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arXiv 2307.08094hep-th
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作为同伦的全息对偶

Holography as Homotopy

Christoph Chiaffrino, Talha Ersoy, Olaf Hohm

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50 pages, 1 figure. v2: references added, typo corrected
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AI总结：

本文用同伦代数解释AdS/CFT全息对偶，证明体理论的循环$L_{\infty}$代数通过同伦收缩等价于边界代数，其作用量对应CFT关联函数的生成泛函。

摘要：

我们用同伦代数解释了AdS/CFT对应形式下的全息对偶。诸如体引力理论这样的场论可以被视为同伦Lie代数或$L_{\infty}$代数。我们将这一对应关系扩展到定义在带有边界的流形（包括AdS的共形边界）上的理论，并考虑了定义带有正确边界项的作用量所需的循环结构。将场投影到其边界值会定义一个同伦收缩，这反过来意味着体理论的循环$L_{\infty}$代数在同伦意义下等价于边界上的循环$L_{\infty}$代数。由此产生的作用量是通过Witten图按惯例计算的“在壳作用量”，根据AdS/CFT，它产生对偶CFT关联函数的生成泛函。这些结果是在关于循环$L_{\infty}$代数的同伦传递的新技术的帮助下建立的。

英文摘要：

We give an interpretation of holography in the form of the AdS/CFT correspondence in terms of homotopy algebras. A field theory such as a bulk gravity theory can be viewed as a homotopy Lie or $L_{\infty}$ algebra. We extend this dictionary to theories defined on manifolds with a boundary, including the conformal boundary of AdS, taking into account the cyclic structure needed to define an action with the correct boundary terms. Projecting fields to their boundary values then defines a homotopy retract, which in turn implies that the cyclic $L_{\infty}$ algebra of the bulk theory is equivalent, up to homotopy, to a cyclic $L_{\infty}$ algebra on the boundary. The resulting action is the `on-shell action' conventionally computed via Witten diagrams that, according to AdS/CFT, yields the generating functional for the correlation functions of the dual CFT. These results are established with the help of new techniques regarding the homotopy transfer of cyclic $L_{\infty}$ algebras.

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