There has been renewed interest in the path-integral computation of the partition function of AdS 3 gravity, both in the metric and Chern-Simons formulations. The one-loop partition function around Euclidean AdS 3 turns out to be given by the vacuum character of Virasoro group. This stems from the work of Brown and Henneaux (BH) who showed that, in AdS 3 gravity with sensible asymptotic boundary conditions, an infinite group of (improper) diffeomorphisms arises which acts canonically on phase space as two independent Virasoro symmetries. The gauge group turns out to be composed of so-called “proper” diffeomorphisms which approach the identity at infinity fast enough. However, it is sometimes far from evident to identify where BH boundary conditions enter in the path integral, and much more difficult to see how the improper diffeomorphisms are left out of the gauge group. In particular, in the metric formulation, Giombi, Maloney and Yin obtained the one-loop partition function around thermal AdS 3 resorting to the heat kernel method to compute the determinants coming from the path integral. Here we identify how BH boundary conditions follow naturally from the usual requirement of square-integrability of the metric perturbations. Also, and equally relevant, we clarify how the quotient by only proper diffeomorphisms is implemented, promoting the improper diffeomorphisms to symmetries in the path integral. Our strategy is general enough to apply to other approaches where square integrability is assumed. Finally, we show that square integrability implies that the asymptotic symmetries in higher dimensional AdS gravity are just isometries.

中文翻译：

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AdS重力配分函数中的渐近边界条件和平方可积性

在度量和陈-西蒙斯公式中，对 AdS 3 引力分配函数的路径积分计算重新产生了兴趣。围绕 Euclidean AdS 3 的单循环分配函数原来是由 Virasoro 群的真空特性给出的。这源于 Brown 和 Henneaux (BH) 的工作，他们表明，在具有合理渐近边界条件的 AdS 3 引力中，出现了无限组（不适当的）微分同胚，它们作为两个独立的 Virasoro 对称性在相空间上规范地起作用。规范群原来是由所谓的“真”微分同胚组成，它们足够快地在无穷远处逼近恒等式。然而，有时很难确定 BH 边界条件进入路径积分的位置，更难看出不正确的微分同胚是如何被排除在规范群之外的。特别是，在度量公式中，Giombi、Maloney 和 Yin 获得了围绕热 AdS 3 的单环分配函数，采用热核方法来计算来自路径积分的行列式。在这里，我们确定了 BH 边界条件如何自然地遵循度量扰动的平方可积性的通常要求。此外，同样重要的是，我们阐明了仅通过适当微分同胚实现的商是如何实现的，将不适当微分同胚提升为路径积分中的对称性。我们的策略足够通用，可以应用于假设平方可积性的其他方法。最后，