arXiv subjects

Kiyoon Eum

Publications and source records attributed to Kiyoon Eum.

5 recordsLinked to original sources

arXiv2026

Moore--Read construction and explicit monodromies of Laughlin states on Riemann surfaces

We revisit the Moore--Read construction of the $\nu=1/k$ Laughlin states on compact Riemann surfaces, deriving their conformal blocks from $U(1)_k$ Chern--Simons theory through the CS/WZW correspondence. We compute their explicit monodromies under quasi-hole transport and Aharonov--Bohm flux insertion, and conjecture that these coincide with the corresponding adiabatic holonomies. Under this identification, we recover classical results on Laughlin states including the flux-averaged Hall conductance $1/k$ via a vector bundle of Laughlin states over $Jac(\Sigma)$. We also relate our construction to the recently developed algebro-geometric approach to higher genus Laughlin states.

math-ph↗

arXiv2025

Asymptotic expansion of the variation of the Quillen metric and its moment map interpretation

In K\"ahler geometry, the Donaldson--Fujiki moment map picture interprets the scalar curvature of a K\"ahler metric as a moment map on the space of compatible almost complex structures on a fixed symplectic manifold. In this paper, we generalize this picture using the framework of equivariant determinant line bundles. Given a prequantization $P=(L,h,\nabla)$ of a compact symplectic manifold $(M,\omega)$, let $\mathcal{G}=\mathrm{Aut}(P)$. For each $k\in\mathbb{N}$, we construct a $\mathcal{G}$-equivariant determinant line bundle $\lambda^{(k)}\rightarrow\mathcal{J}_{int}$ on the space of integrable compatible almost complex structures, equipped with the $\mathcal{G}$-invariant Quillen metric. The curvature form of $\lambda^{(k)}$ admits an asymptotic expansion whose coefficients yield a sequence of $\mathcal{G}$-invariant closed $2$-forms $\Omega_j$ on $\mathcal{J}_{int}$ and corresponding moment maps $\mu_j:\mathcal{J}_{int}\rightarrow C^\infty(M)$. Each $\mu_j$ arises from the asymptotic expansion of the variation of the logarithm of the Quillen metric with respect to K\"ahler potentials, with the complex structure held fixed. This provides a natural generalization of the Donaldson--Fujiki moment map interpretation of scalar curvature. Moreover, we show that $\mu_j$ coincide with the $Z$--critical equations introduced by Dervan--Hallam, and we state a generalization of Fujiki's fiber integral formula.

math.DG↗

arXiv2025

Partition functions of determinantal point processes on polarized K\"ahler manifolds

In this paper, we study the full asymptotic expansion of the partition functions of determinantal point processes defined on a polarized K\"ahler manifold. We show that the coefficients of the expansion are given by geometric functionals on K\"ahler metrics satisfying the cocycle identity, whose first variations can be expressed through the TYZ expansion coefficients of the Bergman kernel. In particular, these functionals naturally generalize the Mabuchi functional in K\"ahler geometry and the Liouville functional on Riemann surfaces. We further show that Futaki-type holomorphic invariants obstruct the existence of critical points of these geometric functionals, extending Lu's formula. We also verify that certain formulas remain valid up to the third coefficient without assuming polarization. Finally, we discuss the relation of our results to the quantum Hall effect (QHE), where the determinantal point process provides a microscopic model. In particular, we recover the higher-dimensional effective Chern-Simons actions derived in the physics literature and confirm a conjecture of Klevtsov on the form of the partition function asymptotics.

math.DG↗

arXiv2024

Limiting behavior of determinantal point processes associated with weighted Bergman kernels

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^n$, and let $\phi$ be a strictly plurisubharmonic function on $\Omega$. For each $k\in\mathbb{N}$, we consider determinantal point process $\Lambda_k$ with kernel $K_{k\phi}$, where $K_{k\phi}$ is the reproducing kernel of infinite dimensional weighted Bergman space $H(k\phi)$ with weight $e^{-k\phi}$. We show that the scaled cumulant generating function for $\Lambda_k$ converges as $k\rightarrow\infty$ to a certain limit, which can be explicitly expressed in terms of $\phi$ and a test function $u$. Note that we need to restrict the type of test function $u$ to those that are $\phi$-admissible.

math.CV↗

arXiv2024

Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^N$, and let $u$ be a continuous plurisubharmonic function on $\Omega$. We construct a sequence of Gaussian analytic functions $f_n$ on $\Omega$, associated with $u$, such that $\frac{1}{n}\log|f_n|$ converges to $u$ in $L^1_{loc}(\Omega)$ almost surely, as $n\rightarrow\infty$. Consequently, the normalized zero currents of $f_n$ converge weakly to $dd^c u$. More generally, for every $1\leq k\leq N$, we prove that the normalized currents of simultaneous zeros of $k$ independent copies of $f_n$ converge almost surely to the Bedford--Taylor product $(dd^c u)^k$. We also give a probabilistic proof of the well-known fact that normalized logarithms of the moduli of holomorphic functions are $L^1_{loc}$-dense in the space of plurisubharmonic functions on a pseudoconvex domain.

math.CV↗