arXiv 2102.042902021-02-08math.DGhep-th

$(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$上$\mathbb{R}$不变Kapustin-Witten方程瞬子解的存在性

The existence of instanton solutions to the $\mathbb{R}$-invariant Kapustin-Witten equations on $(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$

Clifford Henry Taubes

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摘要：

$(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$上的$\mathbb{R}$不变Kapustin-Witten方程的非负整数标记模型解集在Edward Witten解释SU（2）规范理论中的有色琼斯多项式或纽结的程序中起着核心作用。本文解释了为什么有$\mathbb{R}$不变的解决方案，这些方程的$(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$，两个模型的解决方案之间的插值$(0,\infty)$参数从0增加到$\infty$，同时尊重$\mathbb{R}^2$因子渐近。对模型解的极限对的唯一约束是：让$m_0$和$m_\infty$表示它们的非负整数标签，则$m_0 - m_\infty$必须是正的偶数。(As如论文中所解释的，这些插值解存在一个$\mathbb{C}^{(m_0 -m_\infty - 2)/2}\times \mathbb{C}^*$模空间。）

英文摘要：

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. The only constraint on the limiting pair of model solutions is this: Letting $m_0$ and $m_\infty$ denote their non-negative integer labels, then $m_0 - m_\infty$ must be a positive, even integer. (As explained in the paper, there is a $\mathbb{C}^{(m_0 -m_\infty - 2)/2}\times \mathbb{C}^*$ moduli space of these interpolation solutions.)

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