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.
Highlights
Where IWDW is the on-shell gravitational action on a region of bulk spacetime called Wheeler-De Witt (WDW) patch
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
In the second method, the null boundaries of the WDW patch are started at the asymptotic boundary of the bulk spacetime at r = ∞, such that the WDW patch is excised by the cutoff surface at r = rmax
Summary
Which is based on [48], we review the CFT state that is dual to the geometry drawn in figure 1. We have two CFTs on the left and right asymptotic boundaries where |Ei L,R are the corresponding energy eigenstates. Notice that in contrast to eq (1.4) the summation is over all of the energy eigenstates of the dual CFT. This state which is a pure state is dual to a two-sided AdS black hole truncated by a dynamical ETW brane. One can obtain it by Euclidean time evolution from a highly excited pure state |B in the CFT. One can obtain the state |ΨB by a Euclidean path integral with a boundary condition at Euclidean time τ. The state |B might be regarded as a boundary state in the CFT [48]
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