Gravitational path integral from the $T^2$ deformation

TL;DR

This paper explores a $T^2$ deformation of large $N$ conformal field theories, revealing a connection to gravitational path integrals and the Wheeler-DeWitt equation, and proposing new insights into holographic dualities and quantum gravity.

Contribution

It introduces a gravitational path integral representation for the $T^2$ deformed partition function and relates it to the Wheeler-DeWitt equation and gauge-invariant phase space quantization.

Findings

The deformed partition function satisfies a diffusion-type flow equation.

The gravitational path integral acts as the diffusion kernel solving the flow equation.

A gauge-invariant relation between the deformed partition function and the radial WDW wave function is established.

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.