在$\mathcal{N}=2$ QCD中使用运行联轴器进行切割和粘合
Cutting and gluing with running couplings in $\mathcal{N}=2$ QCD
摘要：
我们考虑具有规范群$SU(2)$和$N_f\leq 3$的渐近自由$\mathcal{N}=2$ QCD的序参量$u=\left<{\rm Tr}ϕ^2\right>$作为运行耦合常数$τ\in \mathbb{H}$的函数.如果$τ$的域被限制为适当的基本域$\mathcal{F}_{N_f}$，则函数$u$是一对一的。我们证明，这些领域包括六个或更少的图像的${\rm SL}(2,\mathbb{Z})$锁孔基本域，与适当的识别的边界。对于特殊选择的群众，$u$不产生分支点和削减，使$u$是一个模函数的同余子群$Γ$的${\rm SL}(2,\mathbb{Z})$和基本域是$Γ\backslash\mathbb{H}$。但是，对于一般体量，存在分支点和切口，并且$\mathcal{F}_{N_f}$的子集在体量变化时被切割和粘合。我们研究这种机制的各种现象，如超多重态的去耦，合并的本地奇点，以及合并的非本地奇点，从而产生超共形的阿盖尔-道格拉斯理论。
英文摘要：
We consider the order parameter $u=\left<{\rm Tr}ϕ^2\right>$ as function of the running coupling constant $τ\in \mathbb{H}$ of asymptotically free $\mathcal{N}=2$ QCD with gauge group $SU(2)$ and $N_f\leq 3$ massive hypermultiplets. If the domain for $τ$ is restricted to an appropriate fundamental domain $\mathcal{F}_{N_f}$, the function $u$ is one-to-one. We demonstrate that these domains consist of six or less images of an ${\rm SL}(2,\mathbb{Z})$ keyhole fundamental domain, with appropriate identifications of the boundaries. For special choices of the masses, $u$ does not give rise to branch points and cuts, such that $u$ is a modular function for a congruence subgroup $Γ$ of ${\rm SL}(2,\mathbb{Z})$ and the fundamental domain is $Γ\backslash\mathbb{H}$. For generic masses, however, branch points and cuts are present, and subsets of $\mathcal{F}_{N_f}$ are being cut and glued upon varying the mass. We study this mechanism for various phenomena, such as decoupling of hypermultiplets, merging of local singularities, as well as merging of non-local singularities which give rise to superconformal Argyres-Douglas theories.
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