Gravitational path integral from the $T^2$ deformation

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.

arXiv comment30 pages, 1 figure; v2 references added

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Cited by in corpus (21)

• A pedagogical review on solvable irrelevant deformations of 2d quantum field theory
• Quantum Extremal Islands Made Easy, PartIII: Complexity on the Brane
• Cauchy Slice Holography: A New AdS/CFT Dictionary
• Holography from the Wheeler-DeWitt equation
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• Geometrizing $T\bar{T}$
• Wheeler-DeWitt states of the AdS-Schwarzschild interior
• Conformal Boundary Conditions from Cutoff AdS${}_3$
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• Building Tensor Networks for Holographic States
• Maximally Supersymmetric RG Flows in 4D and Integrability
• $T\bar{T}$ and $J\bar{T}$ Deformations in Quantum Mechanics
• $T\overline{T}$-deformation of $q$-Yang-Mills theory
• $T\bar{T}+{\Lambda }_2$ from a 2d gravity path integral
• Holographic boundary conformal field theory with $T\bar{T}$ deformation
• Phase Transition in JT Gravity and $T\bar{T}$ Deformation
• Holographic and QFT Complexity with angular momentum
• Cost of holographic path integrals
• On the Spread of Entanglement at Finite Cutoff