论文标题
$ \ MATHCAL {N} = 4 $ SYM，ARGYRES-DOUGLAS理论和确切的分级向量空间同构
$\mathcal{N}=4$ SYM, Argyres-Douglas Theories, and an Exact Graded Vector Space Isomorphism
论文作者
论文摘要
在两篇论文中的第一篇论文中，我们详细解释了显然截然不同的理论之间一系列更广泛关系的最简单示例。我们的示例将$ \ mathfrak {su}（2）$ $ $ \ mathcal {n} = 4 $ super yang-mills（sym）与我们称为“ $（3,2）$的理论”。后一种理论是一个准确的边缘对角线$ su（2）$ a $ d_3（su（2））$ argyres-douglas（ad）理论。我们首先要观察到这两种理论的Schur指数是通过代数转换相关的，该代数转换令人惊讶地让人联想到索引转换，描述了希格斯分支上自发对称性破坏的索引转换。但是，这种转换打破了Sym理论的超对称性，以及其完整的$ \ Mathcal {n} = 2 $ $ $ su（2）_f $ take Symmetry。 Moreover, it does so in an interesting way when viewed through the lens of the corresponding 2D vertex operator algebras (VOAs): affine currents of the small $\mathcal{N}=4$ super-Virasoro algebra at $c=-9$ get mapped to the $\mathcal{A}(6)$ stress tensor and some of its conformal descendants, while the extra $ \ MATHCAL {N} = 4 $侧面上的超对称电流被映射到$ \ Mathcal {a}（6）$ side上的高维效率电流及其后代。我们证明这些关系是这两个VOA之间精确分级的矢量空间同构（GVSI）的方面。此GVSI尊重父4D理论的$ U（1）_r $费用。我们简要概述了更多的一般$ \ mathfrak {su}（n）$ $ $ $ \ mathcal {n} = 4 $ SYMERIES通过我们的示例的概括与无限的AD理论类别有关。最后，我们表明，在这类理论中，$ \ Mathcal {a}（6）$ voa对强生成器数量的新不平等饱和。
In this first of two papers, we explain in detail the simplest example of a broader set of relations between apparently very different theories. Our example relates $\mathfrak{su}(2)$ $\mathcal{N}=4$ super Yang-Mills (SYM) to a theory we call "$(3,2)$." This latter theory is an exactly marginal diagonal $SU(2)$ gauging of three $D_3(SU(2))$ Argyres-Douglas (AD) theories. We begin by observing that the Schur indices of these two theories are related by an algebraic transformation that is surprisingly reminiscent of index transformations describing spontaneous symmetry breaking on the Higgs branch. However, this transformation breaks half the supersymmetry of the SYM theory as well as its full $\mathcal{N}=2$ $SU(2)_F$ flavor symmetry. Moreover, it does so in an interesting way when viewed through the lens of the corresponding 2D vertex operator algebras (VOAs): affine currents of the small $\mathcal{N}=4$ super-Virasoro algebra at $c=-9$ get mapped to the $\mathcal{A}(6)$ stress tensor and some of its conformal descendants, while the extra supersymmetry currents on the $\mathcal{N}=4$ side get mapped to higher-dimensional fermionic currents and their descendants on the $\mathcal{A}(6)$ side. We prove these relations are facets of an exact graded vector space isomorphism (GVSI) between these two VOAs. This GVSI respects the $U(1)_r$ charge of the parent 4D theories. We briefly sketch how more general $\mathfrak{su}(N)$ $\mathcal{N}=4$ SYM theories are related to an infinite class of AD theories via generalizations of our example. We conclude by showing that, in this class of theories, the $\mathcal{A}(6)$ VOA saturates a new inequality on the number of strong generators.
