We build a bridge between two algebraic structures in superconformal field theories (SCFT): a vertex operator algebra (VOA) in the Schur sector of 4d \(\mathcal {N}=2\) theories and an associative algebra in the Higgs sector of 3d \(\mathcal {N}=4\). The natural setting is a 4d \(\mathcal {N}=2\) SCFT placed on \(S^3\times S^1\): by sending the radius of \(S^1\) to zero, we recover the 3d \(\mathcal {N}=4\) theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle. We prove that: (1) the Higgs branch operators remain in the cohomology; (2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on the \(S^1\); (3) no new cohomology classes are added. We show that the algebra in 3d is given by the quotient \(\mathcal {A}_H = \mathrm{Zhu}_{s}(V)/N\), where \(\mathrm{Zhu}_{s}(V)\) is the non-commutative Zhu algebra of the VOA V (for \({s}\in \mathrm{Aut}(V)\)), and N is a certain ideal. This ideal is the null space of the (s-twisted) trace map \(T_{s}: \mathrm{Zhu}_{s}(V) \rightarrow \mathbb {C}\) determined by the torus 1-point function in the high temperature (or small complex structure) limit. It therefore equips \(\mathcal {A}_H\) with a non-degenerate (twisted) trace, leading to a short star-product according to the recent results of Etingof and Stryker. The map \(T_{s}\) is easy to determine for unitary VOAs, but has a much subtler structure for non-unitary and non-\(C_2\)-cofinite VOAs of our interest. We comment on relation to the Beem-Rastelli conjecture on the Higgs branch and the associated variety. A companion paper will explore further details, examples, and some applications of these ideas.

我们在超共形场理论（SCFT）中的两个代数结构之间建立了一座桥梁：4d \（\ mathcal {N} = 2 \）理论的Schur扇区中的顶点算子代数（VOA）和3d \（\ mathcal {N} = 4 \）。自然的设置是放置在\（S ^ 3 \ times S ^ 1 \）上的4d \（\ mathcal {N} = 2 \） SCFT ：通过将\（S ^ 1 \）的半径发送为零，我们可以恢复3d \（\ mathcal {N} = 4 \）理论，圆环上相应的VOA退化为圆上的关联代数。我们证明：（1）希格斯分支算子仍然是同调的；（2）所有非希格斯类型的Schur运算符都由包裹在\（S ^ 1 \） ; （3）没有添加新的同调类。我们证明3d代数由商\（\ mathcal {A} _H = \ mathrm {Zhu} _ {s}（V）/ N \）给出，其中\（\ mathrm {Zhu} _ {s} （V）\）是VOA V的非可交换的Zhu代数（对于\（{s} \ in \ mathrm {Aut}（V）\）），并且N是某个理想值。这个理想是（s扭曲）轨迹图\（T_ {s}：\ mathrm {Zhu} _ {s}（V）\ rightarrow \ mathbb {C} \）的零空间，由圆环1点确定在高温（或小型复杂结构）极限内起作用。因此，它装备\（\ mathcal {A} _H \）根据Etingof和Stryker的最新结果，其具有非简并的（扭曲的）痕迹，导致了短星状产物。映射\（T_ {s} \）易于确定单一VOA，但对于我们感兴趣的非单一和非\（C_2 \）有限VOA具有微妙的结构。我们评论了希格斯分支上的Beem-Rastelli猜想及其相关变体的关系。随附的论文将探讨这些思想的更多细节，示例和一些应用。