arXiv research

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

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.

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.

This thesis was motivated by a desire to understand the natural geometry of hyperbolic monopole moduli spaces. We take two approaches. Firstly we develop the twistor theory of singular hyperbolic monopoles and use it to study the geometry of their charge 1 moduli spaces. After this we introduce a new way to study the m…

Constructs an asymptotic metric for moduli space of centred hyperbolic monopoles.

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…

We discuss the Euclidean limit of hyperbolic SU(2)-monopoles, framed at infinity, from the point of view of pluricomplex geometry. More generally, we discuss the geometry of hypercomplex manifolds arising as limits of pluricomplex manifolds.

We review the theory of JNR, mass 1/2 hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that that rational map of a JNR monopole monopole arises by scattering using results of Ati…

We prove the conjecture that a monopole in three-dimensional anti-de Sitter space can be completely determined by its ``holographic'' image on the conformal boundary two-sphere.

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

We determine the group of conformal automorphisms of the self-dual metrics on n#CP^2 due to LeBrun for n>2, and Poon for n=2. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space H^3 minus a finite number of points, called monopole points. We show that for n>2 connected sums, any con…

The paper studies hyperbolic three-manifolds and their geometric constraints.

Study Floer theory of hyperbolic three-manifolds using Dirac spectral flow.

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-b…

Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a 2-sphere of complex structures, but they no longer behave like unit imaginary qua…

We consider the self-dual conformal classes on n#CP^2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the poin…

Study Froyshov invariants and closed geodesics in hyperbolic 3-manifolds.

We prove that the space of gauge equivalence classes of U(1)-invariant connections on some SU(2)-principle bundles over the 4-sphere S^4 is weakly homotopy equivalent to a component of the second loop space of the 2-sphere S^2.

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 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…

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.

$G_2$-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…

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…

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.

New 4-manifolds found without certain Einstein metrics, using Seiberg-Witten theory.

Study monopole h-invariants from a topological viewpoint.

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…

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…

Researchers prove Seifert-Weber dodecahedral space is an L-space.

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…

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…

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…