arXiv subjects

Jeehoon Park

Publications and source records attributed to Jeehoon Park.

At least 19 recordsLinked to original sources

arXiv2026

A perturbative algorithm for flat F-manifolds associated with Landau-Ginzburg models

We develop a perturbative algorithm for constructing formal flat $F$-manifold structures on the cohomologies of dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebras associated with Landau-Ginzburg models. As an application, this approach provides a perturbative construction of formal flat $F$-manifold structures on two important objects: the Jacobian algebra of a homogeneous polynomial with an isolated singularity at the origin, and the primitive cohomology of smooth projective Calabi-Yau complete intersections.

math.AG↗

arXiv2026

Arithmetic BF theory and the Cassels-Tate pairing

We give a systematic treatment of the arithmetic BF theory, introduced by Carlson and Kim. We observe that the Cassels-Tate pairing can be naturally interpreted as an arithmetic BF functional.

math.NT↗

arXiv2025

Bounded Geometries on Hybrid Landau-Ginzburg models of Calabi-Yau complete intersections and $L^2$-Hodge Theory

Given a Calabi-Yau smooth projective complete intersection variety $V$ over $\mathbb{C}$, a hybrid Landau-Ginzburg (LG) model may be associated using the Cayley trick. This hybrid LG model comprises a non-compact Calabi-Yau manifold $X_{CY}$, and a holomorphic function $W$, defined on $X_{CY}$, such that the critical locus of $W$ is isomorphic to $V$. We construct a complete K\"ahler metric $\mathfrak{g}$ and a bounded Calabi-Yau volume form ${\Omega}$ on $X_{CY}$ such that $(X_{CY},\mathfrak{g}, {\Omega})$ is a bounded Calabi-Yau geometry (in fact, $(X_{CY},\mathfrak{g})$ is an asymptotically conical manifold) and the function $W$ is strongly elliptic; this enables us to apply the $L^2$-Hodge theory of Li-Wen \cite{LW} to $(X_{CY},\mathfrak{g}, {\Omega})$ and $W$, which leads to a Frobenius manifold structure on the twisted de Rham cohomology associated to $(X_{CY},W)$. Furthermore, we prove that this twisted de Rham cohomology is isomorphic to the de Rham cohomology $H(V;\mathbb{C})$, which results in a new $L^2$-Hodge theoretic construction of a Frobenius manifold structure on $H(V;\mathbb{C})$. This paper provides the first explicit geometric verification of Li-Wen's theory for genuine non-isolated, compact critical loci using hybrid Landau-Ginzburg models.

math.AG↗

arXiv2025

$L^2$-Hodge theoretic construction of Frobenius manifolds for Calabi-Yau smooth projective hypersurfaces

We provide a new $L^2$-Hodge theoretic construction of a Frobenius manifold structure on the cohomology of a Calabi-Yau smooth projective hypersurface $V$, using Li-Wen's $L^2$-Hodge theory [9] of a Landau-Ginzburg model with compact critical locus $V$. We also give a precise comparison result between the current construction and Barannikov-Kontsevich's construction [2] of the Frobenius manifold structure on the cohomology of $V$.

math.AG↗

arXiv2025

Comparison of Frobenius algebra structures on Calabi-Yau toric hypersurfaces

We establish an isomorphism between two Frobenius algebra structures, termed CY and LG, on the primitive cohomology of a smooth Calabi--Yau hypersurface in a simplicial Gorenstein toric Fano variety. As an application of our comparison isomorphism, we observe the existence of a Frobenius manifold structure on a finite-dimensional subalgebra of the Jacobian algebra of a homogeneous polynomial which may exhibit a non-compact singularity locus.

math.AG↗

arXiv2024

Algebraic description of complex conjugation on cohomology of a smooth projective hypersurface

We describe complex conjugation on the primitive middle-dimensional algebraic de Rham cohomology of a smooth projective hypersurface defined over a number field that admits a real embedding. We use Griffiths' description of the cohomology in terms of a Jacobian ring. The resulting description is algebraic up to transcendental factors explicitly given by certain periods.

math.AG↗

arXiv2023

Entanglement entropies in the abelian arithmetic Chern-Simons theory

The notion of {\em entanglement entropy} in quantum mechanical systems is an important quantity, which measures how much a physical state is entangled in a composite system. Mathematically, it measures how much the state vector is not decomposable as elements in the tensor product of two Hilbert spaces. In this paper, we seek its arithmetic avatar: the theory of arithmetic Chern-Simons theory with finite gauge group $G$ naturally associates a state vector inside the product of two quantum Hilbert spaces and we provide a formula for the {\em von Neumann entanglement entropy} of such state vector when $G$ is a cyclic group of prime order.

math.NT↗

arXiv2023

Twisted de Rham complex for toric Calabi-Yau complete intersections and flat $F$-manifold structures

We describe the primitive middle-dimensional cohomology $\mathbb{H}$ of a compact simplicial toric complete intersection variety in terms of a twisted de Rham complex. Then this enables us to construct a concrete algorithm of formal flat $F$-manifold structures on $\mathbb{H}$ in the Calabi-Yau case by using the techniques of \cite{Park23}, which turn the twisted de Rham complex into a quantization dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra and seek for an algorithmic solution to an associated \textit{weak primitive form.}

math.AG↗

arXiv2023

BF path integrals for elliptic curves and $p$-adic $L$-functions

We prove an arithmetic path integral formula for the inverse $p$-adic absolute values of the $p$-adic $L$-functions of elliptic curves over the rational numbers with good ordinary reduction at an odd prime $p$ based on the Iwasawa main conjecture and Mazur's control theorem. This is an elliptic curve analogue of \cite{CCKKPY}.

math.NT↗

arXiv2023

Modular Symbols with Values in Beilinson-Kato Distributions

For each integer $n\geq 1$, we construct a $\operatorname{GL}_n(\mathbb Q)$-invariant modular symbol $\bm\xi_n$ with coefficients in a space of distributions that takes values in the Milnor $K_n$-group of the modular function field. The Siegel distribution $\bm\mu$ on $\mathbb Q^2$, with values in the modular function field, serves as the building block for $\bm\xi_n$; we define $\bm\xi_n$ essentially by taking the $n$-Steinberg product of $\bm\mu$. The most non-trivial part of this construction is the cocycle property of $\bm\xi_n$; we prove it by using an induction on $n$ based on the first two cases $\bm\xi_1$ and $\bm\xi_2$; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor $K_2$-group modulo torsion satisfy the Manin relations.

math.NT↗

arXiv2023

Arithmetic Chern-Simons theory with real places

The goal of this paper is two-fold: we generalize the arithmetic Chern-Simons theory over totally imaginary number fields studied in [Kim15, CKK+16] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern-Simons invariant with coefficient $\mathbb{Z}/n\mathbb{Z}$ $(n \geq 2)$ associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [Mil06]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient $\mathbb{Z}/2\mathbb{Z}$ in [CKK+16] and the abelian cyclic gauge group with coefficient $\mathbb{Z}/n\mathbb{Z}$ in [BCG+18]. Our non-trivial examples (with non-abelian gauge group and general coefficient $\mathbb{Z}/n\mathbb{Z}$) will be given by a simple twisting argument based on examples of [BCG+18].

math.NT↗

arXiv2022

Charge-Density-Wave Proximity Effects in Graphene

Certain layered transition metal dichalcogenides (TMDCs), such as 1T-TaS2, show a rich collection of charge density wave (CDW) phases at different temperatures, and their atomic structures and electron conductions have been widely studied. However, the properties of CDW systems that are integrated with other electronic materials have not yet been investigated. Here, we incorporate the CDW properties of TMDCs into the electronic transport of graphene for the first time. During CDW phase transitions, anomalous transport behaviors that are closely related to the formation of correlated disorder in TMDCs were observed in the graphene sample used in this study. In particular, the commensurate CDW phase forms a periodic charge distribution with potential fluctuations, and thus constitutes correlated charged impurities, which decreases resistivity and enhances carrier mobility in graphene. The CDW-graphene heterostructure system demonstrated here paves the way to controlling the temperature-dependent carrier mobility and resistivity of graphene and to developing novel functional electronic devices such as graphene-based sensors and memory devices.

cond-mat.mes-hall↗

arXiv2022

Path Integrals and p-adic L-functions

We prove an arithmetic path integral formula for the inverse p-adic absolute values of the Kubota-Leopoldt p-adic L-functions at roots of unity.

math.NT↗

arXiv2021

Batalin-Vilkovisky formalism in the $p$-adic Dwork theory

The goal of this article is to develop BV (Batalin-Vilkovisky) formalism in the $p$-adic Dwork theory. Based on this formalism, we explicitly construct a $p$-adic dGBV algebra (differential Gerstenhaber-Batalin-Vilkovisky algebra) for a smooth projective complete intersection variety $X$ over a finite field, whose cohomology gives the $p$-adic Dwork cohomology of $X$, and its cochain endomorphism (the $p$-adic Dwork Frobenius operator) which encodes the information of the zeta function $X$. As a consequence, we give a modern deformation theoretic interpretation of Dwork's theory of the zeta function of $X$ and derive a formula for the $p$-adic Dwork Frobenius operator in terms of homotopy Lie morphisms and the Bell polynomials.

math.NT↗

arXiv2021

A basis of algebraic de Rham cohomology of complete intersections over a characteristic zero field

Let $\Bbbk$ be a field of characteristic 0. Let $X$ be a smooth complete intersection over $k$ of dimension $n-k$ in the projective space $\mathbf{P}^n_{k}$, for given positive integers $n$ and $k$. When $k=\mathbb{C}$, Terasoma (\cite{Ter90}) and Konno (\cite{Ko91}) provided an explicit representative (in terms of differential forms) of a basis for the primitive middle-dimensional algebraic de Rham cohomology $H_{dR,\operatorname{prim}}^{n-k}(X;\mathbb{C})$. Later Dimca constructed another explicit representative of a basis of $H_{dR,\operatorname{prim}}^{n-k}(X;\mathbb{C})$ in \cite{Dim95}. Moreover, he proved that his representative gives the same cohomology class as the previous representative of Terasoma and Konno. The goal of this article is to examine the above two different approaches without assuming that $k=\mathbb{C}$ and provide a similar comparison result for any field $k$. Dimca's argument depends heavily on the condition $k=\mathbb{C}$ and our idea is to find appropriate Cech-de Rham complexes and spectral sequences corresponding to those two approaches, which work without restrictions on $k$.

math.AG↗

arXiv2021

Polynomial realizations of period matrices of projective smooth complete intersections and their deformation

Let $X$ be a smooth complete intersection over $\mathbb{C}$ of dimension $n-k$ in the projective space $\mathbf{P}^n_{\mathbb{C}}$, for given positive integers $n$ and $k$. For a given integral homology cycle $[γ] \in H_{n-k}(X(\mathbb{C}),\mathbb{Z})$, the period integral is defined to be a linear map from the de Rham cohomology group to $\mathbb{C}$ given by $[ω] \mapsto \int_γω$. The goal of this article is to interpret this period integral as a linear map from the polynomial ring with $n+k+1$ variables to $\mathbb{C}$ and use this interpretation to develop a deformation theory of period integrals of $X$. The period matrix is an invariant defined by the period integrals of the \textit{rational} de Rham cohomology, which compares the \textit{rational} structures ($\mathbb{Q}$-subspace structures) of the de Rham cohomology over $\mathbb{C}$ and the singular homology with coefficient $\mathbb{C}$. As a main result, when $X'$ is another projective smooth complete intersection variety deformed from $X$, we provide an explicit formula for the period matrix of $X'$ in terms of the period matrix of $X$ and the Bell polynomials evaluated at the deformation data. Our result can be thought of as a modern deformation theoretic treatment of the period integrals based on the Maurer-Cartan equation of a dgla (differential graded Lie algebra).

math.AG↗

arXiv2020

Smooth projective Calabi-Yau complete intersections and algorithms for their Frobenius manifolds and higher residue pairings

The goal of this article is to provide an explicit algorithmic construction of formal $F$-manifold structures, formal Frobenius manifold structures, and higher residue pairings on the primitive middle-dimensional cohomology $\mathbb{H}$ of a smooth projective Calabi-Yau complete intersection variety $X$ defined by homogeneous polynomials $G_1(\underline x), \dots, G_k(\underline x)$. Our main method is to analyze a certain dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra $\mathcal{A}$ obtained from the twisted de Rham complex which computes $\mathbb{H}$. More explicitly, we introduce a notion of \textit{a weak primitive form} associated to a solution of the Maurer-Cartan equation of $\mathcal{A}$ and the Gauss-Manin connection, which is a weakened version of Saito's primitive form (\cite{Saito}). In addition, we provide an explicit algorithm for a weak primitive form based on the Gröbner basis in order to achieve our goal. Our approach through the weak primitive form can be viewed as a unifying link (based on Witten's gauged linear sigma model, \cite{W93}) between the Barannikov-Kontsevich's approach to Frobenius manifolds via dGBV algebras (non-linear topological sigma model, \cite{BK}) and Saito's approach to Frobenius manifolds via primitive forms and higher residue pairings (Landau-Ginzburg model, \cite{ST}).

math.AG↗

arXiv2019

An algorithm for a Massey triple product of a smooth projective plane curve

We provide an explicit algorithm to compute a Massey triple product relative to a defining system for a smooth projective plane curve $X$ defined by a homogeneous polynomial $G(\underline x)$ over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for $H^1(X,\mathbb{C})$ in terms of the multiplications inside the Jacobian ring of $G(\underline x)$ and the Cech-deRham complex of $X$. Our algorithm gives a criterion whether a Massey triple product vanishes or not in $H^2(X)$ under a particular non-trivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in $H^2(X)$ of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program.

math.AG↗