标题： 边界共形流形上的高阶结构：高阶贝里相位与边界共形场论 显示英文标题

标题： Higher Structures on Boundary Conformal Manifolds: Higher Berry Phase and Boundary Conformal Field Theory

摘要： 我们在(1+1)d共形场理论(CFT)的共形边界条件空间中引入了更高贝里连接和曲率的概念，这些边界条件通过精确边缘变形相互关联，形成一个"边界共形流形"。 我们的定义建立在之前关于张量网络的研究基础上，例如矩阵乘积态(MPS)，其中三重内积或多波函数重叠在几何上起关键作用。 一方面，我们对更高贝里相位的边界共形场论(BCFT)表述为研究凝聚态系统中可逆相族提供了一种新的分析工具。 另一方面，它揭示了共形边界条件模空间中的新几何结构，超越了通过扎莫洛德奇科夫度规定义的通常黎曼结构。 当边界共形流形被解释为D膜的位置模空间时，我们的更高贝里连接与弦理论中的NS-NS$B$-场一致。 一般定义不需要这种解释，并且纯粹以场论方式表述，涉及边界条件改变(bcc)算符的相关函数。 我们还探讨了更高贝里连接与边界共形流形环空间中的泛函贝里连接之间的关系。 显示英文摘要

摘要： We introduce the notion of higher Berry connection and curvature in the space of conformal boundary conditions in (1+1)d conformal field theories (CFT), related to each other by exactly marginal boundary deformations, forming a "boundary conformal manifold." Our definition builds upon previous works on tensor networks, such as matrix product states (MPS), where the triple inner product or multi-wavefunction overlap plays the key geometric role. On the one hand, our boundary conformal field theory (BCFT) formulation of higher Berry phase provides a new analytic tool to study families of invertible phases in condensed matter systems. On the other hand, it uncovers a new geometric structure on the moduli space of conformal boundary conditions, beyond the usual Riemannian structure defined through the Zamolodchikov metric. When the boundary conformal manifold has an interpretation as the position moduli space of a D-brane, our higher Berry connection coincides with the NS-NS $B$-field in string theory. The general definition does not require such an interpretation and is formulated purely field-theoretically, in terms of correlation functions of boundary-condition-changing (bcc) operators. We also explore a connection between higher Berry connections and functional Berry connections in the loop spaces of boundary conformal manifolds.