Anthony Small
The geometry of minimal surfaces generated by charge 2 Bogomolny monopoles on 3-dimensional Euclidean space is described in terms of the moduli parameter k. We find that the distribution of Gaussian curvature on the surface reflects the monopole structure. This is elucidated by the behaviour of the Gauss maps of the minimal surfaces.
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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 the strongly centred monopoles, and show they form a geodesic submanifold of the $k$-monopole moduli space. The space of strongly centred $k$-monopoles invariant under the cyclic group of rotations about a fixed axis, $C_k$, is shown to consist of several surfaces of revolution, g…
We study Lagrangian points on smooth holomorphic curves in T${\mathbb P}^1$ equipped with a natural neutral Kähler structure, and prove that they must form real curves. By virtue of the identification of T${\mathbb P}^1$ with the space ${\mathbb L}({\mathbb E}^3)$ of oriented affine lines in Euclidean 3-space ${\mathbb E}^3$, these Lagrangian curves give rise to ruled surfaces in ${\mathbb E}^3$, which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in ${\mathbb E}^3$, called the edge of regression of the ruled surface. We give an alternative …
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.
In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bundle. The singularities of this automorphism correspond to the singularities of the singular monopole. We then consider the complex geometry of the moduli space; in particular, we compute dimensions, both from the complex geometric and the gauge theoretic point of view.
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.
There are few known computable examples of non-abelian surface holonomy. In this paper, we give several examples whose structure 2-groups are covering 2-groups and show that the surface holonomies can be computed via a simple formula in terms of paths of 1-dimensional holonomies inspired by earlier work of Chan Hong-Mo and Tsou Sheung Tsun on magnetic monopoles. As a consequence of our work and that of Schreiber and Waldorf, this formula gives a rigorous meaning to non-abelian magnetic flux for magnetic monopoles. In the process, we discuss gauge covariance of surface holonomies for spheres fo…
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-dimensional supersymmetric quantum field theories on $\mathbb{R}^3\times T^2$ and, on the other hand, with moduli spaces of local Calabi-Yau metrics on the canonical bundle of a del Pezzo surface. We explore the asymptotic metric of these moduli spaces and compare our results with Seib…
The moduli space of multi-monopoles on a Riemann surface
本研究针对紧黎曼曲面上含N个旋量的Seiberg-Witten方程解的模空间，完成其欧拉示性数计算并推导有理同调，为3-流形计数不变量及代数曲线几何研究提供支撑。
The Seiberg-Witten equation with multiple spinors generalises the classical Seiberg-Witten equation in dimension three. In contrast to the classical case, the moduli space of solutions $\mathcal{M}$ can be non-compact due to the appearance of so-called Fueter sections. In the absence of Fueter sections we define a signed count of points in $\mathcal{M}$ and show its invariance under small perturbations. We then study the equation on the product of a Riemann surface and a circle, describing $\mathcal{M}$ in terms of holomorphic data over the surface. We define analytic and algebro-geometric com…
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.
Compact multi-monopole moduli spaces on Kähler surfaces
本文研究Kähler曲面上多旋量Seiberg-Witten方程的解模空间紧致性，并利用此结果计算解的数量，得出一个独立的代数几何推论。
Gauge Symmetry Breaking in the Asymptotic Analysis of Self Dual Yang-Mills-Higgs $SU(2)$ Monopoles
研究SU(2)自双Yang-Mills-Higgs Lagrangian在三维空间中的渐近行为，通过添加规范质量项打破规范对称性，证明极小值收敛到谐映射，并探讨不同渐近情形下的能量收敛特性。
We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a Kähler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifol…
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 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…
In this note, we study submanifold geometry of the Atiyah-Hitchin manifold, the double cover of the $2$-monopole moduli space. When the manifold is naturally identified as the total space of a line bundle over $S^2$, the zero section is a distinguished minimal $2$-sphere of considerable interest. In particular, there has been a conjecture by Micallef and Wolfson [Math. Ann. 295 (1993), Remark on p.262] about the uniqueness of this minimal $2$-sphere among all closed minimal $2$-surfaces. We show that this minimal $2$-sphere satisfies the "strong stability condition" proposed in our earlier wor…
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.
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