Abstract
We construct a particular flow in the space of 2D Euclidean QFTs on a torus, which we argue is dual to a class of solutions in 3D Euclidean gravity with conformal boundary conditions. This new flow comes from a Legendre transform of the kernel which implements the T\( \overline{T} \) deformation, and is motivated by the need for boundary conditions in Euclidean gravity to be elliptic, i.e. that they have well-defined propagators for metric fluctuations. We demonstrate equivalence between our flow equation and variants of the Wheeler de-Witt equation for a torus universe in the so-called Constant Mean Curvature (CMC) slicing. We derive a kernel for the flow, and we compute the corresponding ground state energy in the low-temperature limit. Once deformation parameters are fixed, the existence of the ground state is independent of the initial data, provided the seed theory is a CFT. The high-temperature density of states has Cardy-like behavior, rather than the Hagedorn growth characteristic of T\( \overline{T} \)-deformed theories.
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A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, T\( \overline{T} \)-deformed 2D Quantum Field Theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with T\( \overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].
E.A. Mazenc, V. Shyam and R.M. Soni, A T\( \overline{T} \) Deformation for Curved Spacetimes from 3d Gravity, arXiv:1912.09179 [INSPIRE].
L. Freidel, Reconstructing AdS/CFT, arXiv:0804.0632 [INSPIRE].
E. Witten, A Note On Boundary Conditions In Euclidean Gravity, arXiv:1805.11559 [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
M.T. Anderson, On boundary value problems for Einstein metrics, Geom. Topol. 12 (2008) 2009 [math/0612647] [INSPIRE].
O. Aharony, S. Datta, A. Giveon, Y. Jiang and D. Kutasov, Modular invariance and uniqueness of T\( \overline{T} \) deformed CFT, JHEP 01 (2019) 086 [arXiv:1808.02492] [INSPIRE].
S. Datta and Y. Jiang, T\( \overline{T} \) deformed partition functions, JHEP 08 (2018) 106 [arXiv:1806.07426] [INSPIRE].
S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, T\( \overline{T} \) partition function from topological gravity, JHEP 09 (2018) 158 [arXiv:1805.07386] [INSPIRE].
A. Hashimoto and D. Kutasov, T\( \overline{T} \), J\( \overline{T} \), T\( \overline{J} \) partition sums from string theory, JHEP 02 (2020) 080 [arXiv:1907.07221] [INSPIRE].
N. Callebaut, J. Kruthoff and H. Verlinde, T\( \overline{T} \) deformed CFT as a non-critical string, JHEP 04 (2020) 084 [arXiv:1910.13578] [INSPIRE].
A.J. Tolley, T\( \overline{T} \) deformations, massive gravity and non-critical strings, JHEP 06 (2020) 050 [arXiv:1911.06142] [INSPIRE].
R.L. Arnowitt, S. Deser and C.W. Misner, The Dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].
V. Moncrief, Reduction of the Einstein equations in 2 + 1 Dimensions to a Hamiltonian System over Teichmüller Space, J. Math. Phys. 30 (1989) 2907.
S. Carlip, Lectures on (2 + 1) dimensional gravity, J. Korean Phys. Soc. 28 (1995) S447 [gr-qc/9503024] [INSPIRE].
P. Caputa, S. Datta and V. Shyam, Sphere partition functions \& cut-off AdS, JHEP 05 (2019) 112 [arXiv:1902.10893] [INSPIRE].
A. Belin, A. Lewkowycz and G. Sarosi, Gravitational path integral from the T2 deformation, JHEP 09 (2020) 156 [arXiv:2006.01835] [INSPIRE].
T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, Holography at finite cutoff with a T2 deformation, JHEP 03 (2019) 004 [arXiv:1807.11401] [INSPIRE].
W. Donnelly and V. Shyam, Entanglement entropy and T\( \overline{T} \) deformation, Phys. Rev. Lett. 121 (2018) 131602 [arXiv:1806.07444] [INSPIRE].
W. Donnelly, E. LePage, Y.-Y. Li, A. Pereira and V. Shyam, Quantum corrections to finite radius holography and holographic entanglement entropy, JHEP 05 (2020) 006 [arXiv:1909.11402] [INSPIRE].
L.V. Iliesiu, J. Kruthoff, G.J. Turiaci and H. Verlinde, JT gravity at finite cutoff, SciPost Phys. 9 (2020) 023 [arXiv:2004.07242] [INSPIRE].
T. Budd and T. Koslowski, Shape Dynamics in 2 + 1 Dimensions, Gen. Rel. Grav. 44 (2012) 1615 [arXiv:1107.1287] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions, and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
W. Cottrell and A. Hashimoto, Comments on T\( \overline{T} \) double trace deformations and boundary conditions, Phys. Lett. B 789 (2019) 251 [arXiv:1801.09708] [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
V. Moncrief, Reduction of the Einstein equations in (2 + 1)-dimensions to a Hamiltonian system over Teichmüller space, J. Math. Phys. 30 (1989) 2907 [INSPIRE].
A. Hosoya and K.-i. Nakao, (2 + 1)-dimensional quantum gravity, Prog. Theor. Phys. 84 (1990) 739 [INSPIRE].
J.W. York, Jr., Gravitational degrees of freedom and the initial-value problem, Phys. Rev. Lett. 26 (1971) 1656 [INSPIRE].
R. Puzio, On the square root of the laplace–beltrami operator as a hamiltonian, Class. Quantum Grav. 11 (1994) 609.
V. Gorbenko, E. Silverstein and G. Torroba, dS/dS and T\( \overline{T} \), JHEP 03 (2019) 085 [arXiv:1811.07965] [INSPIRE].
S. Chakraborty, A. Giveon and D. Kutasov, T\( \overline{T} \), J\( \overline{T} \), T\( \overline{J} \) and String Theory, J. Phys. A 52 (2019) 384003 [arXiv:1905.00051] [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, T\( \overline{\mathrm{T}} \) and LST, JHEP 07 (2017) 122 [arXiv:1701.05576] [INSPIRE].
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Coleman, E., Shyam, V. Conformal boundary conditions from cutoff AdS3. J. High Energ. Phys. 2021, 79 (2021). https://doi.org/10.1007/JHEP09(2021)079
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DOI: https://doi.org/10.1007/JHEP09(2021)079