从Lie 2-代数胚视角看引力与广义几何
Gravity and generalised geometry from a Lie 2-algebroid perspective
- Rudjer Bošković Institute（鲁德·博什科维奇研究所）
- Imperial College（帝国理工学院）
- London Institute for Mathematical Sciences, Royal Institution（伦敦数学科学研究所，皇家学会）
- Dublin Institute for Advanced Studies（都柏林高等研究院）
- Galatasaray Üniversitesi（加拉塔萨雷大学）
- Constructor University（康斯特克特大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文从Lie 2-代数胚视角重新定义广义联络的挠率与曲率，消除歧义，证明唯一相容广义联络存在，并重构闭弦有效作用量。
AI中文摘要：
我们重新审视了为广义联络定义自然挠率和曲率张量的问题，并利用这些张量从广义几何中重构闭弦无质量模式的有效作用量。出发点是库朗特代数胚作为微分分次（dg）辛流形的描述。对分裂dg流形的一种系统化程序将广义李括号作为更大的Lie 2-代数胚结构的一部分单独挑出，该结构具有正确定义挠率和黎曼曲率张量及其相应比安基恒等式所需的性质。这一框架消除了文献中先前方法的结构性循环和歧义问题。我们证明，对于任何分裂dg流形的普通联络，存在唯一的与广义度量相容的广义联络且具有固定的（共）挠率，同时所有以这种方式获得的Lie 2-代数胚在$L_{\infty}$拟同构意义下等价。对于给定的Lie 2-代数胚，我们证明，在规范选择挠率后，唯一相容的广义联络基于新的广义里奇张量通过几何作用量重现了闭弦有效作用量。
英文摘要：
We revisit the problem of defining natural torsion and curvature tensors for generalised connections and of using them to reconstruct the effective action for the massless modes of the closed string from generalised geometry. The starting point is the description of a Courant algebroid as a differential graded (dg) symplectic manifold. A systematic procedure for split dg manifolds singles out a generalised Lie bracket as part of a larger Lie 2-algebroid structure, with the right properties to unambiguously define torsion and Riemann curvature tensors together with their corresponding Bianchi identities. This framework eliminates the structural circularity and ambiguity problems of previous approaches in the literature. We show that a unique generalised metric compatible generalised connection with fixed (con)torsion exists for any choice of ordinary connection that splits the dg manifold, while at the same time all Lie 2-algebroids obtained in this way are equivalent up to $L_{\infty}$ quasi-isomorphisms. For a given Lie 2-algebroid we demonstrate that the unique compatible generalised connection reproduces the effective closed string action from a geometric action based on the new generalised Ricci tensor upon making a canonical choice of torsion.
