Paul Norbury
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.
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Hyperbolic monopoles with continuous symmetries
本文建立连续对称双曲单极子的分类框架与结构定理，将球对称情形归结为表示论问题，并给出结构群约束，进而构造新的球对称 $\mathrm{Sp}(n)$ 双曲单极子。
Certain hyperbolic monopoles and all hyperbolic vortices can be constructed from SO(2) and SO(3) invariant Euclidean instantons, respectively. This observation allows us to describe a large class of hyperbolic monopoles as hyperbolic vortices embedded into $\mathbb{H}^3$ and yields a remarkably simple relation between the two Higgs fields. This correspondence between vortices and monopoles gives new insight into the geometry of the spectral curve and the moduli space of hyperbolic monopoles. It also allows an explicit construction of the fields of a hyperbolic monopole invariant under a $\math…
We show that many hyperbolic monopoles can be distinguished from each other via their asymptotic values in contrast to the case of Euclidean monopoles.
The moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass, and this leads to a nontrivial mass dependence of the holomorphic data (spectral curves, rational maps, holomorphic spheres) associated to hyperbolic multi-monopoles. In this paper, we obtain an explicit description of this dependence for general hyperbolic monopoles of magnetic charge two. In addition, we show how to compute the monopole mass of higher charge spectral curves with tetrahedral and octahedral symmetries. Spectral curves of euclidean monopoles are recovered from our results via an infinite-mass l…
Hyperbolic monopole motion is studied for well separated monopoles. It is shown that the motion of a hyperbolic monopole in the presence of one or more fixed monopoles is equivalent to geodesic motion on a particular submanifold of the full moduli space. The metric on this submanifold is found to be a generalisation of the multi-centre Taub-NUT metric introduced by LeBrun. The one centre case is analysed in detail as a special case of a class of systems admitting a conserved Runge-Lenz vector. The two centre problem is also considered. An integrable classical string motion is exhibited.
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, existence was known only for monopoles with integral mass (since these arise from U(1) invariant instantons on Euclidean 4-space). The method of proof is based on Taubes' gluing procedure, using well-separated, explicit, charge one monopoles. The analysis is carried out in a weighted Sob…
The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space
本文在点粒子近似下构造了两个中心双曲单极子模空间的渐近度量，通过考虑对径构型克服了双曲情形下无法分解质心运动的困难，得到了负质量Taub-NUT度量的双曲类比，并建立了其与Hitchin构造的双曲Atiyah-Hitchin度量渐近形式的关系。
Magnetic monopoles in hyperbolic space are in correspondence with certain algebraic curves in mini-twistor space, known as spectral curves, which are in turn in correspondence with rational maps between Riemann spheres. Hyperbolic monopoles correspond to circle-invariant Yang-Mills instantons, with an identification of the monopole and instanton numbers, providing the curvature of hyperbolic space is tuned to a value specified by the asymptotic magnitude of the Higgs field. In previous work, constraints on ADHM instanton data have been identified that provide a non-canonical realization of the…
In a paper of Braam and Austin, $\text{SU}(2)$ magnetic monopoles in hyperbolic space $H^{3}$ were shown to be the same as solutions to matrix-valued difference equations called the discrete Nahm equations. Here, I discover the $(N-1)$-interval discrete Nahm equations and show that their solutions are equivalent to $\text{SU}(N)$ hyperbolic monopoles. These discrete time evolution equations on an interval feature a jump in matrix dimensions at certain points in the evolution, which are given by the mass data of the corresponding monopole. I prove the correspondence with higher rank hyperbolic …
In this work we study the dimensional reduction of smooth circle invariant Yang-Mills instantons defined on 4-manifolds which are non-trivial circle fibrations over hyperbolic 3-space. A suitable choice of the 4-manifold metric within a specific conformal class gives rise to singular and smooth hyperbolic monopoles. A large class of monopoles is obtained if the conformal factor satisfies the Helmholtz equation on hyperbolic 3-space. We describe simple configurations and relate our results to the JNR construction, for which we provide a geometric interpretation.
Instantons with continuous conformal symmetries: Hyperbolic and singular monopoles and more, oh my!
本文通过引入线性约束简化非线性约束，全面研究了具有各类连续共形对称性的瞬子，发现了大量新型瞬子并提供了分类框架，同时探讨了其与双曲、奇异磁单极子等规范理论对象的联系。
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.
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.
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-bundle over the spectral curve, such that the discrete-time DN evolution corresponds to walking in the Jacobian of the spectral curve in a straight line through the line-bundle with steps of a fixed size. Some of the implications for hyperbolic monopoles are also discussed.
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…
Hyperbolic Monopoles, (Semi-)Holomorphic Chern-Simons Theories, and Generalized Chiral Potts Models
本文通过（半）全纯场论建立双曲磁单极子与广义手征波茨模型的对应，推广至SU(n)群，提出4d CS构造技术，并揭示其源于10维全纯CS理论的维数约化，统一为同一物理的两种体现。
The space of all possible boundary conditions that respect self-adjointness of Hamiltonian operator is known to be given by the group manifold $U(2)$ in one-dimensional quantum mechanics. In this paper we study non-Abelian Berry's connections in the space of boundary conditions in a simple quantum mechanical system. We consider a system for a free spinless particle on a circle with two point-like interactions described by the $U(2) \times U(2)$ family of boundary conditions. We show that, for a certain $SU(2) \subset U(2) \times U(2)$ subfamily of boundary conditions, all the energy levels bec…
Monopole Operators and Bulk-Boundary Relation in Holomorphic Topological Theories
本研究针对三维N=2超对称场论的全纯扭曲，构造阿贝尔规范理论的非微扰体局域算子并验证，检验体边代数猜想关系，发现磁单极算子的相关出现，还探讨了代数的高阶结构。
We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.
The Dirac monopole on a three-dimensional torus is considered as a solution to the Bogomolny equation with non-trivial boundary conditions. The analytical continuation of the obtained solution is shown to be a three-dimensional generalization of the Kronecker series. It satisfies the corresponding functional equation and is invariant under modular transformations.
Spherical-DAHA predictions for boundary monopole bubbling of non-minuscule 't Hooft lines in $\mathcal N=4$ SYM
研究正则Nahm极边界条件下四维$\mathcal N=4$ $U(N)$超杨-米尔斯理论中非极小磁荷的半BPS边界't Hooft线，计算鼓泡修正的缺陷半指标并识别为球面DAHA元，通过Macdonald核证明与S对偶Neumann Wilson线半指标相等。
In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary
本文研究带类时边界的整体双曲流形上Friedrichs系统的柯西问题，通过施加容许边界条件证明强解的存在唯一性，且双曲型Friedrichs系统的柯西问题在Hadamard意义下适定，并给出相关实例。
We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then using local methods developed by Lax, Phillips and Rauch, Massey to show smoothness of the solutions. Our proof actually works for a slightly more general class of local boundary conditions.
A boundary value problem related to a parabolic higher order operator with a small parameter is analized. When the small parameter tends to zero, the reduced operator is hyperbolic. When t tends to infinity a parabolic hyperbolic boundary layer appears. In this paper a rigorous asymptotic approximation uniformly valid for all t is established.
We study topological phases in the hyperbolic plane using noncommutative geometry and T-duality, and show that fractional versions of the quantised indices for integer, spin and anomalous quantum Hall effects can result. Generalising models used in the Euclidean setting, a model for the bulk-boundary correspondence of fractional indices is proposed, guided by the geometry of hyperbolic boundaries.
Homology stability for asymptotic monopole moduli spaces
该研究证明了带框狄拉克磁单极和理想磁单极两类渐近模空间的同调稳定性，通过配置空间的一般同调稳定性结果实现，并揭示了它们作为经典模空间部分紧化边界超曲面的结构。
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