# Complexity growth and shock wave geometry in AdS-Maxwell-power-Yang–Mills theory

## Abstract

We study effects of non-abelian gauge fields on the holographic characteristics for instance the evolution of computational complexity. To do so we choose Maxwell-power-Yang–Mills theory defined in the AdS space-time. Then we seek the impact of charge of the YM field on the complexity growth rate by using \(complexity=action\) conjecture. We also investigate the spreading of perturbations near the horizon and the complexity growth rate in local shock wave geometry in presence of the YM charge. At last we check validity regime of Lloyd bound.

## 1 Introduction

*G*is the Newton‘s coupling constant,

*L*is a length scale of the system which is given by the AdS radius for large black holes and the horizon radius for small black holes.

*V*is the volume of spacelike slice or the Einstein–Rosen bridge (ER) with connected points \(t_L\) and \(t_R\) corresponding to the left and the right boundaries, respectively. (II) At the newer conjecture, quantum complexity is proportional to classical action in the bulk which is defined in “Wheeler-DeWitt”’ (WD) patch, or CA [5, 6]. The privilege of this conjecture rather than the older one is needlessness to any length scale chosen by hand, such as “L” or the event horizon radius,

*WDW*patch is given by the summation of the action and all boundary terms of this patch which is defined between the times \(t_L\) and \(t_R\) on the boundaries and at late time approximation could be restricted by the Lloyd bound [7] as follows:

*E*is the excited energy of the boundary quantum state. It is now understood that satisfying the Lloyd bound in holographic theories are due to the orthogonality of quantum states and so in general does not need to be hold [8, 9]. To obtain the growth rate of complexity on the boundary we must calculate the time derivative of this action on the boundary by attention to conjecture of “CA” given by (1.2) as

*CA*conjecture for the

*WDW*patch at late time approximation in

*AdS*black holes is bounded as follows [5, 6]:

at which \(+\) and − stand for the states of the most outer and inner horizon, respectively. Equality satisfies for stationary AdS black holes in Einstein gravity and charged AdS black hole in Gauss–Bonnet gravity. In general non-stationary cases inequality would be expected. In the other words the work [10] shows that there is a universal formula for the action growth of stationary black holes for which the Lloyd bound is independent of the charged black hole size and so would be satisfied for any arbitrary size of charged black hole.

In this work we use non-rotating case of this universal form of the action growth and seek relation between this bound and the non-abelian charges. To do so we choose the Yang–Mills (YM) field given in the Maxwell-power-Yang–Mills theory propagating in AdS spacetime. There are two important motivations which encourage us to consider YM field in our study: at first we can find YM equations in the low energy limit of some string theory models which leads to certain revisions of the no-hair theory of black hole physics. Secondly, the unification of general relativity and quantum mechanics in high energy regime of the most of string theory models is possible when it predicts a non-abelian gauge field.

*W*(

*t*) which are separated in spatial coordinates such that [14]

## 2 The complexity growth

*G*in general

*d*-dimensional theory and \(A_\mu ^{(a)}\) is the \(SO(d-1)\) gauge group Yang–Mills potentials. According to Wu-Yang ansatz the YM invariant \({\mathcal {F}}\) reduces to the following form [23].

*f*(

*r*) like an equipotential surface \(f(r)=constant\) the first law of the black hole thermodynamics could be derived for which

## 3 The complexity growth in a shock wave geometry

*W*(

*t*) acts on the boundary at the same time. It will be effective on the initial state of black hole which is a thermofield double state (

*TFD*), so the old state changes to \(W(t_w)|TFD\rangle \). For a local shock wave this operator depends on transverse coordinates and localizes on the boundary at

*x*. \(W(t_w,x)\) grows in this spatial direction vs the time which leads to the growth of action due to the perturbation on the boundary. In the other word action growth depends on the growth velocity of perturbation on the boundary in spatial direction which is called “butterfly velocity”. To see the evolution of action growth in the presence of a local shock wave it would be useful to study this perturbation in more details. Shock wave perturbs the black hole solution by injection of a small amount of energy from the boundary of AdS spacetime towards the horizon. This perturbation grows by raising the time due to the back reaction effects and so propagates on the horizon. By attention to the work presented by Dary and t‘Hooft [26], we study the problem in Kruskal null coordinates (

*u*,

*v*) as ,

*r*which is defined by \(dr_*=\frac{dr}{f(r)}\). The effect of shock wave geometry is considered as the effect of a massless particle at \(u=0\) which moves in the direction of

*v*with the speed of light. So geometry for \(u<0\) stays unchanged like (2.6) and in Kruskal form will be [20] :

*u*we can determine new coordinates system from the old ones by using the well known step function \(\theta (u)\) such as follows.

*S*and it is a delay time on the action growth due to the “switchback effect”. In other side when the shock wave is local, the shift function depends on the transverse coordinates. By solving the equations in this case we have an extra term in the above exponential part. If there is just one transverse coordinate, named

*x*, so the shift function yields:

*f*(

*r*) defined in (2.7) or (2.8) the butterfly velocity reads

## 4 Conclusion and summary

In this work we used a black hole metric solution containing the electric and the Yang–Mills charges and calculate corresponding complexity growth rate by applying conjecture of “complexity=action” [5, 6]. We obtained that the Lloyd bound is saturated only for \(\gamma \ge 1\) in late time approximation, but not for values less than one. In the other side, when the boundary is disturbed by a small amount of energy and so the spacetime takes form of a shock wave geometry [4, 10], then the spreading of perturbation near the horizon affects on the complexity growth rate via the butterfly velocity. We show that the existence of the Yang–Mills field causes to increase the butterfly velocity and it decreases by raising the \(\gamma \) factor of the YM field. This is in an opposition direction of [29] at which the violation of Lloyd bound is correlated to the exceeding of butterfly velocity from the speed of light. It is shown that in large shift condition the action of WDW patch raises as linearly by increasing the butterfly velocity \(v_B\).

## Notes

### Acknowledgements

Authors should thank to the editor and anonymous referees for their comments and suggestions which cause to improve this work for readers. They (E. Y. and M. F.) have appreciate also for hospitality and generosity behavior of the Lorentz Institute of theoretical physics at Leiden University of Netherlands.

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