Paul Norbury
Kapustin and Witten associate a Hecke modification of a holomorphic bundle over a Riemann surface to a singular monopole on a Riemannian surface times an interval satisfying prescribed boundary conditions. We prove existence and uniqueness of singular monopoles satisfying prescribed boundary conditions for any given Hecke modification data confirming the underlying geometric invariant theory principle.
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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 to the case when $b^2(X) \neq 0$.
$G_2$-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 $\mathbb{S}^4$ and $\mathbb{CP}^2$. These are the first nontrivial examples of $G_2$-monopoles. Associated with each monopole there is a parameter $m \in \mathbb{R}^+$, known as the mass of the monopole. We prove that under a symmetry assumption, for each given $m \in \mathbb{R}^+$ t…
$G_2$-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 noncompact case) of non-Abelian monopoles with Dirac type singularities, and examples of monopoles whose singularities are not of that type. We also give an existence result for Abelian monopoles with Dirac type singularities on compact manifolds. This should be one of the building block…
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.
Bogomolny Monopoles on Asymptotically Cylindrical 3-Manifolds
该研究在特定渐近圆柱三维流形上，通过拼接方法构造了两类不同的单极子，拓展了相关数学理论。
We prove the existence of non-trivial irreducible SU(2)-monopoles with Dirac singularities on any rational homology 3-sphere, equipped with any Riemannian metric, using a gluing construction.
We prove an L^2-estimate involving Ricci curvature and a harmonic 1-form on a closed oriented Riemannian 3-manifold admitting a solution of any rescaled Seiberg-Witten equations. We also give a necessary condition to be a monopole class on some special connected sums.
In this paper we study spherically symmetric monopoles, which are critical points for the Yang-Mills-Higgs functional over a disk in 3 dimensions, with prescribed degree and covariant constant at the boundary. This is a 3-dimensional gauge-theory generalization of the Ginzburg-Landau model in 2 dimensions.
We calculate Perelman's invariant for compact complex surfaces and a few other smooth four-manifolds. We also prove some results concerning the dependence of Perelman's invariant on the smooth structure.
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 restriction of the space of monopoles to the boundary is a submersion onto a Lagrangian submanifold of the space of connections and spinors on the boundary. Both these spaces are infinite dimensional, even modulo gauge, since no boundary conditions are specified for the Seiberg-Wit…
We study the asymptotic behavior of finite energy $\rm{SU}(2)$ monopoles, and general critical points of the $\rm{SU}(2)$ Yang--Mills--Higgs energy, on asymptotically conical $3$-manifolds with only one end. Our main results generalize classical results due to Groisser and Taubes in the particular case of the flat $3$-dimensional Euclidean space $\mathbb{R}^3$. Indeed, we prove the integrality of the monopole number, or charge, of finite energy configurations, and derive the classical energy formula establishing monopoles as absolute minima. Moreover, we prove that the covariant derivative of …
We study moduli spaces of Seiberg-Witten monopoles over spin^c Riemannian 4-manifolds with long necks and/or tubular ends. This first part discusses compactness, exponential decay, and transversality. As applications we prove two vanishing theorems for Seiberg-Witten invariants.
We propose the study of some kind of monopole equations directly associated with a contact structure. Through a rudimentary analysis about the solutions, we show that a closed contact 3-manifold with positive Tanaka-Webster curvature and vanishing torsion must be either not symplectically semifillable or having torsion Euler class of the contact structure.
The virtual dimensions of both framed and unframed SU(2) magnetic monopoles on asymptotically conic 3-manifolds are obtained by computing the index of a Fredholm extension of the associated deformation complex. The unframed dimension coincides with the one obtained by Braam for conformally compact 3-manifolds. The computation follows from the application of a Callias-type index theorem.
This article investigates the asymptotics of $\rm{G}_2$-monopoles. First, we prove that when the underlying $\rm{G}_2$-manifold is nonparabolic (i.e. admits a positive Green's function), finite intermediate energy monopoles with bounded curvature have finite mass. The second main result restricts to the case when the underlying $\rm{G}_2$-manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian--Yang--Mills connection over the asymptotic cone. Finally, our last result exhibits a Fredholm setup…
We introduce the notion of a special monopole class on a four-manifold. This is used to prove restrictions on the smooth structures of Einstein manifolds. As an application we prove that there are Einstein four-manifolds which are simply connected, spin, and satisfy the strict Hitchin--Thorpe inequality, and which are homeomorphic to manifolds without Einstein metrics. In the Appendix, written with J. Wehrheim, we use special monopole classes to discuss the classification of Ricci-flat four-manifolds.
We investigate the $\mathrm{Pin}^-(2)$-monopole invariants of symplectic $4$-manifolds and Kähler surfaces with real structures. We prove the nonvanishing theorem for real symplectic $4$-manifolds which is an analogue of Taubes' nonvanishing theorem of the Seiberg-Witten invariants for symplectic $4$-manifolds. Furthermore, the Kobayashi-Hitchin type correspondence for real Kähler surfaces is given.
We establish a gluing theorem for monopoles over 4--manifolds containing long necks. The theorem is stated in terms of an ungluing map defined explicitly in terms of data that appear naturally in applications. Orientations of moduli spaces are handled using Benevieri--Furi's concept of orientations of Fredholm operators of index 0.
On a smooth closed oriented 4-manifold $M$ with a smooth action by a finite group $G$, we show that a $G$-monopole class gives the $L^2$-estimate of the Ricci curvature of a $G$-invariant Riemannian metric, and derive a topological obstruction to the existence of a $G$-invariant nonsingular solution to the normalized Ricci flow on $M$. In particular, for certain $m$ and $n$, $m\Bbb CP_2 # n\bar{\Bbb CP}_2$ admits an infinite family of topologically equivalent but smoothly distinct non-free actions of $\Bbb Z_d$ such that it admits no nonsingular solution to the normalized Ricci flow for any in…
Constraints on families of smooth 4-manifolds from $\mathrm{Pin}^{-}(2)$-monopole
本文利用$\mathrm{Pin}^{-}(2)$-单极方程推广了Baraglia关于单连通闭光滑4-流形微分同胚与同胚群弱同伦不等价的结论，并给出了群同态非满射的新实例。
On a smooth closed oriented $4$-manifold $M$ with a smooth action by a compact Lie group $G$, we define a $G$-monopole class as an element of $H^2(M;\Bbb Z)$ which is the first Chern class of a $G$-equivariant Spin$^c$ structure which has a solution of the Seiberg-Witten equations for any $G$-invariant Riemannian metric on $M$. We find $\Bbb Z_k$-monopole classes on some $\Bbb Z_k$-manifolds such as the connected sum of $k$ copies of a 4-manifold with nontrivial mod 2 Seiberg-Witten invariant or Bauer-Furuta invariant, where the $\Bbb Z_k$-action is a cyclic permutation of $k$ summands. As an …
Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining four-manifold invariants by counting $SU(2)$ instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group. This is a ``dual'' equation in which the gauge group is the dual of the maximal torus of $SU(2)$. The new viewpoint suggests many new results about the Donaldson invariants.
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.
Using quantum field-theoretic arguments, Witten has established a relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. In this survey article, we describe the program to prove this relation using a moduli space of PU(2) = SO(3) monopoles as a cobordism between the Donaldson moduli space of anti-self-dual SO(3) connections and moduli spaces of U(1) monopoles. We provide an overview of some of our transversality and Uhlenbeck compactness results for PU(2) monopoles, along with some of our calculations of Donaldson invariants in terms of Seiberg-Witten invariants…
We refine some classical estimates in Seiberg-Witten theory, and discuss an application to the spectral geometry of three-manifolds. In particular, we show that on a rational homology three-sphere $Y$, for any Riemannian metric the first eigenvalue of the laplacian on coexact one-forms is bounded above explicitly in terms of the Ricci curvature, provided that $Y$ is not an $L$-space (in the sense of Floer homology). The latter is a purely topological condition, and holds in a variety of examples. Performing the analogous refinement in the case of manifolds with $b_1>0$, we obtain a gauge-theor…
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 leading order asymptotic of the metric generalizes the one obtained by Gibbons, Manton and Bielawski in the case of complete decomposition into monopoles of unit charge. From the structure of the compactifications, we prove part of Sen's conjecture for the L2 cohomology of the s…
Some aspects of the construction of SW Floer homology for manifolds with non-trivial rational homology are analyzed. In particular, the case of manifolds that are obtained as zero-surgery on a knot in a homology sphere, and for torsion spinc structures. We discuss relative invariants in the case of torsion spinc structures.
Contact Monopoles in Dimension Five: Rigidity and Homogeneous Classification
本文在五维接触度量流形上引入接触单极子，证明紧致情形下其具有平行Higgs场，并利用Wang定理对单位切球丛$T_1S^3$上的齐次接触单极子进行分类，得到非平坦$U(1)$及非交换$SO(3)$解。
Gluing $\mathbb Z_2$-Harmonic Spinors and Seiberg-Witten Monopoles on 3-Manifolds
本文通过构造一个参数族的双旋量Seiberg-Witten单极，研究Z₂-调和旋量在规范化的极限下的收敛性，采用交替方法克服无限维障碍束的困难。
A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canonical operator takes values in sections of a nontrivial line bundle. The constructed approximation is demonstrated for the Dirac magnetic monopole on the two-dimensional sphere.
We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible SO(3) monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710…
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.
Almost Hermitian structures on moduli spaces of non-Abelian monopoles and applications to the topology of symplectic four-manifolds
本文通过Morse理论研究非阿贝尔单极模空间的几乎正交结构，证明了辛四维流形上Bogomolov-Miyaoka-Yau不等式成立。
We prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given four-manifold. In a previous article, dg-ga/9710032, we proved transversality using an extension of the holonomy-perturbation methods of Donaldson, Floer, and Taubes, together with the existence of an Uhlenbeck compactification for the perturbed moduli space. However, it remained an important and interesting question to see whether there were simpler, more intrinsic alternatives to th…
In this paper we develop analysis of the monopole maps over the universal covering space of a compact four manifold. We induce a property on local properness of the covering monopole map under the condition of closeness of the AHS complex. In particular we construct a higher degree of the covering monopole map when the linearized equation is isomorphic, which induces a homomorphism between K group of the group C^* algebras. It involves non linear analysis on the covering space, which is related to L^p cohomology. We also obtain various Sobolev estimates on the covering spaces. As a possible ap…
We introduce a general notion of twistorial map and classify twistorial harmonic morphisms with one-dimensional fibres from self-dual four-manifolds. Such maps can be characterised as those which pull back Abelian monopoles to self-dual connections. In fact, the constructions involve solving a generalised monopole equation, and also the Beltrami fields equation of hydrodynamics, and lead to constructions of self-dual metrics.
This is the first of two articles in which we give a proof - for a broad class of four-manifolds - of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the four-manifold. This article is a revision of sections 1-3 of an earlier version of the article dg-ga/9712005, now split into two parts, while a revision of sections 4-7 of that earlier version appears in a recently updated dg-ga/9712005. In the present article, we construct virtual…
We prove a generalization of Bennequin's inequality for Legendrian knots in a 3-dimensional contact manifold (Y,xi), under the assumption that Y is the boundary of a 4-dimensional manifold M and the version of Seiberg-Witten invariants introduced by Kronheimer and Mrowka [Invent. Math. 130 (1997) 209-255] is nonvanishing. The proof requires an excision result for Seiberg-Witten moduli spaces; then the Bennequin inequality becomes a special case of the adjunction inequality for surfaces lying inside M.
Existence of multi-monopoles on mapping tori
本文研究了在映射射影上多单极的存在性问题，通过证明一个绝热极限定理，构造了多种参数下的多单极解，并在不同 chambers 中提供了首次显式构造。
I show that flat PSL(2;R)-connections on three-manifolds satisfying certain 'stability condition' can be interpreted as solutions of the Seiberg-Witten equations with two spinors. This is used to construct explicit examples of the Seiberg-Witten moduli spaces. Also, I show that in this setting blow up sets satisfy certain non-trivial topological restrictions.
Towards a monopole Fueter Floer homology I: a compactness theorem
受 Donaldson-Segal 猜想启发，本文为定义三维流形上基于超凯勒丛 Fueter 截面计数的单极子 Floer 理论迈出第一步，并在电荷 k=2 情形证明紧致性定理，以克服非紧致性困难。
In this paper, we study the Seiberg-Witten equations on the product R x Y, where Y is a compact 3-manifold with boundary. Following the approach of Salamon and Wehrheim in the instanton case, we impose Lagrangian boundary conditions for the Seiberg- Witten equations. The resulting equations we obtain constitute a nonlinear, nonlocal boundary value problem. We establish regularity, compactness, and Fredholm properties for the Seiberg- Witten equations supplied with Lagrangian boundary conditions arising from the monopole spaces studied in [20]. This work therefore serves as an analytic foundati…
The Seiberg-Witten equations are defined on certain complex line bundles over smooth oriented four manifolds. When the base manifold is a complex Kahler surface, the Seiberg-Witten equations are essentially the Abelian vortex equations. Using known non-abelian generalizations of the vortex equations as a guide, we explore some non-abelian versions of the Seiberg-Witten equations. We also make some comments about the differences between the vortex equations that have previously appeared in the literature and those that emerge as Kahler versions of Seiberg-witten type equations.
In this article we complete the proof---for a broad class of four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the four-manifold. This article is a revision of sections 4--7 of an earlier version, while a revision of sections 1--3 of that earlier version now appear in a separate companion article (math.DG/0007190). Here, we use our computations of Chern classes for the virtual normal bundles for the Seiberg-Witten …
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