We prove generic regularity and Uhlenbeck-type compactification theorems for the moduli spaces of PU(2)-monopoles. Generic regularity is NOT obtained in the usual way (by applying Sard theorem to a smooth parameterized moduli space), since the parameterized moduli space can be a priori singular. We explain why, using the standard order 0-perturbations, one cannot get the smoothness of the parameterized moduli space. (The same difficulty arises in the case of ASD-Spin^c moduli spaces, so the definition of the Spin^c-polynomial invariants should be revised.) We show that the singular locus of the parameterized moduli space is contained in a closed subspace which is (in a certain sense) of infinite codimension, and we apply Sard theorem to the complement of this subspace. Our approach to prove Uhlenbeck compactness follows closely the strategy developed in the instanton case by Donaldson and Kronheimer. Similar results, with different methods, were obtained by Feehan - Leness and Feehan.
本文的目标是（扰动）PU（2）-单极的模空间的S^1-等变横截问题和紧化问题。题为“PU（2）-Monopoles的模空间（修订版）”的实质性改进版本给出了横截性结果的更简单、更清楚的证明，该版本已于1999年6月在arxiv上发表，并出现在Asian J. Math中，参见模空间of PU（2）-Monopoles，Asian J.Math.Vol.4，No.2（2000），391-436。
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 Euclidean three-space cut out by a plane arrangement associated with each monowall.
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…
Let $X=\mathbb{C}\timesΣ$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $X$ with finite analytic energy. The spin bundle $S^+\to X$ splits as $L^+\oplus L^-$. When $2-2g\leq c_1(S^+)[Σ]<0$, the moduli space is in bijection with the moduli space of pairs $((L^+,\bar{\partial}), f)$ where $(L^+,\bar{\partial})$ is a holomorphic structure on $L^+$ and $f: \mathbb{C}\to H^0(Σ, L^+,\bar{\partial})$ is a polynomial map. Moreover, the solution has analytic energy $-4π^2d\cdot c_1(S^+)[Σ]$ if $f$ …
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…
