In this work we propose a simple and systematic framework for including edge modes in gauge theories on manifolds with boundaries. We argue that this is necessary in order to achieve the factorizability of the path integral, the Hilbert space and the phase space, and that it explains how edge modes acquire a boundary dynamics and can contribute to observables such as the entanglement entropy. Our construction starts with a boundary action containing edge modes. In the case of Maxwell theory for example this is equivalent to coupling the gauge field to boundary sources in order to be able to factorize the theory between subregions. We then introduce a new variational principle which produces a systematic boundary contribution to the symplectic structure, and thereby provides a covariant realization of the extended phase space constructions which have appeared previously in the literature. When considering the path integral for the extended bulk + boundary action, integrating out the bulk degrees of freedom with chosen boundary conditions produces a residual boundary dynamics for the edge modes, in agreement with recent observations concerning the contribution of edge modes to the entanglement entropy. We put our proposal to the test with the familiar examples of Chern-Simons and BF theory, and show that it leads to consistent results. This therefore leads us to conjecture that this mechanism is generically true for any gauge theory, which can therefore all be expected to posses a boundary dynamics. We expect to be able to eventually apply this formalism to gravitational theories.

中文翻译：

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扩展动作、边缘模式的动力学和纠缠熵

在这项工作中，我们提出了一个简单而系统的框架，用于在具有边界的流形的规范理论中包含边缘模式。我们认为，为了实现路径积分、希尔伯特空间和相空间的可分解性，这是必要的，并且它解释了边缘模式如何获得边界动力学并可以对诸如纠缠熵之类的可观察量做出贡献。我们的构建从包含边缘模式的边界动作开始。例如，在麦克斯韦理论的情况下，这等效于将规范场耦合到边界源，以便能够分解子区域之间的理论。然后我们引入了一个新的变分原理，它对辛结构产生了系统的边界贡献，从而提供了先前出现在文献中的扩展相空间结构的协变实现。当考虑扩展体 + 边界作用的路径积分时，将体自由度与选定的边界条件相结合会产生边缘模式的残余边界动力学，这与最近关于边缘模式对纠缠熵的贡献的观察一致。我们用熟悉的陈-西蒙斯和 BF 理论的例子来测试我们的建议，并表明它会导致一致的结果。因此，这使我们推测这种机制对于任何规范理论都是普遍正确的，因此可以预期所有规范理论都具有边界动力学。我们希望最终能够将这种形式主义应用于引力理论。