The Kapustin-Witten equations on R^4 are equations for a pair of connection on the product principle SU(2) bundle and 1-form with values in the product Lie algebra bundle. The 1-form is the Higgs field. A dichotomy is proved to the effect that either the averaged norm of the Higgs field on large radius spheres grows faster than a power of the radius, or its 1-form components everywhere pairwise commute.
In this article, we study the Kapustin-Witten equations on a closed, simply-connected, four-dimensional manifold which were introduced by Kapustin and Witten. We use the Taubes' compactness theorem in arXiv:1307.6447v4 to prove that if $(A,ϕ)$ is a smooth solution of Kapustin-Witten equations and the connection $A$ is closed to a $generic$ ASD connection $A_{\infty}$, then $(A,ϕ)$ must be a trivial solution. We also prove that the moduli space of the solutions of Kapustin-Witten equations is non-connected if the connections on the compactification of moduli space of ASD connections are all $ge…
This paper supplies a new characterization of the Kapustin Witten equation solutions on $(0,\infty) \times \mathbb{R}^2 \times \mathbb{R}$ that play a key role in Edward Witten's program to obtain the Jones polynomial knot invariants using solutions to the same equation on $(0,\infty) \times S^3$.
This is a written version of lectures that I would have given myself about aspects of the differential operator that is obtained from the linearized Kapustin-Witten equations on the product of the half-line with a compact, oriented, Riemannian 3-manifold. These lectures concern for the most part certain instances of much more general theorems of R. Mazzeo and E. Witten. There is also a 'lecture series' about the asymptotics of solutions to the same Kapustin-Witten equations as the half-line parameter limits to infinity.
We consider solutions of Kapustin-Witten equation with Nahm pole boundary on $S^3\times \mathbb{R}^+$. These solutions are usually called Nahm pole solutions. In this paper, we will prove that there exists a constant $C>0$ such that $\|F_A\|_{L^2}\leq C$ for any Nahm pole solution $(A,ϕ)$.
For a 3-manifold $X$ and compact simple Lie group $G$, we study the expansions of polyhomogeneous Nahm pole solutions to the Kapustin-Witten equations over $X\times (0,+\infty)$. Let $y$ be the coordinate of $(0,+\infty)$, we prove that the sub-leading terms of a polyhomogeneous Nahm pole solution is smooth to the boundary when $y\to 0$ if and only if $X$ is an Einstein 3-manifold.
In this note, we classify all solutions to the $\mathrm{SU(n)}$ Kapustin-Witten equations on $S^1\timesΣ\times \mathbb{R}^+$, where $Σ$ is a compact Riemann surface, with Nahm pole singularity at $S^1\timesΣ\times \{0\}$. We provide a similar classification of solutions with generalized Nahm pole singularities along a simple divisor (a "knot") in $S^1\timesΣ\times \{0\}$.
We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 \times \mathbb{S}^1$ and the structure group is $\mathrm{SU} (2)$.
This paper describes the behavior of sequences of solutions to the Kapustin-Witten equations with Nahm pole asymptotics on the product of the half-line with a compact, oriented, Riemannian 3-manifold. These sequences have sub-sequences that either converge to another solution after acting term-wise by an automorphism of the principle bundle, or they converge after renormalization to a (weak) Z/2 harmonic 1-form from the 3-manifold; it is independent of the half-line coordinate in the product structure.
This article finds a structure of singular sets on compact Kahler surfaces, which Taubes introduced in the studies of the asymptotic analysis of solutions to the Kapustin-Witten equations and the Vafa-Witten ones originally on smooth four-manifolds. These equations can be seen as real four-dimensional analogues of the Hitchin equations on Riemann surfaces, and one of common obstacles to be overcome is a certain unboundedness of solutions to these equations, especially of the "Higgs fields". The singular sets by Taubes describe part of the limiting behaviour of a sequence of solutions with this…
A non-negative integer labeled set of model solutions to the $\mathbb{R}$-invariant Kapustin-Witten equations on $(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$ plays a central role in Edward Witten's program to interpret the colored Jones polynomial or a knot in the context of SU(2) gauge theory. This paper explains why there are $\mathbb{R}$-invariant solutions to these equations on $(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$ that interpolate between two model solutions as the $(0,\infty)$ parameter increases from 0 to $\infty$ while respecting the $\mathbb{R}^2$ factor asymptotics. Th…
Arising from a topological twist of $\mathcal{N} = 4$ super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrized by $t\in\mathbb{P}^1$. The parameter corresponds to a linear combination of two super charges in the twist. When $t=0$ and the four-manifold is a compact Kähler surface, the equations become the Simpson equations, which was originally studied by Hitchin on a compact Riemann surface, as demonstrated independently in works of Nakajima and the third-named author. At the same time, there is a notion of $λ$-connection …
In this paper we prove existence for model knot solutions to the dimensionally reduced twisted Kapustin-Witten equations on $\mathbb{R}^3_+$ for any twisting parameter $t\in(0,\infty)$. We start with the explicit solutions for $t = 1$ derived in \cite{WFivebranes} and perform a continuity argument in $t$. This corroborates a prediction of Gaiotto and Witten \cite[][p. 961]{GW}.
In this paper, we study the dimensionally reduced twisted Kapustin-Witten equations on the product of a compact Riemann surface $Σ$ with $\mathbb{R}^+$. The main result is a Kobayashi-Hitchin type correspondence between the space of tilted Nahm pole solutions and the moduli space of Beilinson-Drinfeld opers. This corroborates a prediction of Gaiotto and Witten \cite[p.971]{gaiotto2012knot}.
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 …
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…
On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations. When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover th…
We establish a Kobayashi-Hitchin correspondence between solutions of the extended Bogomolny equation with a Dirac type singularity and Hecke modifications of Higgs bundles. This correspondence was conjectured by Witten and plays an important role in the physical description of the the geometric Langlands program in terms of S-duality for N=4 super Yang-Mills theory in four dimensions.
Kähler流形$X$上的扭曲Higgs丛是由全纯向量丛$E$和某个全纯向量丛$M$的全纯丛态射$ϕ\colon M\otimes E \to E$组成的对$(E,ϕ)$。当$X$是一条曲线，$M$是$X$的切丛时，Hitchin首先考虑了这样的对象，Simpson也考虑了高维基。希钦-小林对应表明$(E,ϕ)$是多稳定的当且仅当$E$允许一个厄密度量解希钦方程。这种对应关系是判断方程是否存在解的有力工具，但它提供的关于实际解的信息很少。在本文中，我们研究了这个问题的量化表示在有限维数据和平衡度量，使近似解希钦方程。本研究的动机来自Donagi-Wijnholt在arXiv：1104.2610上关于Vafa-Witten方程的平衡度量的工作。
本文的目的是研究一个方程组，它同时推广了Yang-Mills-Higgs理论中的涡旋方程和Gromov赝全纯曲线理论中的全纯性方程。我们从这两个理论的一些结果和定义扩展到一个共同的设置。我们引入一个功能推广杨米尔斯希格斯功能，其极小值与我们的方程的解决方案。我们证明了一个Hitchin-Kobayashi对应允许研究的解决方案的方程在Kaehler的情况下。给出了方程（扰动）解（的规范等价类）集的光滑流形结构（即所谓的模空间）。给出了模空间的紧化，推广了Gromov关于全纯曲线模的紧化。最后，我们利用模空间定义了（在一定条件下）紧辛流形的不变量，其中S ^1具有Hamilton几乎自由作用。这些不变量推广了Gromov-Witten不变量。这是作者的博士学位。论文它的一个章节包含在论文数学DG/9901076中。1999年4月提交论文后，笔者了解到K。Cieliebak，A. R. Gaio和D.萨拉蒙也（从不同的角度）得出了同样的方程，并制定了一个非常类似的程序（见math.SG/9909122）。
