Regularizations of action-complexity for a pure BTZ black hole microstate
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Springer
Received: April 28, 2020 Accepted: June 15, 2020 Published: July 3, 2020
Farzad Omidi School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
E-mail: [email protected] Abstract: In the action-complexity proposal there are two different methods to regularize the gravitational on-shell action, which are equivalent in the framework of AdS/CFT. In this paper, we want to study the equivalence of them for a pure BTZ black hole microstate. The microstate is obtained from a two-sided BTZ black hole truncated by a dynamical timelike ETW brane. Moreover, it is dual to a finite energy pure state in a two-dimensional CFT. We show that if one includes the timelike counterterms inspired by holographic renormalization as well as the Gibbons-Hawking-York term on the timelike boundary of the WDW patch, which exists in one of the regularizations, the coefficients of the UV divergent terms of action-complexity in the two methods become equal to each other. Furthermore, we compare the finite terms of action-complexity in both regularizations, and show that when the UV cutoff surface is close enough to the asymptotic boundary of the bulk spacetime, action-complexities in both regularizations become exactly equal to each other. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 2004.11628
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)020
JHEP07(2020)020
Regularizations of action-complexity for a pure BTZ black hole microstate
Contents 1 Introduction
1
2 Setup 2.1 CFT picture 2.2 Holographic picture 2.3 Action-complexity
4 4 5 7 8 11 12 12 15 17 19 19 21 22 22 24 26 26 27
4 Discussion
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1
Introduction
One of the most fruitful information theoretic concepts which has been extensively explored in the context of AdS/CFT [1] is computational complexity where is proved to be very helpful in understanding the interior of black holes [2–7]. Computational complexity of a given state is defined as the minimum number of simple unitary operations, i.e. gates, to prepare the state form an initial reference state [8]. In the framework of AdS/CFT, there are different proposals including volume-complexity (CV) [2, 3, 9], action-complexity (CA) [5, 6], and the second version of the volume-complexity proposal, dubbed CV2.0 [10]. In the action-complexity proposal, the complexity C of a state on a time slice Σ in the CFT, is defined by IWDW C(Σ) = , (1.1) π~
–1–
JHEP07(2020)020
3 Comparison of regularization methods 3.1 Boundaries of WDW patch 3.2 T > 0 and rh < LT rmax 3.2.1 Regularization 1 3.2.2 Regularization 2 3.2.3 Surface terms for the timelike boundary 3.3 T > 0 and rh > LT rmax 3.3.1 Regularization 1 3.3.2 Regularization 2 3.4 T < 0 3.4.1 Regularization 1 3.4.2 Regularization 2 3.5 T = 0 3.5.1 Regularization 1 3.5.2 Regularization 2
C reg.1 = C reg.2 .
(1.2)
It was observed in refs. [16, 17] that the structures of the UV divergent terms in the two regularizations are the same. However, their coeffi
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