Abstract
The holographic complexity is UV divergent. As a finite complexity, we propose a “regularized complexity” by employing a similar method to the holographic renor-malization. We add codimension-two boundary counterterms which do not contain any boundary stress tensor information. It means that we subtract only non-dynamic back-ground and all the dynamic information of holographic complexity is contained in the regularized part. After showing the general counterterms for both CA and CV conjectures in holographic spacetime dimension 5 and less, we give concrete examples: the BTZ black holes and the four and five dimensional Schwarzschild AdS black holes. We propose how to obtain the counterterms in higher spacetime dimensions and show explicit formulas only for some special cases with enough symmetries. We also compute the complexity of formation by using the regularized complexity.
Highlights
B tL rh rh tL can be characterized by the codimension-two surface at fixed times t = tL and t = tR at the two boundaries of the AdS black hole [10, 11]
We studied how to obtain the finite term in a covariant manner from the holographic complexity for both CV and CA conjectures when the boundary geometry is not deformed by relevant operators
Inspired by the recent results that the divergent terms are determined only by the boundary metric and have no relationship to the stress tensor and bulk matter fields, we showed that such divergences can be canceled by adding codimension-two boundary counterterms
Summary
To regularize the complexity we may try the same method as the entanglement entropy case, for example, in refs. [30,31,32] i.e. find out the divergent behavior and just discard. To propose a well defined subtraction for the regularized complexity, we follow the procedure of the holographic renormalization [23,24,25,26] In this procedure, the divergences are canceled by adding covariant local boundary surface counterterms determined by the near-boundary behaviour of bulk fields. As the α and β can not be determined by theory itself, such terms cannot be written as the covariant geometrical quantities of the boundary metric This results show that it is necessary to add the term Iλ into the action (2.11) to obtain an covariant regularized complexity. We see that this is just the value shown in eq (2.3) In this coordinate system, the surface counterterm is as the same as the background subtraction term and we find the regularized complexity is just as the same as one shown in eq (2.5).
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