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Minh Lam Nguyen

Publications and source records attributed to Minh Lam Nguyen.

7 recordsLinked to original sources
arXiv2026

On the Existence of Solutions to the Seiberg-Witten Vortex Equations with Exponential Decay on the Plane

Clifford Taubes showed that the moduli space of the variational equations of the Yang-Mills-Higgs functional on the plane is non-empty, and its elements correspond to "vortices". Inspired by this result, in this paper, we completely characterize exponential decay solutions to the Seiberg-Witten vortex equations, where the equations are derived by applying a Hitchin-type dimensional reduction. We show that the existence of Seiberg-Witten ``vortices'' is not provided by the exponential decay solutions. This is in contrast to the Yang-Mill-Higgs vortex equations.

math.AP↗
arXiv2025

A Fueter operator for 3/2-spinors

We show the non-compactness of moduli space of solutions of the monopole equations for $3/2$-spinors on a closed 3-manifold is equivalent to the existence of `3/2-Fueter sections' that are solutions of an overdetermined non-linear elliptic differential equation. These are sections of a fiber bundle whose fiber is a special 4-dimensional submanifold of the hyperkähler manifold of center-framed charged one $SU(2)$-instantons on $\mathbf{R}^4$. This fiber bundle does not inherit a hyperkähler structure.

math.DG↗
arXiv2024

Spectral invariants and equivariant monopole Floer homology for rational homology three-spheres

In this paper, we study a model for $S^1$-equivariant monopole Floer homology for rational homology three-spheres via a homological device called $\mathcal{S}$-complex. Using the Chern-Simons-Dirac functional, we define an $\mathbf{R}$-filtration on the (equivariant) complex of monopole Floer homology $HM$. This $\mathbf{R}$-filtration fits $HM$ into a persistent homology theory, from which one can define a numerical quantity called the spectral invariant $ρ$. The spectral invariant $ρ$ is tied with the geometry of the underlying manifold. The main result of the papers shows that $ρ$ provides an obstruction to the existence of positive scalar curvature metric on a ribbon homology cobordism.

math.GT↗
arXiv2023

The three-dimensional Seiberg-Witten equations for 3/2-spinors: a compactness theorem

The Rarita-Schwinger-Seiberg-Witten (RS-SW) equations are defined similarly to the classical Seiberg-Witten equations, where a geometric non-Dirac-type operator replaces the Dirac operator called the Rarita-Schwinger operator. In dimension four, the RS-SW equation was first considered by the second-named author. The variational approach will also give us a three-dimensional version of the equations. The RS-SW equations share some features with the multiple-spinor Seiberg-Witten equations, where the moduli space of solutions could be non-compact. In this note, we prove a compactness theorem regarding the moduli space of solutions of the RS-SW equations defined on 3-manifolds.

math.GT↗
arXiv2023

Pin(2)-equivariance of the Rarita-Schwinger-Seiberg-Witten Equations

We define a variant of the Seiberg-Witten equations using the Rarita-Schwinger operators for closed simply connected spin smooth 4-manifold X. The moduli space of solutions to the system of non-linear differential equations consist of harmonic 3/2-spinors and connections satisfying certain curvature condition. Beside having an obvious U(1)-symmetry, these equations also have a symmetry by Pin(2). We exploit this additional symmetry to perform finite dimensional approximations for the eigenvalue problem of the 3/2-monopole map and show that under a certain topological assumption, the moduli space of solutions is always non-compact, and thus non-empty.

math.DG↗
arXiv2023

An abelian gauge-theoretic variant of the Seiberg-Witten equations for multiple-spinors

We consider a variant of the Seiberg-Witten equations for multiple-spinors. The moduli space of solutions to our generalized Seiberg-Witten equations in the setting of Kähler surfaces has a direct relation with ASD connections of holomorphic vector bundle. Also in Kähler setting, we construct a numerical invariant from the equations that detects a notion of $ϕ-$stability of $SU(n)-$holomorphic vector bundles where $ϕ$ is some prescribed non-trivial holomorphic section.

math.DG↗