We study a $T^2$ deformation of large $N$ conformal field theories, a higher dimensional generalization of the $T\bar T$ deformation. The deformed partition function satisfies a flow equation of the diffusion type. We solve this equation by finding its diffusion kernel, which is given by the Euclidean gravitational path integral in $d+1$ dimensions between two boundaries with Dirichlet boundary conditions for the metric. This is natural given the connection between the flow equation and the Wheeler-DeWitt equation, on which we offer a new perspective by giving a gauge-invariant relation between the deformed partition function and the radial WDW wave function. An interesting output of the flow equation is the gravitational path integral measure which is consistent with a constrained phase space quantization. Finally, we comment on the relation between the radial wave function and the Hartle-Hawking wave functions dual to states in the CFT, and propose a way of obtaining the volume of the maximal slice from the $T^2$ deformation.

中文翻译：

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来自 T 2 变形的引力路径积分

我们研究了大 $N$ 共形场理论的 $T^2$ 变形，这是 $T\bar T$ 变形的高维概括。变形的分配函数满足扩散类型的流动方程。我们通过找到它的扩散内核来解决这个方程，它由两个边界之间的 $d+1$ 维的欧几里得引力路径积分给出，度量的狄利克雷边界条件。考虑到流动方程和 Wheeler-DeWitt 方程之间的联系，这是很自然的，我们通过给出变形分配函数和径向 WDW 波函数之间的规范不变关系提供了一个新的视角。流动方程的一个有趣的输出是重力路径积分测量，它与受约束的相空间量化一致。最后，