Gravitational path integral from the $T^2$ deformation

TL;DR

This work analyzes a $T^2$ deformation of large $N$ CFTs, a higher-dimensional analog of TTbar, showing that the deformation flow is diffusion-like with a kernel given by a Euclidean gravitational path integral in $d{+}1$ dimensions between two Dirichlet boundaries. It establishes a gauge-invariant connection between the deformed partition function $Z^{(\lambda)}$ and the radial Wheeler–DeWitt wavefunction $\\Psi$, and clarifies the role of counterterms and the gravitational path-integral measure, including a Ricci potential that yields the Einstein–Hilbert action. The authors then relate the radial wavefunction to Hartle–Hawking states and discuss York time, the Lorentzian continuation, and a formula linking the extremal-volume of the maximal slice to derivatives of the HH wavefunction, hinting at a complexity=volume-type interpretation. The discussion highlights open issues such as the uniqueness of the flow, the possibility of a fake bulk, and connections to the Freidel kernel in $d=2$, outlining future directions for validating the gravity interpretation of finite-cutoff holography in higher dimensions.

Abstract

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.

Figures (1)

• Figure 1: Three different Dirichlet problems.