at Stokes phases. On the resurgent side, we analyze the Borel singularities of the saddle expansions and use alien operators to recover the same Stokes coefficients. These examples serve as explicit finite-dimensional test cases for the dictionary between thimble wall-crossing and alien calculus.

math-ph↗

arXiv2026

Path homology of circulant digraphs

We organize and extend a set of computations and structural observations about the Grigoryan--Lin--Muranov--Yau (GLMY) path complex of circulant digraphs $\vec{C}_n^S$ and circulant graphs $C_n^S$. Using the shift automorphism $\tau$ and a Fourier decomposition, we reduce many rank computations for the GLMY boundary maps to finite-dimensional $\tau$-eigenspaces. This provides a reusable "symbol-matrix" recipe that highlights (i) the dependence on prime versus composite $n$ and (ii) stability phenomena for certain natural choices of connection sets $S$. Several fully worked examples are included, together with a discussion of how the additive structure of $S$ governs low-dimensional chains and Betti numbers.

math.CO↗

arXiv2025

Contact Term Algebras and Dijkgraaf's Master Equation

This paper is devoted to study integrable deformations of chiral conformal field theories on elliptic curves from the viewpoint of contact algebra. We introduce the relevant integrable condition within the framework of conformal vertex algebra, and derive the contact term relations among certain local operators. We investigate three versions of genus one partition functions and derive the contact equations. This leads to a rigorous formulation of Dijkgraaf's master equation \cite{Dijk1996master} for chiral deformations.

math.QA↗

arXiv2024

The Cellular Homology of Digraphs

In \cite{TY}, we investigate the pair $(P, \Supp(P))$ of minimal path $P$ and its supporting sub-digraph $\Supp(P)$ in the path complex of a digraph $G$ under the strongly regular condition. In this paper, first, we consider the special minimal path $P$ specified by the admissible condition (Definition \ref{admpair}), which means that $(P,\Supp(P))$ admits a singular cubical realization. Based on such a subset, we systematically introduce the definitions of cellular chain complex associated to $G$ and prove the well-definedness. Then we study several properties of such cellular homologies. Finally, we present several intriguing examples as well as some important observations.

math.CO↗

arXiv2022

Minimal Path and Acyclic Models in the Path Complex

In this paper, firstly, we will study the structure of the path complex $(\Omega_*(G;\Z),\partial)$ of a digraph $G$ via the $\Z$-generators of $\Omega_*(G,\Z)$ under strongly regular condition, which is called the minimal path in \cite{HY}. In particular, we will study various examples of the minimal $3$-paths. Secondly, we will show that the supporting sub-digraph of minimal path has acyclic path homologies. Thirdly, we will consider the applications of such an acyclic model.

math.AT↗

arXiv2022

Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and $tt^*$ Structures

The aim of this paper is to rigorously establish the Calabi-Yau/Landau-Ginzburg (CY/LG) correspondence for the $tt^*$ geometry structure--a generalized version of variation of Hodge structures. Although it is well-known that there exists a map between Hodge structures on the LG and CY's sides that preserves the Hodge filtration and bilinear form, it remains unclear whether the real structures are also preserved. In our paper, we conduct a detailed analysis of two period integrals on the LG's side. Based on this analysis, we modify the real structure proposed by Cecotti on LG's side, and show that the aforementioned map is also preserved under the modified real structure. As a result, we establish full CY/LG correspondence for $tt^*$ structures.

math-ph↗

arXiv2019

Dispersionless Integrable Hierarchy via Kodaira-Spencer Gravity

We explain how dispersionless integrable hierarchy in 2d topological field theory arises from the Kodaira-Spencer gravity (BCOV theory). The infinitely many commuting Hamiltonians are given by the current observables associated to the infinite abelian symmetries of the Kodaira-Spencer gravity. We describe a BV framework of effective field theories that leads to the B-model interpretation of dispersionless integrable hierarchy.

math-ph↗

arXiv2017

$tt^*$ Geometry, Singularity Torsion and Anomaly Formulas

This paper is concerned with the Schr\"odinger operators $\Delta_{f_0}$ and $\Delta_f$ attached to a pair $(\mathbb{C}^n, f_0)$ and its deformation $(\mathbb{C}^n, f)$, where $f_0$ is a non-degenerate and quasi-homogeneous polynomial on $\mathbb{C}^n$ and $f$ is its relevant or marginal deformation. We give the $tt^*$ geometry structure on the Hodge bundle associated to $\Delta_f$, which describes the genus 0 anomaly. Next we study the corresponding singularity torsion type invariants and give the anomaly formulas for the 2nd torsion type invariant.

math-ph↗