高阶S-对偶与Shephard-Todd群
Higher S-dualities and Shephard-Todd groups
摘要：
Seiberg和Witten证明了在N=2的SQCD中，$N_f=2N_c=4$的S-对偶群PSL（2，Z）通过triality作用于Spin（8）的味荷。还有其他N=2的SCFT，其中SU（2）SYM耦合到强相互作用的非拉格朗日物质：它们的物质电荷是$E_6$，$E_7$和$E_8$的重量，而不是Spin（8）。S-对偶群PSL（2，Z）作用于这些权：用什么代替$E_6,E_7,E_8$根格的Spin（8）三重性？在本文中，我们回答这个问题。PSL（2，Z）（的有限中心扩张）对物质荷的作用通过特殊Shephard-Todd群$G_4$和$G_8$的作用进行因式分解，这两个群应该被看作是通常的三元群$\mathfrak{S}_3\simeq \mathrm{Weyl}(A_2)$的复杂类似物。我们的分析是基于SU（2）规范SCFT的S-对偶与管状加权射影线簇范畴的自同构群的识别。
英文摘要：
Seiberg and Witten have shown that in N=2 SQCD with $N_f=2N_c=4$ the S-duality group PSL(2,Z) acts on the flavor charges, which are weights of Spin(8), by triality. There are other N=2 SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of $E_6$, $E_7$ and $E_8$ instead of Spin(8). The S-duality group PSL(2,Z) acts on these weights: what replaces Spin(8) triality for the $E_6,E_7,E_8$ root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL(2,Z) factorizes trough the action of the exceptional Shephard--Todd groups $G_4$ and $G_8$ which should be seen as complex analogs of the usual triality group $\mathfrak{S}_3\simeq \mathrm{Weyl}(A_2)$. Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type.
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