In this note, we study a simplified variant of the familiar holographic duality between supergravity on \(\hbox {AdS}_3\times S^3\times T^4\) and the SCFT (on the moduli space of) the symmetric orbifold theory \(Sym^N(T^4)\) as \(N \rightarrow \infty \). This variant arises conjecturally from a twist proposed by the first author and Si Li. We recover a number of results concerning protected subsectors of the original duality working directly in the twisted bulk theory. Moreover, we identify the symmetry algebra arising in the \(N\rightarrow \infty \) limit of the twisted gravitational theory. We emphasize the role of Koszul duality—a ubiquitous mathematical notion to which we provide a friendly introduction—in field theory and string theory. After illustrating the appearance of Koszul duality in the “toy” example of holomorphic Chern-Simons theory, we describe how (a deformation of) Koszul duality relates bulk and boundary operators in our twisted setup, and explain how one can compute algebra OPEs diagrammatically using this notion. Further details, results, and computations will appear in a companion paper.

在本说明中，我们研究了\（\ hbox {AdS} _3 \ times S ^ 3 \ times T ^ 4 \）上的超重力与对称球面的SCFT（在其模空间上）之间熟悉的全息对偶性的简化变体理论\（Sym ^ N（T ^ 4）\）为\（N \ rightarrow \ infty \）。该变体是由第一作者和李思（Si Li）提出的一种扭曲而产生的。我们获得了有关扭曲二重理论中直接作用于原始二元性的受保护子部分的许多结果。此外，我们确定了扭曲引力理论的\（N \ rightarrow \ infty \）极限中出现的对称代数。我们强调Koszul对偶性的作用场论和弦论是一种普遍存在的数学概念，我们对此提供了友好的介绍。在说明了全同性的Chern-Simons理论的“玩具”示例中出现了Koszul对偶性之后，我们描述了Koszul对偶性（的变形）如何在扭曲的设置中关联体和边界算符，并解释了如何使用图形化地计算代数OPE。这个概念。进一步的细节，结果和计算将出现在随附的论文中。