关于膜、映射与层的实验
Experiments with membranes, maps and sheaves
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中文总结 AI 辅助
该研究受M理论存在的推测启发，针对带环面作用的卡拉比-丘五维流形，建立并数值检验了其格罗莫夫-威滕不变量与三维子流形的潘德哈里潘德-托马斯不变量通过膜指标实现的对应，还给出局部射影空间低次膜指标的推测公式并验证几何工程化结论。
中文摘要 AI 辅助
受M理论推测存在的启发，我们研究了带环面作用的卡拉比-丘五维流形Z的计数不变量之间的对应网络，其中包括五维格罗莫夫-威滕（Gromov-Witten）不变量与Z中三维子流形X的K理论潘德哈里潘德-托马斯（Pandharipande-Thomas）不变量之间的对应，该对应由所谓的膜指标（generalized Gopakumar-Vafa不变量）介导。我们针对受限环面作用的带形几何建立了该对应，并针对一般环面作用对其进行了数值检验；还为闭顶点、局部曲面及局部射影空间提供了进一步的数值证据，其中针对局部射影空间，我们给出了其低次膜指标的推测公式。当五维流形是仿射平面与合适的环簇的乘积时，我们证明了几何工程化，并将潘德哈里潘德-托马斯不变量的生成级数与对应的瞬子配分函数等同，同时通过数值检验确认了其与格罗莫夫-威滕级数的等价性。
英文摘要
Motivated by the conjectural existence of M-theory, we investigate a web of correspondences between enumerative invariants of a Calabi-Yau fivefold $Z$ with a torus action. This includes a correspondence between the fivefold Gromov-Witten invariants and K-theoretic Pandharipande-Thomas invariants of a threefold $X \subset Z$, mediated by so-called membrane indices, which generalise Gopakumar-Vafa invariants. We establish the correspondence for strip geometries for restricted torus actions and test it numerically for general torus actions. Further numerical evidence is provided for the closed vertex, local surfaces and local projective spaces. For the latter we present conjectural formulae for their low-degree membrane indices. When the fivefold is the product of the affine plane with a suitable toric variety, we prove geometric engineering and equate the generating series of Pandharipande-Thomas invariants with the corresponding instanton partition function while equality with the Gromov-Witten series is probed numerically.
发表机构
- ISTA（奥地利科学与技术研究院）
- ETH Zurich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。
