4d N = 2 QFT中的同调S-对偶
Homological S-Duality in 4d N=2 QFTs
摘要：
4d $\mathcal{N}=2$超对称理论$\mathcal{F}$的$S$-对偶群$\mathbb{S}(\mathcal{F})$与其簇范畴$\mathscr{C}(\mathcal{F})$模平凡作用于物理量的子群的三角形自等价群相一致。$\mathbb{S}(\mathcal{F})$是一个离散群，它可与西格尔模群$Sp(2g,\mathbb{Z})$的子群相对应（$g$是库仑分支的维数）。这种识别减少了一个给定的$\mathcal{N}=2$理论的$S$-对偶群的确定同调代数中的一个问题。在本文中，我们描述的技术，使计算简单的一大类$\mathcal{N}=2$ QFT。群$\mathbb{S}(\mathcal{F})$自然地表示为广义辫群。$S$-对偶群通常比预期的要大。在某些模型中，$S$-对偶性的增强是相当壮观的。例如，具有巨大$S$-对偶群的QFT是具有规范群$SO(8)\times SO(5)^3\times SO(3)^6$和双旋量和三旋量表示中的半超多重态的拉格朗日SCFT。我们集中在四个家庭的例子：$\mathcal{N}=2$ SCFT的形式$(G,G^\prime)$，$D_p(G)$和$E_r^{(1,1)}(G)$，以及渐近自由的理论$(G,\widehat{H})$（其中包含$\mathcal{N}=2$ SQCD作为一个特殊情况）。对于$E_r^{(1,1)}(G)$模型，我们证实了$PSL(2,\mathbb{Z})$ $S$-对偶群的存在，该对偶群由Del Zotto、Vafa和Xie预测，但对于该类中的大多数模型，$S$-对偶被增强到一个更大的群。
英文摘要：
The $S$-duality group $\mathbb{S}(\mathcal{F})$ of a 4d $\mathcal{N}=2$ supersymmetric theory $\mathcal{F}$ is identified with the group of triangle auto-equivalences of its cluster category $\mathscr{C}(\mathcal{F})$ modulo the subgroup acting trivially on the physical quantities. $\mathbb{S}(\mathcal{F})$ is a discrete group commensurable to a subgroup of the Siegel modular group $Sp(2g,\mathbb{Z})$ ($g$ being the dimension of the Coulomb branch). This identification reduces the determination of the $S$-duality group of a given $\mathcal{N}=2$ theory to a problem in homological algebra. In this paper we describe the techniques which make the computation straightforward for a large class of $\mathcal{N}=2$ QFTs. The group $\mathbb{S}(\mathcal{F})$ is naturally presented as a generalized braid group. The $S$-duality groups are often larger than expected. In some models the enhancement of $S$-duality is quite spectacular. For instance, a QFT with a huge $S$-duality group is the Lagrangian SCFT with gauge group $SO(8)\times SO(5)^3\times SO(3)^6$ and half-hypermultiplets in the bi- and tri-spinor representations. We focus on four families of examples: the $\mathcal{N}=2$ SCFTs of the form $(G,G^\prime)$, $D_p(G)$, and $E_r^{(1,1)}(G)$, as well as the asymptotically-free theories $(G,\widehat{H})$ (which contain $\mathcal{N}=2$ SQCD as a special case). For the $E_r^{(1,1)}(G)$ models we confirm the presence of the $PSL(2,\mathbb{Z})$ $S$-duality group predicted by Del Zotto, Vafa and Xie, but for most models in this class $S$-duality gets enhanced to a much larger group.
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