论文标题
$ q $ - nonabelianization for Line缺陷
$q$-nonabelianization for line defects
论文作者
论文摘要
我们考虑$ Q $ -Nonabelianization Map，该地图将3个manifold $ m $中的$ l $链接到链接到$ \ widetilde {l} $ in Branched $ n $ n $ - fold cover $ \ wideTilde {m} $。在量子场理论术语中，$ q $ -nonabelianization是相关的两个不同的缺陷的UV-IR地图：在紫外线中，我们有六维$（2,0）$ $ \ mathfrak {gl}（gl}（gl}（n）$ m \ mathbb in $ \ mathbb {r} $ at $} $的超级结构性字段理论\ {x^4 = x^5 = 0 \} $;在IR中，我们有$（2,0）$ type $ \ mathfrak {gl}（1）$的$（2,0）$ oon $ \ widetilde {m} \ times \ times \ mathbb {r}^{2,1} $，并将缺陷放在$ \ \ \ \ \ wideDilde {l} \ times {l} \ times \ times \ times \ times \ times \ times \ \ x^x上在情况下，$ M = \ Mathbb {r}^3 $，$ q $ -nonabelianizians计算链接的琼斯多项式，或与组$ u（n）$相关的类似物。在情况下，$ M = C \ times \ Mathbb {r} $，当$ l $对$ c $的投影是一个简单的非合同环路时，$ q $ -nonabelianization计算4D $ \ natcal {n} = 2 $类$ $ s $ s $ s $ $ s $ s $ $ s $ $ s $ s $的框架bps状态的受保护的旋转字符。在情况下，$ n = 2 $和$ m = c \ times \ mathbb {r} $，我们给出了$ q $ -nonabelianization Map的具体构造。该构建使用与覆盖$ \ widetilde {c} \ to c $的Holomorphic相关的WKB叶子的数据。
We consider the $q$-nonabelianization map, which maps links $L$ in a 3-manifold $M$ to links $\widetilde{L}$ in a branched $N$-fold cover $\widetilde{M}$. In quantum field theory terms, $q$-nonabelianization is the UV-IR map relating two different sorts of defect: in the UV we have the six-dimensional $(2,0)$ superconformal field theory of type $\mathfrak{gl}(N)$ on $M \times \mathbb{R}^{2,1}$, and we consider surface defects placed on $L \times \{x^4 = x^5 = 0\}$; in the IR we have the $(2,0)$ theory of type $\mathfrak{gl}(1)$ on $\widetilde{M} \times \mathbb{R}^{2,1}$, and put the defects on $\widetilde{L} \times \{x^4 = x^5 = 0\}$. In the case $M = \mathbb{R}^3$, $q$-nonabelianization computes the Jones polynomial of a link, or its analogue associated to the group $U(N)$. In the case $M = C \times \mathbb{R}$, when the projection of $L$ to $C$ is a simple non-contractible loop, $q$-nonabelianization computes the protected spin character for framed BPS states in 4d $\mathcal{N}=2$ theories of class $S$. In the case $N=2$ and $M = C \times \mathbb{R}$, we give a concrete construction of the $q$-nonabelianization map. The construction uses the data of the WKB foliations associated to a holomorphic covering $\widetilde{C} \to C$.
