Naichung Conan Leung, Ryosuke Takahashi
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 $(A,ϕ)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Geometric aspects of the Kapustin-Witten equations
本文介绍了Kapustin-Witten方程。讨论了复Yang-Mills方程与Kapustin-Witten方程之间的联系。此外，我们还讨论了Kapustin-Witten方程、矩映射条件和梯度Chern-Simons流之间的关系。本文的新结果对应于KW方程的解的估计，给出了连接的复部分的估计。这就留下了在连接的复杂部分上获得全局估计的问题。
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.
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 $S^1\timesΣ\times \{0\}$.
In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing the two solutions together along the cylindrical end. In addition, we establish a local Kuranishi model for this gluing picture. As an application, we show that over any compact four-manifold with $S^3$ or $T^3$ boundary, there exists a Nahm pole solution to the obstruction pertur…
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 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…
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 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.
In the present paper, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin-Witten equations in 4-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin-Witten equations. The imaginary parts of the solutions are singular. By rescaling, we find some limit behavior for these singular solutions. In addition, for any integer $k$, we can construct a 5$|k|$ dimensional family of $C^1$ solutions to the Kapustin-Witten equations on Euclidean space, again wi…
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 $(0,\infty) \times S^3$.
In this article, we consider the Kapustin-Witten equations on a closed $4$-manifold. We study certain analytic properties of solutions to the equations on a closed manifold. The main result is that there exists an $L^{2}$-lower bound on the extra fields over a closed four-manifold satisfying certain conditions if the connections are not ASD connections. Furthermore, we also obtain a similar result about the Vafa-Witten equations.
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.
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 …
Growth of the Higgs Field for Kapustin-Witten solutions on ALE and ALF gravitational instantons
本文研究了Ricci-flat ALX空间上$θ$-Kapustin-Witten二阶方程的解，建立了Taubes二分法：Higgs场平均范数快速增长或换位子处处为零。由此证实了ALE和ALF引力瞬子上有限能量解满足特定平坦条件。
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}.
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}.
A Family of Instanton-Invariants for Four-Manifolds and Their Relation to Khovanov Homology
本文基于物理论证，提出四流形上的一个参数族Haydys-Witten瞬子Floer同调群，通过维度约简和边界条件，建立与Khovanov同调的关系。
Growth of spinors in the generalized Seiberg-Witten equations on $\mathbb R^4$ and $\mathbb R^3$
本文研究三维和四维空间上广义Seiberg-Witten方程的旋量性质，通过曲率衰减假设等分析，证明旋量平均L²范数增长特性及有限能量解的旋量无穷远收敛性，部分推广了已有相关结果。
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 $L …
Mod r Vanishing Theorem of Seiberg-Witten Invariant for 4-Manifolds acted by Cyclic Group Z_r
本文提出并证明了一个消灭定理。如果一个4-流形$M$允许一个循环群$\mathbb{Z}_r$的光滑作用，那么给定$M$上的一个$\mathbb{Z}_r$ -等变$Spin^c$ -结构$\mathcal{C}$，在一些微小的假设下，Seiberg-Witten不变量$SW\mathcal{(C)}$模$r$为零.这里$r$可以是任何正整数。这个定理是F. Fang证明的$\mathbb{Z}_p$是素数阶循环时模p消灭定理的推广.
Remarks on $\mathrm{Sp}(1)$-Seiberg-Witten equation over $3$-manifolds
研究证明在闭合双曲3流形上，超流体1群Seiberg-Witten方程总有一个由双曲度量诱导的canonical不可约解，并探讨其模空间切空间与局部等距扁结构的关联，以及在特定条件下解的刚性。
Solutions to the Seiberg-Witten equations in all dimensions
本文构造了高维广义 Seiberg-Witten 方程的显式解，涵盖维度 5、6 和 8，将解与涡旋关联，并揭示了解空间非紧性与奇维调和形式的关系。
Gluing $\mathbb Z_2$-Harmonic Spinors and Seiberg-Witten Monopoles on 3-Manifolds
本文通过构造一个参数族的双旋量Seiberg-Witten单极，研究Z₂-调和旋量在规范化的极限下的收敛性，采用交替方法克服无限维障碍束的困难。
A gerbe-like construction in gauge theory II: the case of homology tori
本文研究同调环面的奇确定性，通过构造反线性Z/4作用，恢复Baraglia(2023)关于闭合spin4流形mod2Seiberg-Witten不变量的部分结果。
Non-convergent sequences of solutions to the massive Vafa-Witten equations with 'interesting' $\mathbb{Z}/2\mathbb{Z}$ self-dual harmonic 2-form limits
本文在2维环面与亏格大于1的Riemann曲面乘积上构造了Vafa-Witten方程的发散解序列，经重整化后收敛于一个包含余维2子流形及实线丛的“有趣”$\mathbb{Z}/2\mathbb{Z}$自对偶调和2-形式数据集。
We consider a generalisation of the Seiberg-Witten invariant to the families Seiberg-Witten invariants of a smooth family of 4-manifolds with fibres diffeomorphic to a 4-manifold $X$. Of particular interest is the special case when the family has a smoothly varying Kähler structure. We obtain a general computation of the invariants when $b_1=0$ in terms of characteristic classes of some vector bundles corresponding to the cohomology groups of holomorphic line bundles of the family. Finally, we apply the formula to some examples of Kähler families where some more further explicit computations c…
Atiyah's classical work on circular symmetry and stationary phase shows how the $\hat{A}$-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable ''antiholomorphic sector'' in the $\mathbb{C}/Λ$-equivariant cohomology of the conformal double loop space $\mathrm{Maps}(\mathbb{C}/Λ,X)$ of a rationally string manifold $X$ produces the Witten genus of $X$. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation o…
Let $X$ be a smooth, compact, oriented $4$-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on $X$ by diffeomorphisms. The obstructions show that certain group actions on $H^2(X , \mathbb{Z})$ preserving the intersection form can not be lifted an action of the same group on $X$ by diffeomorphisms. Using our obstructions, we construct numerous examples of group actions which can be realised continuously but can not be realised smoothly for any differenti…
A Wess--Zumino--Witten type equation in the space of Kähler potentials in terms of Hermitian--Yang--Mills metrics
本文证明 Kähler 位势空间中 Wess-Zumino-Witten 型方程的解可由 Hermitian-Yang-Mills 度量一致逼近，核心是 Berndtsson 定理的新版本。
Virtual refinements of the Vafa-Witten formula
本文提出 Vafa-Witten 公式的虚拟精细化猜想，将秩 2 层模空间的虚拟 $χ_y$-亏格用 descendent Donaldson 不变量和 Seiberg-Witten 不变量表示，并通过局部化计算验证该猜想。
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…
Concentrating Local Solutions of the Two-Spinor Seiberg-Witten Equations on 3-Manifolds
针对带有奇异集Z0的Z2-调和旋量，在Z0的管状邻域上构造一族依赖于参数ε的双旋量Seiberg-Witten方程局部解，这些解在ε→0时曲率L2范数发散，且重整化后局部收敛到原Z2-调和旋量。
Concentrating Dirac Operators and Generalized Seiberg-Witten Equations
本文研究了带零阶扰动的Dirac算子在极限下的解集中性质，并将其其推广至非线性Dirac方程，进而应用于广义Seiberg-Witten方程解的模空间提供紧致性证明，并将弱收敛改进为局部光滑收敛。
Probabilistic construction of the $\mathbb{H}^3$-Wess-Zumino-Witten conformal field theory and correspondence with Liouville theory
该研究严格构造了闭合黎曼曲面上带任意光滑规范场扭曲的$\n\n\nmathbb{H}^3$-WZW模型的概率路径积分，证明了其主场关联函数与刘维尔共形场论的对应关系，推广了已有结果，为量子解析朗兰兹对应提供了概率视角的支撑。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。