WdW-patches in AdS$_{3}$ and complexity change under conformal transformations II

We study the null-boundaries of Wheeler-de Witt (WdW) patches in three dimensional Poincare-AdS, when the selected boundary timeslice is an arbitrary (non-constant) function, presenting some useful analytic statements about them. Special attention will be given to the piecewise smooth nature of the null-boundaries, due to the emergence of caustics and null-null joint curves. This is then applied, in the spirit of our previous paper arXiv:1806.08376, to the problem of how complexity of the CFT$_2$ groundstate changes under a small local conformal transformation according to the action (CA) proposal. In stark contrast to the volume (CV) proposal, where this change is only proportional to the second order in the infinitesimal expansion parameter $\sigma$, we show that in the CA case we obtain terms of order $\sigma$ and even $\sigma\log(\sigma)$. This has strong implications for the possible field-theory duals of the CA proposal, ruling out an entire class of them.


I n t r o d u c t i o n
In th e past years, a consensus has form ed th a t quan tu m in form ation th e o ry has an im . . = .
w herein a length scale L has to be in trod u ced into eq u a tion ( 1.2) fo r dim en sion al reasons w h ich is usually picked to b e th e A d S scale [8][9][10][11] . T h e secon d p rop osa l is th e action proposal [10,11]  ca lcu la tin g co m p le x ity w here investigated in [13][14][15][16][17][18][19][20][21][22] follow in g the g eom etric ideas o f [1 , 2], in [23][24][25][26] follow in g p ath integral m eth od s and in [2 7 , 28] follow in g an a x io m a tic app roach .
A fascin atin g co n n e ctio n w ith grou p th e o ry was investigated in [29] . See also [30][31][32][33][34][35] for oth er relevant works. C om p arison s betw een field th e o ry ca lcu la tion s and h olog ra p h ic ca lcu la tion s o f co m p le x ity w here a ttem p ted in [36][37][38][39][40][41]  T h is, and th e fact th a t w ith cu rren tly d evelop ed techn iqu es th e ca lcu la tion o f co m p le x ity in field-theories som etim es requires th e assu m p tion o f w eak cou p lin g or even free theories, th e com p a rison s a ttem p ted in [36][37][38][39][40][41] are som ew h at lim ited t o a rather qu a lita tive level.
F or exam p le, in [40] we studied h ow in A d S 3/ C F T 2 com p lexity, a cco rd in g in trod u ction o f som e relevant co n ce p ts and n ota tion , and will b e used as a reference for ou r later m ore n on -triv ia l results. S ection 3 is d e v o te d to an exp la n a tion o f h ow we w ill stu dy con form a l tran sform ation s in A d S 3/ C F T 2, follow in g the lines o f ou r p reviou s p ap er [40].
O u r novel results th en start in section 4 , w here we discuss th e features o f generic W d Wpatches in P o in ca re -A d S 3. B ased on this, we will th en ca lcu la te con trib u tion s to the a ction on th e W d W -p a tc h term b y term , starting w ith th e bulk term in section 5 , and then m ovin g on t o th e surface term s (section 6) , th e p aram etriza tion o f the null-rays con stitu tin g th e n u ll-bou n daries in section 7 , jo in t-te rm s in section s 8 and 9 , and finally th e so called cou n ter term s in section 10. W e close w ith a su m m ary and co n clu sio n in section 11. Further techn ica l details w ill b e relegated to th e a pp en dices A and B .
1 Another conjectured holographic dual of bulk volumes is the so called fidelity susceptibility [43], see however [44] for a recent critique of this proposal. t o th e future and th e past, respectively. In ord er t o avoid divergen ces, a c u to ff surface has t o be im p osed near th e b o u n d a ry at z = e. Similarly, a c u to ff ca n b e im p osed at z = z max, w ith z max ^ to . A s p oin ted ou t in [48] , it is gen erically n ot p ossible t o ca lcu la te the con trib u tion s from a null b o u n d a ry t o th e a ction via a lim itin g p roced u re from fam ily o f tim elike or spacelike b oun d aries, w ith th e e x ce p tio n b ein g th e case w here th e n u ll-b ou n d a ry in q u estion is a K illin g horizon . T h is is th e case for th e P o in ca re-h orizon . A n o th e r intricacy arises in defining th e W d W -p a tc h in th e presence o f a sm all c u to ff e, see a p p e n d ix D .4 o f [46] and also [49]. and are hence intersected b y th e c u to ff surface. W e will pick th e latter con ven tion , w hich appears to b e th e overall m ore co m m o n on e in th e literature. It was also show n in [4 6 , 49] 2In contrast to the notations and convention of [40,45], we are using a coordinate z instead of A, with A = 1/ z 2.
3We use this term instead of lightsheet, as a priori the lightfronts bounding a WdW-patch do not have to satisfy the necessary requirements to be lightsheets according the the definition of [47]. F ig u r e 1. W dW -patch for the t = 0 boundary slice in the Poincare-patch. Technically, the W dWpatch would be the lightly shaded square region between the lightfronts t = ± z and the Poincarehorizon. However, we introduce the field-theory U V cutoff z = e and the IR cutoff z = zmax near the Poincare-horizon, shown as dashed (blue) lines. Hence, the integration-domain W for the action proposal, which we will still refer to as W dW -patch, is the darkly shaded region. We also mark the A s u ltim a tely w orked ou t in detail in [48] , th e a ctio n is (see also [50][51][52][53][54][55][56][57] , we m ostly follow [4 6 , 4 9 , 57]4) w here we have inclu ded the a p p rop ria te surface, b ou n d ary, jo in t and cou n ter term s. O f cou rse, G N stands for N e w to n ' s con sta n t. W e will n ow g o th rou g h these term s on e b y one.
4See also footnote 7 in [58] for a remark on the sign of the term <x k. A u rfa " = s n c N ( L l^d t + / . z " " , l zmax ) = 2 z g n (7 -z m a x ) ■ ( 2 -10) 5In fact, t = is a timelike Killing vector, which defines the units in which we measure boundary time. This kind of consideration also played a role in [59]. Lastly, we are d ealin g w ith th e term Acounter °log(|0^c |)dA^pdx, (2.17) 8 n G N N Ni w h ich has been in trod u ced already in [48], but th e im p orta n ce o f w h ich was p oin ted ou t in [57] (see also [58,60] for th e im p o rta n ce o f these term s, b u t also [61]). A gain , th e null and th e m easure j p d x com es from integrating over all th e different null rays co n stitu tin g th e lightfront. T h e reason w h y these term s are called counter-term s is th at th ey m ake sure th at th e value o f th e a ction rem ains th e sam e u nder reparam etrisation s o f th e affine p aram eter A param etrisin g the lightrays th a t m ake u p th e null bou n d aries [4 8 , 57] . A s we ca n see, how ever, this com es at th e price o f in trod u cin g an a rbitrary lengthscale £c .6 W ith the equations o f a p p en d ix A in m ind, we co u ld n ow p ro ce e d t o d ire ctly evalu ate ( 2 .17 ) , how ever, we will first sim p lify th e expression follow in g [57] . T o d o so, we rem ind ourselves firstly that th e expansion 9 is given b y (see a p p e n d ix A ) (2. 18) v P H ence ( 2 .17) ca n be rew ritten as T h is is as far as [57] went, but we ca n m ake an ad d ition al step b y using R a y ch a u d h u ri' s eq u a tion ( A .1 4 ) , w hich in a 2 + 1-dim ensional vacu u m bu lk -sp acetim e boils d ow n t o ^ --9, and hence, using ( 2 .18 ) again, 6One might be tempted to set this lengthscale equal to the AdS-scale L as e.g. [57], however in general this tends to simplify the results for complexity almost too much. So we leave l c to be arbitrary in this paper.

E n d resu lt
Tak ing th e results from the previou s su bsection s togeth er, we find 7

Conformal transformations in A d S 3/ C F T 2 3.1 Solution generating diffeom orphism s
Let us revise some o f the details about how to implement local conform al transformations in A dS 3/C F T 2, discovered in [63], but following the outline and notation o f [40,45]. W e start with equation (2.1) . Local conform al transformations can now be implemented by applying global bulk diffeom orphism s which act nontrivially near the boundary [63], see also [45].
These diffeom orphism s map solutions o f the equations o f 2 + 1 dimensional AdS gravity to new solutions which will be physically inequivalent, hence describing distinct CFT-states.
T hey can thus be called solution generating diffeomorphisms (SG D s) [45]. For example, holographically calculating the expectation value o f the energy-m om entum tensor o f the boundary theory by the m ethod o f [64] after applying an SGD will give a result different from zero (which we would get from the metric ( 2.1) ), which however agrees with the formula for the energy-m om entum tensor o f a C F T after a conform al transform ation due to the Schwarzian derivative [45]. The resulting metrics, due to their discovery in [63], are called Banados geom etries and have been studied in more detail for example in [45,[65][66][67][68][69].
7See also [62] for related, but more general results. T h e S G D s are o f cou rse on ly defined u p to a residual d iffeom orp h ism w h ich is trivial at th e b ou n d ary. F ollow in g [45], we will w rite th em as8 z = z^J G + ( X + ) G -( x -), (3.1) x + = G + (X + ), (3.2) x -= G -( X -), (3.3) w here G ± are som e fu n ction s w ith G ± > 0. T h e line elem ent in the new co ord in a tes Z, X ± is [45] 1 (3.7) T h is m otivates th e secon d (equivalent) p ersp ective th a t we can take, nam ely th at in th e old coord in a tes o f th e P oin ca re-p a tch , th e SG D s a ctively shift th e p osition o f th e c u to ff surface a ccord in g to ( 3 .7 ) , w hich in th e h olog ra p h ic ca lcu la tio n o f C F T quantities th en leads to th e changes ex p e cte d for a con form a l tra n sform a tion [45].9 T h is is show n in figure 2 . In th e coord in a tes o f ( 2 .1) , the induced line elem ent o n this c u to ff surface ( 3 .6) th en reads e2 d x + d xw h ich is o f cou rse con sisten t w ith th e w ay th e m etric tran sform s u nder con form a l trans form a tion s, acquirin g an overall p refactor. T h ro u g h o u t this paper, we will sw itch betw een these tw o p erspectives, d ep en d in g on w hat is easier for th e given task at the tim e.
8This is different from the convention used in [63]. The convention used here and in [40,45], while leading to a somewhat more involved expression for the line element, has the advantage of presenting the SGDs in a simpler form. This will not affect our physical endresults.
9Something similar happens in AdS2-holography: there, the family of physically inequivalent solutions to the bulk equations is given by the set of curves defining different cutoff-surfaces near the boundary of AdS2 [70]. Following [40], we will again consider a small SGD with the expansion parameter a ^ 1. Just as in [40], we will throughout the paper assume that the functions g ± as well as their derivatives are sm ooth, bounded, and fall off to zero at infinity. The line-element (3.4) can similarly be expanded, yielding where we have switched from lightcone coordinates X± to standard coordinates t X on the boundary. In this paper, as in [40], we will be interested in terms up to and including order as they are rather cum bersom e. It is a trivial exercise to derive them from ( 3.4) . x + = G + (X + ) = X+ + a g + (X + ),
H ow ever, w hen w orking p ertu rb a tiv ely in a , we ca n m ake use o f th e inverse tran sform ation s x + = G (| T 1) (x + ) w x +a g + ( x + ) + a 2g + ( x + ) g + (x + ) + O ( a 3), (3.12) (3.14) C onsequently, th e eq ua l-tim e b ou n d a ry -slice in th e new coord in a tes, t = 2 ( x + + x _ ) = to , z = 0, w hen m a p p ed back to th e o ld P o in ca re -p a tch co o rd in a te s takes th e (a p p ro x i m a te) form (3. 16) F rom now on, unless ex p licitly specified oth erw ise, we will gen erally assum e to = 0. Secondly, due to th e con form a l flatness o f ( 2.1) , the lightrays th at foliate th e surface t + (z, x ) are straight lines o f unit slope in th e co o rd in a te system spanned b y t, z, x . H ence, a lon g each o f these lightrays, th e expressions dzt + ( z , x ) and dxt + ( z , x ) will b e con stan t.
D raw in g th e lines in the z, x -p la n e alon g w hich these quantities are con sta n t 11 will hence b e an easy m eth od to draw th e p ro je ctio n s t o th e z, x -p la n e o f th e lightrays w h ich foliate the null fron t, given a num erical solu tion o f t + (z , x ) . In figure 3 , we show th e corresp on d in g figures for som e sim ple ch oices o f t bdy(x ).
Th irdly, apart from num erical a pproach es, we can also try t o solve ( 4 .1) iteratively in a , starting w ith th e a = 0 result t + ( z , x ) = + z . T o secon d order, this y ie ld s 12 10Of course the treatment of the past boundary will be almost identical, so we will not spell it out in every step in the following.
11For example using the C on to u rP lo t[...] command of Wolfram Mathematica. 12 Similar expansions of general lightfronts in the z coordinate were done for example in [49,57,71].  In section 2 .1 , see also figure 1, we in trod u ced a tim elike IR -c u to ff surface z = z max near

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Consequently the replacement zmax ^ zmid. As can be seen from and so for the moment we obtain (2) A series exp an sion o f t (z , x ) -t (z , x ) -2z in z shows th a t th e term Abuik 1 will not con trib u te any divergen ces in th e lim it e ^ to . T h is is as g o o d as ou r general a pp roa ch gets.
F or specific exam ples sim ilar to th e ones evaluated in [40] , we find (keeping in m ind ( 3. 18) and (3 .17 ) and takin g e ^ 0) which we have now spelled out in (untilded) Poincare-coordinates. Again, we will argue that this does not contribute at order O ( a 2), in the following way: as said above, the region W 2 is bounded by the surfaces z = zmid,t = t + , t = t _ , z = zmax. W hen replacing A bulk(W 2, a) with A bulk(W 2, a = 0), we are instead integrating (the same integrand) over the region bounded by the surfaces z = zmid, t = + z , t = -z, z = zmax. How big is the error that we make by changing the integral bounds? This can be estimated by integrating over -1 7 -

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th e gray-sh aded areas in figure 4 . D u e to th e b ou n d s ( 5.1) , ( 5.2) , th e error E 1 in trod u ced b y rep lacin g t = t + ( z , x ) w ith t = + z and t = t -(z , x ) w ith t = -z is at m ost o f ord er Similarly, th e error E 2 d u e t o integrating from z = z mid instead o f z = z mid ^ z = z mid/ y / G + / ( x + ) G -/ ( x -) (w h ere we have used ( 3 .14) ) is estim ated by 13 . T h e term at z = z max is the easiest t o deal w ith , w h ich we d o in P o in ca re-coord in a tes.
T h en , ju st as in section 2 .3 , we find K = -2 and ^7 = 1 /zm ax. So

J -<x
\ z max \ z m a x // 13Below, we do not specify the integral bounds in the f dt integral explicitly, but it is enough to know that by (5.1) , (5.2) , | t| < O(zmid). The dependence of the exact integration bounds on the other coordinates does not play a role to lowest order in a, so we can assume that the integration bounds of the t-integral are independent of x and z below.

A ffin e p a r a m e tr is a tio n o f lig h tra y s a n d n o r m a lisa tio n
In order to com pute the remaining terms, namely the null-surface term, the joint terms and the counter terms, we need to discuss the normalisation o f the null normals k^ for the lightfronts in question. W ith ou t loss o f generality, we will focus on the future lightfront, described by the function t + ( z ,x ) in Poincare-coordinates. Generalising section 2.3, the null-normal k^ is given by the equation Herein, the function $ (a , t, x, z) is meant to allow for general local rescalings which of course d o n 't affect the orthogonality o f k^ to the lightfront or the condition k^k^ = 0, which is equivalent to (dz t + (z , x ) ) 2 + (dxt + (z , x ) ) 2 = 1. (4.1) 14Compared to [64], we changed the sign o f the extrinsic curvature, to conform with our conventions of appendix A . in order to calculate k . Ideally, we would like to be able to set k = 0, as was also the case in section 2.3. Calculating the Christoffel symbols and covariant derivative in Poincarecoordinates is an easy exercise, and in fact in the special case $ = 1 we find k = 0 as a consequence o f (4.1) and ( 7.1) . In the more general case, we obtain (again using (4.1) )

T im e lik e -n u ll jo in ts
The types o f timelike-null joints that we might have to deal with for nonzero a will be similar to the joint-term s already studied in section 2.4 for the a = 0 case. At the IRcu toff surface z = zmax, we will again have a volume element ^/p ~ 1 /z max and an integrand n ~ log (|k ■ s|) with at most a logarithmic divergence, so these terms will again vanish in the limit zmax ^ to.
W e are left with the timelike-null joints at the cu toff surface z = e. For simplicity, we will focus on the joint between the cu toff surface and the future lightfront t + ( z ,x ) , the 16Strictly speaking, because o f this limit <h cannot have an arbitrary x-dependence, but should be only a function of g'+ (x), <h (a, x) = <h (a, g+ (x)), because as visible in (7.5) this is how dxt + (z ,x ) and dzt + ( z ,x ) depend on x in this limit. We will see shortly that this is indeed satisfied, at least to second order in a. This is not surprising, as g+ (x) ~ t bdy/(x), and at the beginning o f section B.3 we will see how some properties of at the boundary are only functions of t bdy/(x). 17Note that in this equation, evaluating the product at the boundary z = 0 also implies setting the t-coordinate in (7.6) to be t = t bdy (x), as this is the time-coordinate as a function of x for which the lightfront emanates from the boundary. Herein, k and k7 are the outward-pointing normal one-forms associated with the two lightfronts that meet on the null-null joint from its two sides. fc1 is an auxiliary vector, colinear 8For the lightray coming to the crease from the other side, we had introduced the coordinate , which has to be a function o f . 19This will apply to all three cases studied in this section. The scalar product turns out to be and the same overall remarks apply as in the previous case: as expected, the quantity is negative and has a term o f order O ( a 0). Consequently 21

C ounter term s near boundary
Just as in section 8 , we will focus on the intersection between the U V -cutoff surface and the future lightfront t + ( z ,x ) . The em bedding and induced volume element on this joint-curve 22Specifically, we will use the last expression in this equation, which is formulated in terms of the em bedding of the joint-curve into the ambient Poincare-space and the null-vector fcM, without the need to apply covariant derivatives to fcM. O f course, all expressions for 6 given in section A.2 are equivalent, but especially for large z when we do not know the lightfronts t ± ( x , z) analytically it is convenient in practice to avoid having to act on with covariant derivatives. (1 0 .7) i.e. th e cou n ter term s p rovid e us w ith co n trib u tio n s at orders a and even a lo g (a ). for a discu ssion on th e op era tors U ± (a g ± ) th a t im plem en t th e co n fo rm a l tran sform ation 24 Another curious fact is that ( 3 -log(64))n ~ -3.64074, so it seems that the change of complexity induced by the conformal transformations g+ = 1+x2 and g+ = 1+x2 is identical subject to the assumptions (3.18) and (3.17) . See also (5.7) and (5.8) . This equivalence was already a feature of the results for the volume proposal [40], but we don't currently understand why this fact should hold generally for any holographic complexity proposal. w ith som e p ositive finite con sta n t K . N ote th a t for sm all a , a K ' < |a log(a )| K for any p ositive con stan ts K , K ', as lim C T 0 da ( -a l o g (a ) ) = + r o . H ence ( 11.5) and ou r results im p ly the follow in g statem ent:

) in A dS3/ C F T 2 with the counter-term s chosen as in ( 2 .6 ) .
T h e existen ce o f th e O ( a l o g (a ) ) term s is th e central result o f this paper: desp ite the fact th a t we were on ly able to ex p licitly co m p u te th em for three co n cre te exam ples, we have p rovid ed argum ents th rou g h ou t the p ap er th at these term s sh ould gen erally be e x p e cte d t o con trib u te w ith the orders th at th ey d o. Let us repeat: for n o n -con sta n t t bdy(x ), we gen erically ex p e ct cau stics and creases t o em erge in th e lightfronts b o u n d in g th e W d Wp atch [73]. T h e focu sin g th eorem im plies th a t th e cau stics will have z-co o rd in a te s o f 25A somewhat similar behaviour of complexity decrease with infinite slope was observed in [72] in the time evolution of complexity in black hole backgrounds.
26See e.g. the appendices of [40] and [45] for how to write the generators of conformal transformations in this form.  [77] or [78,79] , a lth ou gh in [78] it was show n th a t th e cau stics w ou ld n ot play a role. 27The paper [75] dealt with complexity of AdS/BCFT models, a topic also studied in [76]. overlap betw een ou r ca lcu la tions a bove and results con cern in g hole-ography and differential entropy [86,87], see especia lly [88,89] . In the n om en clatu re o f [88] , th e fu n ctio n z tl (x ) was th e outer envelope o f a given set o f intervals th a t ca n b e derived from t bdy(x ) and t 1.
A lso, the sw allow -tail like featu re show n in figure 6b o f [89] is related t o th e em ergen ce o f a ca u stic and null-null jo in t in th e case t bdy (x ) = 1 + 2 w hich we stu d y th ro u g h o u t this paper, see e.g. figure 3 , u p p er left corn er. W e leave it to th e future t o stu d y in m ore gen erality the possible relations betw een differential e n trop y and W d W -p a tc h e s , resp ectively com plexity. For t bdy(x ) = const., we find s 1 = 0, and for the boosted case t bdy(x ) = co n st1x + con st2 (|const1| < 1) it is easy to derive (B .10) explicitly from the analytical solution o f ( 4.1) which can be found in this case. Now, for general sm ooth t bdy(x ), if we zoom in close enough around any x f , the setup should be well approxim ated by t bdy(x) = con st1x + con st2, and hence (B .10) is the general result.