几何朗兰兹与Nahm方程和Bogomolny方程

The Extended Bogomolny Equations and Generalized Nahm Pole Boundary Condition

In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $Σ\times \RP$ with Nahm pole singularity at $Σ\times \{0\}$ and the Hitchin component of the stable $SL(2,\mathbb{R})$ Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop a partial Kobayashi-Hitchin correspondence for solutions with a knot singularity in this program, corresponding to the non-Hitchin components in the moduli space of stable $SL(2,\mathbb{R})$ Higgs bundles. We also prove existence and uniqueness of solutions with knot singulariti…

The extended Bogomolny equations with generalized Nahm pole boundary conditions, II

We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the product of a compact Riemann surface $Σ$ with ${\mathbb R}^+_y$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as $y \to \infty$, and certain holomorphic data consisting of effective triplets $(\mathcal E, φ, L)$ where $(\mathcal E, φ)$ is a stable $\mathrm{SL}(n+1,\mathbb C)$ Higgs pair and $L …

高电荷周期单极子

Higher charge periodic monopoles

我们考虑SU（2）Bogomolny方程的单周期解，并利用Nahm变换生成一类电荷k>2的单极子，从而推广了较低电荷链的已知结果.给出了一些简单的散射过程，并与$\mathbb{R}^3$中单极子的测地线运动进行了比较。

周期磁单极的Nahm变换与N=2超杨-米尔斯理论

Nahm Transform For Periodic Monopoles And N=2 Super Yang-Mills Theory

在$R^2\times S^1$上研究Bogomolny方程.虽然他们不承认非平凡的有限能量的解决方案，我们表明，有有趣的无限能量的解决方案与希格斯场的几何增长在无穷远。我们称这些解为周期性单极子。利用Nahm变换，我们证明了在无限远处Higgs场呈指数增长的圆柱体上，周期性单极子与希钦方程的解是一一对应的。周期单极子的模空间属于一类新的超凯勒流形，并在量子规范理论和弦论中有应用。例如，我们证明了$k$周期单极子的模空间提供了规范群$SU(k)$在任意半径的圆上紧化的${\cal N}=2$ super Yang-Mills理论的精确解。

G2 monopoles

We investigate some aspects of Bogomolny-Prasad-Sommerfield monopole solutions in the Yang-Mills-Higgs theory with exceptional gauge group $G_2$ spontaneously broken to $U(1)\times U(1)$. Corresponding homotopy group is $π_2\left(G_2/U(1)\times U(1)\right)$ and similar to the $SU(3)$ theory, the $G_2$ monopoles are classified by two topological charges $(n_1,n_2)$. In fundamental representation these yield a subset of $SO(7)$ monopole configurations. Through inspection of the structure of $Alg(G_2)$, we propose an extension of the Nahm construction to the $(n,1)_{G_2}$ monopoles. For $(1,1)_{G…

Periodic Monopoles from Spectral Curves

We consider $\text{SU}(2)$ Bogomolny equations on $\mathbb{R}^2\times\hat{S}^1$ and use the spectral curve defined by the holonomy in the periodic direction to approximate the fields in the limit of large size to period ratio. Symmetries of the Nahm transform allow a study of the effective two dimensional dynamics, which is compared with known results on the full moduli space. The techniques are applied to systems of higher charge and higher rank gauge group, allowing a direct comparison to other periodic Yang-Mills systems.