arXiv research

We study periodic monopoles satisfying some mild conditions, called of GCK type. Particularly, we give a classification of periodic monopoles of GCK type in terms of difference modules with parabolic structure, which is a kind of Kobayashi-Hitchin correspondence between differential geometric objects and algebraic obje…

We construct periodic monopoles (with singularities), i.e. monopoles on $\mathbb{R}^{2} \times \mathbb{S}^{1}$ possibly singular at a finite collection of points, by gluing methods.

Relative moduli spaces of periodic monopoles provide novel examples of Asymptotically Locally Flat hyperkahler manifolds. By considering the interactions between well-separated periodic monopoles, we infer the asymptotic behavior of their metrics. When the monopole moduli space is four-dimensional, this construction yi…

We study Bogomolny equations on $R^2\times S^1$. Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using Nahm transform, we show that periodic monop…

Classifies charge-3 monopoles with symmetry, identifying new spectral curves.

We show that for generic choices of parameters the moduli spaces of periodic monopoles (with singularities), i.e. monopoles on $\mathbb{R}^{2} \times \mathbb{S}^{1}$ possibly singular at a finite collection of points, are either empty or smooth hyperkähler manifolds. Furthermore, we prove an index theorem and therefore…

The main result is a computation of the Nahm transform of a SU(2)-instanton over RxT^3, called spatially-periodic instanton. It is a singular monopole over T^3, a solution to the Bogomolny equation, whose rank is computed and behavior at the singular points is described.

An interesting theme in complex differential geometry is to find a correspondence between algebraic objects and differential geometric objects. One of the most attractive is the non-abelian Hodge theory of Simpson. In this paper, pursuing an analogue of the non-abelian Hodge theory in the context of $q$-difference modu…

The low velocity dynamic of a doubly periodic monopole, also called a monopole wall or monowall for short, is described by geodesic motion on its moduli space. This moduli space is hyperkaehler and non-compact. We establish a relation between the Kaehler potential of this moduli space and the volume of a region in Eucl…

We explain the correspondences between twisted monopoles with Dirac type singularity and polystable twisted mini-holomorphic bundles with Dirac type singularity on a 3-dimensional torus. We also explain that they are equivalent to polystable parabolic twisted difference modules on elliptic curves.

Study Berry connections for 2d GLSMs, linking to cohomology theories.

We construct the Nahm transform from finite energy instantons on the product of a real line and a three dimensional torus to Dirac-type singular monopoles on the dual torus. Moreover, we show the correspondence between the data which handle the asymptotic behavior of instantons at infinity and one of monopoles at singu…

We prove the existence of a Hermitian-Einstein metric on holomorphic vector bundles with a Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying Kähler manifolds. We also study the curvature decay of the Hermitian-Einstein metrics. It is useful for the study of the class…

This paper establishes that the Nahm transform sending spatially periodic instantons (instantons on the product of the real line and a three-torus) to singular monopoles on the dual three-torus is indeed a bijection as suggested by the heuristic. In the process, we show how the Nahm transform intertwines to a Fourier-M…

Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkähler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimens…

Study of 2d gauged linear sigma models to derive difference equations and spectral data.

We discuss $SU(2)$ Bogomolny monopoles of arbitrary charge $k$ invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry…

The study connects monopole chains to Higgs bundles and classifies symmetric chains.

Study on JNR monopoles, focusing on their spectral curves and energy density.

Study of large mass limits of G2 and Calabi-Yau monopoles on specific manifolds.

We define and study certain hyperkaehler manifolds which capture the asymptotic behaviour of the SU(2)-monopole metric in regions where monopoles break down into monopoles of lower charges. The rate at which these new metrics approximate the monopole metric is exponential, as for the Gibbons-Manton metric.

We discuss the spectral curves and rational maps associated with $SU(2)$ Bogomolny monopoles of arbitrary charge $k$. We describe the effect on the rational maps of inverting monopoles in the plane with respect to which the rational maps are defined, and discuss the monopoles invariant under such inversion. We define t…

We classify hyperbolic monopoles with continuous symmetries and construct new examples.

Study shows instantons and monopoles decompose into U(1) components.

$G_2$-monopoles are solutions to gauge theoretical equations on $G_2$-manifolds. If the $G_2$-manifolds under consideration are compact, then any irreducible $G_2$-monopole must have singularities. It is then important to understand which kind of singularities $G_2$-monopoles can have. We give examples (in the noncompa…

Researchers create BPS monopoles with any desired symmetry breaking.

Proves a new skein exact triangle for real monopole Floer homology.

In this paper, we explore the interplay between contact structures and sutured monopole Floer homology. First, we study the behavior of contact elements, which were defined by Baldwin and Sivek, under the operation of performing Floer excisions, which was introduced to the context of sutured monopole Floer homology by …

Existence of non-trivial monopoles on 3-spheres proven.

New gauge condition fixes $L^2$ metric divergence in hyperbolic monopole spaces.

We show that many hyperbolic monopoles can be distinguished from each other via their asymptotic values in contrast to the case of Euclidean monopoles.

$G_2$-Monopoles are solutions to gauge theoretical equations on noncompact $7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$ Bryant-Salamon manifolds. We construct examples of $G_2$-monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the $\ma…

We associate to an SU(2) hyperbolic monopole a holomorphic sphere embedded in projective space and use this to uncover various features of the monopole.

Develops invariants for webs and foams using Seiberg-Witten theory.

Singular monopoles are nonabelian monopoles with prescribed Dirac-type singularities. All of them are delivered by the Nahm's construction. In practice, however, the effectiveness of the latter is limited to the cases of one or two singularities. We present an alternative construction of singular monopoles formulated i…

On a complete manifold, such as Euclidean 3-space or hyperbolic 3-space, the limit at infinity of the norm of the Higgs field is called the mass of the monopole. We show the existence, on hypebolic 3-space, of monopoles with given magnetic charge and arbitrary mass. Previously, aside from charge one monopoles, existenc…

Study monopole h-invariants from a topological viewpoint.

Derives a formula for fermion dimensions in spherically symmetric monopole backgrounds.

In this expository review we discuss various aspects of gauge theory. While the focus is on mathematics, wherever possible we make contact with theoretical high energy physics. Particular emphasis is placed on instantons and monopoles, which admit physical interpretation, and yield interesting and nontrivial mathematic…

The study examines sequences of 3D generalised monopoles with uniform bounds.

We construct monopoles in any asymptotically conical (AC) $3$-manifold $X$ with $b^2(X)=0$. For sufficiently large mass, our construction covers an open set in the moduli space of monopoles. We also give a more general construction of Dirac monopoles in any AC manifold, which may be useful for generalizing our result t…

We construct the first nontrivial examples of Calabi-Yau monopoles. Our main interest on these, comes from Donaldson and Segal's suggestion \cite{Donaldson2009} that it may be possible to define an invariant of certain noncompact Calabi-Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metri…

A quasiclassical method approximates magnetic monopole eigenvalues.

Study of continuous symmetries in Nahm data and BPS monopoles.

The study examines asymptotic properties of G2-monopoles on nonparabolic G2-manifolds.

Study SO(3) vortices over orbifold Riemann surfaces and monopoles.

In this paper, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Solutions to these equations are called monopoles. Under some simple topological assumptions, we show that the solution space of all monopoles is a Banach manifold in suitable function space topologies. We then prove that the re…

We describe compactifications of the moduli spaces of SU(2) monopoles on R3 as manifolds with corners, with respect to which the hyperKaehler metrics admit asymptotic expansions up to each boundary face. The boundary faces encode monopoles of charge k decomposing into widely separated monopoles of lower charge, and the…