从非点状和RG流的指标
The index from a nonpoint and RG flows
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- Texas A&M University（德州农工大学）
- Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）
- Korea Advanced Institute of Science and Technology（韩国科学技术院）
- California Institute of Technology（加州理工学院）
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中文总结 AI 辅助
本文扩展了双滤过仿射簇构造，通过普适势和syzygy最大化确定Argyres--Douglas理论的Higgs分支簇，并利用拉回实现RG流，从而系统计算Macdonald指标。
中文摘要 AI 辅助
我们将双滤过仿射簇的代数几何构造扩展到具有非平凡Higgs分支的理论，该构造编码了4维$\mathcal{N}=2$超共形场论的Higgs分支和Macdonald指标。对于Argyres--Douglas理论$\mathcal{D}_p(SU(N))$，我们发现一个由$\mathfrak{sl}(N)$的Casimir不变量函数给出的普适势决定了该簇。该簇由'加胖'的幂零轨道闭包给出，编码了4维理论的OPE退耦结构。我们称之为syzygy最大化的过程在广泛的例子中确定了精确的势，所得的簇再现了4维数据。底层的分次环给出了相关顶点算子代数的Zhu的$C_2$-代数的猜想表示，而双滤过通过保留R-对称信息细化了Arakawa的关联簇。我们发现Higgs分支重整化群流通过将相同的普适势拉回到相应的Slodowy切片来实现，包括红外Higgs分支为一点的流。特别是，我们的构造为确定$\mathcal{D}_p(SU(N), O)$理论的Macdonald指标提供了一种系统方法，此前这些理论没有已知的一般表达式。
英文摘要
We extend the algebro-geometric construction of bifiltered affine schemes that encode both the Higgs branch and the Macdonald index of 4d $\mathcal{N}=2$ superconformal field theories to theories with non-trivial Higgs branches. For the Argyres--Douglas theories $\mathcal{D}_p(SU(N))$, we find that a universal potential, given by a function in the Casimir invariants of $\mathfrak{sl}(N)$, determines the scheme. The scheme is given by a `fattened' nilpotent orbit closure, encoding the OPE decoupling structure of the 4d theory. A procedure we call syzygy maximization fixes the precise potential in a broad range of examples, and the resulting schemes reproduce the 4d data. The underlying graded ring gives conjectural presentations of Zhu's $C_2$-algebra of the associated vertex operator algebra, while the bifiltration refines Arakawa's associated scheme by retaining the R-symmetry information. We find that Higgs branch renormalization group flows are realized by pulling back the same universal potential to the corresponding Slodowy slices, including flows for which the infrared Higgs branch is a point. In particular, our construction provides a systematic method for determining the Macdonald indices of the $\mathcal{D}_p(SU(N), O)$ theories, for which no general expressions were previously known.
