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11-8-2016 Holographic Renormalization of Einstein-Maxwell- Dilaton Theories Bom Soo Kim University of Kentucky, [email protected] Right click to open a feedback form in a new tab to let us know how this document benefits oy u.

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Repository Citation Kim, Bom Soo, "Holographic Renormalization of Einstein-Maxwell-Dilaton Theories" (2016). Physics and Astronomy Faculty Publications. 403. https://uknowledge.uky.edu/physastron_facpub/403

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Notes/Citation Information Published in Journal of High Energy Physics, v. 2016, no. 11, article 44, p. 1-43.

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Digital Object Identifier (DOI) https://doi.org/10.1007/JHEP11(2016)044

This article is available at UKnowledge: https://uknowledge.uky.edu/physastron_facpub/403 JHEP11(2016)044 finite Springer graphy and ry condition aturating the November 2, 2016 November 8, 2016 September 15, 2016 : : : 10.1007/JHEP11(2016)044 Accepted Published Received doi: re prevalent in a fixed charge ator can be a function of the t fully fixed due to the s through the dilaton coupling, e boundary value problem. In stein-Maxwell-Dilaton theories. y, We study the on-shell action and Published for SISSA by . 3 1608.06252 that manifest in the massless scalar or the scalar with mass s The Authors. AdS-CFT Correspondence, Black Holes in String Theory, Holo c

We generalize the boundary value problem with a mixed bounda , [email protected] Department of Physics andLexington, Astronomy, KY University 40506, of U.S.A. Kentuck E-mail: Open Access Article funded by SCOAP the stress energy tensorthe to note presence practical of importances theboundary of terms scalar th fields, physicalBreitenlohner-Freedman quantities bound. are no Keywords: that involves the gauge and scalar fieldsIn in the particular, context of the Ein expectationexpectation value value of of the theensemble dual current because operator. the scalar conserved charge oper The is sharedwhich properties by is both a also field responsible for non-Fermi liquid properties. Abstract: Bom Soo Kim Holographic renormalization of Einstein-Maxwell-Dilaton theories condensed matter physics (AdS/CMT) ArXiv ePrint: JHEP11(2016)044 5 7 8 2 4 21 22 23 24 25 26 27 28 28 29 30 31 32 32 34 36 11 11 12 13 14 15 16 17 19 37 20 31 11 = 0 = 0 = 1 = 1 F F c c – 1 – F F c c φ k e φ ∼ ∼ ) ) φ φ ( ( W W = 0 6= 0 ) F F φ c c ( W background background 5 4 5.1.2 EMD theories 4.3.3 Case with 5.1.1 Examples in holography 4.2.2 Grand4.2.3 canonical ensemble Semi canonical ensemble 4.3.1 On-shell4.3.2 action Case with 4.1.1 On-shell4.1.2 action Grand4.1.3 canonical ensemble Semi canonical ensemble 4.2.1 On-shell action 3.2.1 Scalar3.2.2 mass above the BF Scalar3.2.3 bound mass saturating the BF Massless bound scalar 3.1.1 Scalar3.1.2 mass above the BF Scalar3.1.3 bound mass saturating the BF Massless bound scalar 5.2 Non-Fermi liquids & charge splitting 5.1 Finite boundary terms 4.3 Interpolating solution 4.2 AdS 4.1 AdS 3.1 Exponential coupling: 3.2 Polynomial coupling: 2.2 Scalar and2.3 vector variations: On-shell coupled action2.4 & stress energy Comparison tensor to minimal coupling 2.1 Scalar or vector variation 6 Conclusion 5 Two physical implications 4 EMD solutions 3 Dilaton coupling Contents 1 Introduction & salient features 2 Generalities JHEP11(2016)044 , ν µν A ] in g (1.1) (1.3) (1.2) (1.4) 12 where 2 ]–[ 6 F ) φ ( g between the W ). Here we investi- φ ( = 0. We are directed . he chemical potential. W rν is the determinant of the ). As we see below, it is  ics and analytic examples. φ g en studied extensively. See . We produce not only the δF ( in addition to a metric ixed boundary condition on the dimensional reductions δφ tinctive physical properties tion . that can be evaluated by the ) r chlet boundary condition has φ is a unit normal vector for a W φ µ low. It is consistent to impose mphasis on the role of this cou- to have a consistent variational g with the form ( has contributions not only from r J n 0 W , J ∂φ . , ν ν log rµ A ∂ δA F rν ) rν rν because of φ F ( F F r brings forth this possibility naturally. The and a dilaton r n δφ + n 2 µ in an action with respect to the gauge field ) gW ) F A rν φ – 2 – 2 φ ) − ( ( φ F = 0 to have a well defined variational problem. √ ( ) δF φ  γW ν ∝ ( γW ) gives ν − gW µ − δA A − J √ 1.2 r gW √ n √ ]. − ) . φ √ 16 ( µ ] and recent ones for condensed matter applications [ A ]–[ 5 γW 13 ]–[ − 1 ]. √ 17 ) is only a function of a radial coordinate r ( µ ) combined with ( A , but also the scalar variation 1.3 rν run for all the coordinates. Compared to the minimal couplin δF is the determinant of a boundary metric, and is the field strength. There is a conserved current γ µ, ν dA Now let us consider an alternative quantization that sets EMD theories have a U(1) gauge field The variation of the term = problem. Thus we seek aboth possibility fields. to impose The a coupling generalized term m natural to couple the variation of the gauge and scalar fields the gauge field, but also from the scalar field through the func gauge field and a scalar, these theories have a scalar couplin F variation of the gauge field when the field This is theHolographic case renormalization of of fixing EMDbeen theories the considered with constant in this [ part Diri of the gauge field, t fixed radius. The notationsthe are Dirichlet systematically explained boundary be condition metric. Upon a close examination, one finds the charge variation of ( where One can easily seevariation a new feature for the variation of this term the holographic context [ gate the holographic renormalization of EMDpling. theories with We e are going to focus on the theories with AdS asymptot where Einstein-Maxwell-Dilaton (EMD) theoriesof are consistent natural string from depending theory. on the They parameters present havesome in earlier provided the literature theory various [ and dis have be 1 Introduction & salient features provides a boundary contribution of the form to add the boundary term JHEP11(2016)044 . in 5.2 and . It k (1.5) (1.6) 4 φ (with 2.1 . ∼ 3.1 ) ), to 2.4 φ ( = 0. As one 1.1 W rν δF y for two different ) (1.7) in section d scalar fields. ld together with that δφ . There we attempt to φ ( e blem can be generalized h this investigation. sented in the beginning o physical implications.  5.1 ν ling in section ∼ A nd the stress energy tensor condition text is the main task of the ) notion of finite radiative cor- ion rν . φ rm, upon variation, gives the implified, using ( n two sections, section on-Fermi liquid properties and l problem actually fit together ( F s energy tensor. reviously developed variational g. It is presented in section  r . ) n W  rν φ r δF ∂φ δφ . ∂ ∂W ( r ) ν φ A ( φn + 4 r φ φ W c ∂φ φ W n L + ) and the polynomial coupling Λ ). We contrast the boundary value problem of log 2 – 3 – ∂ φ 4.2 + 4.3 ) ). We consider two possible boundary terms for φ ) + 4 L φ ν 2 Λ r φ Jackiw. We also try to survey earlier literature with δφ A r ∂  ( unfixed. Then putting the variations together, we get ν r ∂ φ ( γ J n φ r δ ( `ala − r ∂ 1) , c r φ √ ) mixes the scalar variation with the gauge field dependent . They are applied to the three examples in section − φn γn φ and section δφ is usually finite at the boundary. φ c ), the exponential coupling 2.3 ( c − ν ν ( φ + ( √ A A 4.1 ν −  W rν J γ F r − n √ ∂φ ∂W ] and in section 18 (with an example in section 2 , with some review. The program is carried out systematicall 3.2 2.2 to include the mixed boundary condition between the gauge an In the context of EMD theories, the boundary variational pro Investigating this variational problem in more general con This term prompts us to consider the variation of the gauge fie Before moving on, let us list some lessens we have learned wit Along the way, our investigations direct us to appreciate tw In parallel, we also carefully examine the on-shell action a • section Note the combination paper. The basics of the variational problem are described i of the scalar.boundary The contribution canonically normalized scalar kinetic te already anticipated, there is an additional term. It can be s the scalar field with appropriateapproaches coefficients [ following the p nicely with the analysis of the on-shell action and the stres compare the holographic finite boundaryrections terms in to Quantum the field similar theory the EMD theories with that of the theories with a minimal coup is pleasant to see that the results of the general variationa section forms of thetwo coupling examples in section term and plays a crucial role. Here we leave the coefficients Λ finite counter terms in holography.splitting Another of implication the is conserved n charge due to the dilaton couplin The first term is expected, and one can impose the variational Note the term One is the finite boundary or counter terms considered in sect of the EMD theories.of The section basics along with some review are pre JHEP11(2016)044 i 4 φ α / 1 and (1.8) (2.1) hO α ∼ i F W Q hO have on that renders , , 4 ) 2 r  ( the parameters, Λ ) on 2 ensor to ADM mass. dr from the gauge field. φ alized with the faster atures of the theories h operator ( ms, whose coefficients al scalar operator ynamics is satisfied. arge between the gauge . The conserved charge 1 ies due to this coupling are possible sub-leading V m the variation problem me conditions on l and interesting physics − tion and the stress energy happens when the bound- W  − ess scalar and for the scalar ··· 2 ) ··· , + ∂φ ( 2 2 ··· 1 2 r L with two illustrations, one for the + , where  +1 dimensional asymptotically AdS − 6 i d 2 + F ··· F Q
) 2 φ hO ( d~x F W + ) = 1 + 2 − c Q – 4 – r ( dt R 2 = )  r i ( , h g α 1 − h hO ··· √ − x  +1 d is a constant. ··· d c ) = 1 + + Z r ( 2 2 2 1 r L 1 κ h 2  = = 2 S expansion. , and ds r πG , of the boundary terms. This can be different from the conditi φ c at the boundary. = 8 2 F κ with the scalar fields. Thiswith is mass especially saturating clear the for the BFfalloff massl bound at when the their boundary. solutions are re are not fixed by the requirements of the theory, are general fe and scalar fields throughcan the be coupling. clarified The with physical propert examples. The first two examples in secti The dilaton coupling provides a way to share the conserved ch and tensor depend on the parameters.can The be condition obtained used fro to renderFurthermore, the the differential mass form evaluated by of the the stress first energy law t of thermod the on-shell action and the stressary energy terms tensor provide finite. finite contributions. This Then the on-shell ac Q Our variational problem reveals that the finite boundary ter is shared by the scalar,such and as the a system non-Fermi exhibits liquid a state. non-trivia with some additional contributions in general. There are so at the boundary, andThe thus other effectively example the gives a entire non-trivial charge boundary comes profile for The general boundary value problem can impose conditions on can be a function of the expectation value of the dual current For the fixed charge ensemble, the expectation value of the du • • • • where These lessens are summarized in conclusion section terms in large We consider the Einstein-Maxwell-Dilaton theory in variational problem and another for the on-shell action. 2 Generalities space JHEP11(2016)044 ]. 26 and . (2.5) – (2.6) (2.3) (2.4) (2.2) (2.7) dt 24 ) 2.4 , r , ( t 18  A ) φ = ( V 2 1 ). We consider the ly. We examine the , + φ dA, A ( 2 . They are contrasted 2 . = L hat can be identified as F W 2 φ 2.3  wo sections, we carefully F . 2 2 oupling in section oundary condition for the m W been studied in [ ng φ . φ =0 te, + L ) rm φ + 4 2 Λ 2

ergy tensor. 2 ) boundary contribution ection δφ
g physical quantities of the dual d ( + 2 φ ∂φ φ F q . r ( , while the general mixed boundary r ) ∂ 1 2 4 1 φ ∂ φ r rt c r ( unfixed. Both terms were considered  ± φ W γ n φn 2 d µν c φ and g − gW F c , we investigate the general boundary vari- = . φ ) + √ − −  φ ± x and ρ γ √ 2.2 ( d ν 2 – 5 – = 0 d − φ V F 4 κ ]. See also a previous work in the context of EMD √ 2 Z µρ 2 x ∂φ 2 23 ∂V − 2 d , λ ∂ d 1 κ ∂φ ]–[ ) − = WF 2 2 + x 2 ] to have a fresh look on the problems. The variation of Z q λ 19 L ( − r 2 F + 2 β 18 and section = 1 κ = φ 2 + 2 φ ν ∂φ ∂W L ) 2.1 − 2 φ δS x φ∂ λ − ) = ( µ m r φ α ∂
, ( 1 2 φ b ν S → ∂ = = 0 φ µν
µν µν gg Rg − 2 1 √ ] that considered the scalar and vector variations separate gW F − µ 17 − ∂ µν g √ R ] mostly with some particular values of Λ − 1 µ The boundary behavior of the scalar The variational principle for scalar and Maxwell fields have The Einstein, Maxwell and scalar equations are In two sections section √ ). Then the second equation gives a finite and constant value t ∂ 18 r ( in [ Note that we introduce the coefficients Λ depends on its mass possible boundary terms, the on-shell action, and stressHere en we review them followingthe [ canonically conjugate scalar kinetic term contains the 2.1 Scalar orHolographic vector renormalization variation is afield central theory part from of the extractin gravity sidetheories [ [ ational problem of therevisit EMD the theories. on-shell action With andto the the the stress results boundary energy value of tensor problem the in of t s the theories with a minimal c To satisfy the equationscalar. of For motion, this it purpose, is we introduce required a to general impose boundary a te b A particular solution depends on the form of the gauge coupli conditions were also advertised. gauge and scalar fields depending only on the radial coordina a charge φ JHEP11(2016)044 l . < < φ 2 4 ) for / (2.8) (2.9) L 2 (2.13) (2.12) (2.11) (2.14) (2.15) (2.10) 2 φ d 2.5 . − ]. For the < m . . 4,  φ / 18 4 c  2 2 d / d α 2 − d − − φ and − 1 ) Λ φ β ≥  . ]. 2 + ass saturating the BF dependence 4 1 δβ 38 L r amined in [ e a boundary condition. 2 φ r + ]–[ m log ) = alues of the parameter Λ og  = 0. Then the expectation rator is given by 30 φ α r ( ( es, α rameters Λ . ns [  W . And the corresponding scalar log 2 2 d β . ]. See also previous discussions on  ˜ κ αδβ ) 2 d + 2 24 x for x . φ ( d/ d − . . r β Λ d r ], and it is possible to impose boundary ) enables us to read off the sub-leading φ 2 2 1 β α Λ ˜ + κ 29 Z ˜ κ  , 2.3 log  r ) − − = − i δβ 2 ··· = 2 d 1 + – 6 – d/ i ) = ) log +1 =0 ) and the variation of boundary term ( + +  r d β φ x α = r ( 1 ( 2 L b 2.4 2 − φ α hO − δα hO 2 followed by setting d d κ  Λ log 4. The last one saturates the Breitenlohner-Freedman δS r 2 L ) / d/ → 2 d 2 + β δα d φ ( = φ − + qκ = 0 provides a well defined variational problem. − x = 1, the variation yields 2 φ 2 d 2 r δS 4, the condition ( α d ˜ κ = φ / . There are two different ways to have a well defined variationa c + 2 log Z +1 µ L d α = 1 2 φ ( L = 2 and m 2 ) = x t 4. The scalar behaves as W d κ φ d/ / A d ( at the boundary ]. For the mass range slightly above the BF bound, 2 b − t d or on a linear combination of them [ = 0 = 2 Z 28 δS A − = 2 φ , 4, and c κ / + φ = ) = 2 27 α, β φ 2 d φ ( L b δS − 2 φ 1 δS m 4, the two falloffs are both normalizable [ < / + 2 2 The variational problem for the gauge field is also closely ex Here we illustrate the variational problem with the scalar m On the other hand, for φ d L 2 φ − δS 1 Below we seek more general possibilities by utilizing the pa Then the variational principleexpectation is value is well defined for fixed The resulting variation is There are more than one possible quantization for different v Second, we consider to impose the condition with a standard l Maxwell action with contribution for special case of the generalized boundary conditions on the scalar variatio bound, condition on m Here we define ˜ problem. First it works for Λ (BF) bound [ There are three different cases with three different mass rang To have a well defined variational problem, one needs to impos Consider the boundary contribution ( value (density for both space and time) of the dual scalar ope Here setting Λ JHEP11(2016)044 φ = t ), A φ (2.17) (2.16) (2.21) (2.19) (2.18) (2.20) ( rt ed, and F W r γ W n − . ), we get to the variation of √ l defined. This fixes 2  4 κ harge? To answer the on of the action with 2 2.19 δφ − ) y condition between the . φ ls of the solution , ) ( ν ν W A . δA ∂φ . This allows more freedom for ( ) rν t log ν . rν F expansion. The variation of the A ∂ r r ν F δA r ∂ n r A ( + n δφ ) = rν rν ) φ ) . There is an alternative quantization, rt ( φ rt t F F φ rt ( r r ( A F ) shows that the variational problem is F δF in EMD theories is not just fixing the sub- . W γW t  ∂φ γ n ν γ n γW A − ν 2.16 ) and the variation of ( − − log − √ A ). The boundary term δA of – 7 – ∂ √ √ r x √ d x x n produces the boundary contribution x µ 2.18 d d d + d 2.17 = 0. This fixes the coefficient of the sub-leading rν d d d A rt Z F rt ) Z Z rν Z 2 F φ 2 δF 2 2 κ 2 ( 2 1 κ 1 κ δF κ 2 2 − γW ) = − produces the boundary contribution − = A ( b √ A A ) combined with ( of the time component x S d δS µ d 2.17 = 0 at the boundary renders the variational problem well defin Z ν 2 , which is the electric charge. 2 t κ δA A are sub-leading contributions in the large = 0 is imposed at the boundary, the variational problem is wel at the boundary. Note that the variation of gauge field is tied ν ··· rt δA F Do the EMD theories have an analogous quantization of fixing c is actually finite. By combining the ( question, let us add the following boundary term and contribution of µq respect to the gauge field 2.2 Scalar andLet vector variations: us examine coupled the Maxwell term of the EMD action. The variati it amounts to fix the chemical potential Then the variation ( fixing the charge, that can be done by adding the boundary term the chemical potential Once needs to be fixed. The boundary condition depends on the detai well defined with the condition which is a direct generalization of the ( leading boundary contribution of the gauge field Again imposing Where the scalar. As wegauge see and below, scalar it fields. is natural to fix a mixed boundar the boundary condition for the gauge field. The combination In general, the alternative quantization for action with respect to the gauge field JHEP11(2016)044 hile mal ) eady φ (2.26) (2.23) (2.22) (2.25) (2.24) r ∂ ( δ r φn ) φ . c δφ ). The boundary (  and the space-like )  ) + r ν rν . ical ensemble. EMD 2.22 δφ A dimensional time-like ( , δF  rν ry terms.  d ( ) ) ν ν F ) rν r dr A A 2 n j r ]. Our formalism utilize two al vector is φ rν δF ( N ( 39 ss energy tensors. δ F ν r . For the rest of the section, we ng to fix the chemical potential, W n φ ∂φ + L a large fixed ∂W 3 A n , is independent of the scalar field F  2 Λ j r , c ) ν 0) dx + 4 W , 1) A ∂φ ) + ), and thus we examine two different W n , ∂W 0 )( φ ν , φ 0 rν ) + 4 F φ ( , dr ν c L i F Λ ). If δA ensemble. r W ··· ( ) + 4 N δA , ··· + δφ φ ( + 4 has the form rν , , in section 0 ( r + , k 0 φ W n φ ν rν ∂ F i r , ( φ ( r φ ). We also define the time-like unit normal of a (1 A F – 8 – δ ∂ δ L ∂M (0 r r Λ Σ dx r F ∼ , r rν r ( n i c N ) W n F + ij N ) x φn 4 1) φ r W n − γ φ ( A φ n = c ( r − − 1) + b = ) + 4 ∂ W µ ) φ + 2 r φ µ − n c ∂φ r δS semi canonical ∂W n ( u δφ ∂ dr F = 1, we get ( r 2 1) c r and φ  F ) + r N φ c − φn φ ∂ γ e ( 2 ( 1) are the coordinates spanning a given time-like surface, w r φ + 4( = b δ . The projections onto these hyper-surfaces require two nor κ − c n 0 ∼ φ ( − 2 2 c √ x δS ) − ds x φ , d  +  + d ( = 0, the gauge and scalar variations are separated, and one alr γ γ d A W ··· − − F , Z c δS √ √ 1 , x x = + d d d d φ = 0 can i δS Z Z − ( s M 2 2 i = 1 1 κ κ δS x 2 2 M For EMD theories with We consider both the grand and semi canonical cases by using ( Thus we consider general boundary contributions and bounda = = δS is the holographic coordinate. The corresponding unit norm , it correspond to fix the charge of the gauge field and to a canon theories are different. We call it classes separately, φ vectors to the surfaces. The ADM form of the metric for the considered the case above. Thisand case is is referred usually as correspondi grand canonical ensemble. hyper-surface homomorphic to boundary 2.3 On-shell actionLet & us stress introduce energy some notations tensor kinds following of Brown hyper-surfaces, and the York time-like [ boundary surface at condition depends on the form of the coupling lay out the general formulas for the on-shell action and stre Here we note the mixing term 4 If one considers surface at a fixed time where space-like surface by r where the components are ordered as ( JHEP11(2016)044 . − d (2.33) (2.29) (2.30) (2.31) (2.27) (2.32) (2.28) #) . The t γ ] A rt 20 . F ) gives r . . ∂M ) 2.2 , ) W n φ + 1)-dimensional 0 ( b F . b d c S dx S ν 2 . b Σ u + ) µ −  N µ ) + u 2 2 n ation of ( A + φ . ν + he surface. ( ollecting all the boundary b φ b 2 L ∇ ν WF S 4 Λ action finite. It contains the n dx + µ es and the ( − + − )( WF ν n 0  1 3 φ V n d r  − µ comes from another ADM decom- by dx ∂ − − R 1 a  Σ r ∇ Σ µν d d 2 ( on the time-like hyper-surface, they 2) 2 g N − N 2 1 φn ), are possible boundary terms for the + . However, if we restrict them to the d − + = φ L − φ 2 2 i, j WF c ( d a µν ) g b g = µν − 2( − − S , and dx has possible boundary terms that make the ∂φ ( ( can be a horizon in the presence of black hole √ µν V + b ) with 2)  2 1 ab 0 d Θ – 9 – S ), we get dr ∂M ( 1 σ r 1 − , and the scalar curvature of the metric ij , the extrinsic curvature, defined as ǫ r R r 0 γ ) and L − + 2 ij d ) + − r L 2.22 A 0 d φ Z d 2( ( ( , b x , σ dx g V d S − 0 µν d ν 1 − ] and Balasubramanian-Kraus counter terms [ Θ + 1 n Θ ), dx  √ − + 1 ν µ Z 40 2 L − Σ -dimensional time-like boundary hyper-surface and the ( j γ n d d 2 d dr γ d N − 2.31 1 κ µ ǫ − i r = 2 0 − √ r γ x µν Z R = = d Θ + g =  j d " 1, spanning the spatial coordinates at the boundary ǫ ij x = γ r dx d shell i Θ − = d − − r µν γ Z √ dx , d on Z 2 are the metric at fixed 2 ij S 1 2 γ κ − d 1 κ ··· , 2 − R = 2 γ = = defines the local flow of time in = 1 is the UV boundary cut-off, and b ds µ and ǫ a S u r shell γ − The projections onto the The last two terms in ( We evaluate the corresponding on-shell action. The first equ on S gauge and scalar fieldsterms that introduced are in used the for the previous variation section. ( C This on-shell action is one central object we consider. or 0 for a zero temperature background. Gibbon-Hawking term is the trace of Θ position Then we have the on-shell action variational principle work andwell counter known terms Gibbons-Hawking that [ yield the where where Here Where 1)-dimensional space-like intersection surface are given Since they are projection operators, they do not have invers have well-defined inverses and can be defined as the metric on t appropriate components, for example, indices are raised and lowered by the metric JHEP11(2016)044 ] i 1 ]) ss ) 00 39 − 43 T h 2.1 , (2.38) (2.37) (2.39) (2.36) (2.40) (2.34) (2.35) , d r 42 ··· , 2 , ,  2 2 = 1 . Let us identify φ dr i φ L 2 2 ij 4 Λ a, b r L T h nserved charge associ- + + . To do so, we convert , has been also considered the stress energy tensor. j φ s such as mass (energy), from the metric in ( e Gibbons-Hawking term r ij φ ∂ dx T r i . j . , can be computed solely from u dx ij φn i b → ∞ . kj ij ] (also used recently in [ ij u φ . T and Λ 2 P L ˜ c g r T i Σ ) bj δ 41 φ δγ j 2  2 j ik c σ r ξ L γ A without the Gibbons-Hawking term. ij σN ai ij . γγ ri γ 2 − b σ ≡ T √ F S i − √ ij x + ab r 2 u 1 T √ σ ( − 2 − dr 2 1 (3) ij σ d ] for particular fixed coefficients. − 2 W n − 2 ij d d G 1 √ d 4 r − →∞ L γ 18 r 2 r x L ), one can compute the corresponding stress d Z 1 Θ − + L − = lim − – 10 – t =
d = = − i d A i 2.33 2 d ij ij kj rt ab Σ − d~x s T ]. The terms with T Z F h h ij r x M ab and trace of pressure + 39 1 = Θ γ = ik σ 2 , − ˜ 1 g 1 ξ ij d M ˜ g W n dt 20 d T L − Q 1 − ij − includes the radial coordinate as well. The field theory stre − 2 d γ d Z to the field theory stress energy tensor κ 2 and metric as p that generates an isometry of the boundary geometry as in [ 2 2 h − i = = ij µ, ν r ij L F ξ 1 are the coordinates of the field theory and T ij c T P γ − M + Θ is identified as the time component of the stress energy tenso ]. It is given by , d can be computed using the relation [ → ∞ − ) 39 −−→ r M i ij , ··· ij , 2 T 2.33 20 1 h , = Θ of the field theory, which can be read off at ds ij g = 0 T is the Lagrangian for the boundary action 2 b κ i, j L There is an equivalent way to extract the energy and pressure Given the stress energy tensor, we can proceed to compute a co Another key objects are the Brown-York conserved quantitie the stress energy tensor the stress energy tensor the metric ˜ where Explicitly for ( of the field theory. The mass density We note that the field theory metric is flat. With this we get In particular, the mass density ated with a killing vector pressure, and angular momentum, whichOnce can one be worked has out theenergy from tensor full [ on-shell action ( in the context of the variational problem in [ where Note that the first fourand the terms counter on terms the for right AdS [ hand side are from th are the spatial ones, while energy tensor JHEP11(2016)044 . . ∼ R µq ) φ (3.1) = ( . The (2.43) (2.41) (2.42) r dt and W ) r ( W t ). The new c φ A ), which were ( . = te radius W i + 1 dimensions? 2.37 . ) W d , c ij δφ  T = )( dA, A ) are the energy density . bj φ φ ) ified due to the dilaton mass saturating the BF r ( i ) = σ φ ng in ( red below and also other ependence of ai V aa undary? The term ∂φ iA F δφ σ T ( including the mass slightly e boundary. ∂W − h − xpressions ( additional term proportional = 2 µq ) φ onal problem with some general r W P ) and and ∂ c ( r Dφ = F i ( r ( c i φ n 1 2 00 . Then, T φ aa h − T ) = W ) + h 2 c . ν e g δφ F 1 4 ( 1 4 δA ν ( A , − φ rν . . ) = rν k e ij ij – 11 – R φ F can be gauged away. Thus we confirm that there φ ( r F T  T r r ∼ j ) : pure gauge (2.44) n g ∼ W n h A u ) i − ) δφ γ u φ ( φ √ ( ( Σ φ − ∂φ x ). We consider in detail the exponential coupling ∂W r √ φ W W γ ) +1 ( σN A x d ), becomes r d − φ √ d W d ( √ in 2.3 = 2 Z Z − 4 κ W 2 E 2 2 1 κ 1 κ F 2 = 2 c . We focus on the solution with i µ − = 00 S iA = T h − ], where the equivalence is established as a function of a fini µ ), after using ( 44 ∂ minimal = 2.22 µ δS D Will there be a qualitative difference for the minimal coupli and the polynomial coupling φ In this section we performforms a of detailed the analysis Dilaton on coupling the variati We consider the exponential coupling 3.1 Exponential coupling: 3 Dilaton coupling is no direct coupling between the gauge and scalar fields at th e Let us consider the action to the charge. 2.4 Comparison toFrom minimal above, coupling we have learnedcoupling. the Especially boundary the value variation problem of is the mod scalar field has an Is there a coupling between the gauge field and scalar at the bo does not have physical effects because boundary terms for the matter are and pressure of the field theory side. They coincide with the e examples in [ evaluated using the stress energy tensor We have checked this equivalence for the examples we conside It is independent ofHere the we details consider of severalabove the different the gauge cases BF field bound forbound, except with the and the two scalar the different d massless mass, normalizable scalar modes, in the turn. The indices are raised or lowered by the metric ˜ where term in ( JHEP11(2016)044 = 0, , the (3.7) (3.2) (3.5) (3.6) (3.3) (3.8) φ calar . This c F are pa- ) c W x F ( c . The need . and F φ αδβ . Hereafter we ) and = 0 − W at the boundary. = 0 and φ F λ F c F φ 1] Q c δQ Q − µq φ en c F [ . and the on-shell action finite. c δα,δβ,δµ,δQ s in the massless case sec- ze that − β F + q . µq ). For general Λ + Q the boundary. In general we ant coupling − λ µ mplicity. δµ φ 2.6 = = . c ) that couples the charge and the i i µq F − µ 1) Q 3.1 , φ = 0 hO = 0 (3.4) − . 2 hO  − ˜ ) κ F ) c λ + (Λ αδβ x x ] Q ( φ ( λ φ F c F c F r + ( Q 2 β , Q + βδα ) α , ) ) δQ − ) – 12 – − x = + αδβ ( . The term ( + λ λ µq φ βδα rt 2 λ c F 1] − and does not contribute. Thus the case at hand is ˜ 2 κ F + [1 1] 3.2 = 1, it is required to fix c − ˜ αδα κ + + − φ − ) − F + , and use the form for the field strength − λ 2 λ − φ Λ c − . ( φ λ ) λ ˜ c κ − λ [ c ( +1 ] x r [ βδα d +2 ( 3.2 = φ µ − d c L 1] = i at the boundary. The dual field theory operators have the − 2 2 δµ α − i r κ − µ β λ 2 − µq φ φ hO ˜ κ = 1 and c c hO [ = 2 1) , one can have the following quantization condition for the s φ ) and − c 2 µ + [1 − + κ φ can be, in general, different due to the non-trivial coupling λ φ α F c F (Λ − (Λ Q − +( +  λ ) has the form x ( d and d ∝ q 2.22 and the next section Z M = 0 with fixed = δS F 3.1.3 M Let us consider some particular examples for simplicity. Wh Here we impose c δS Then rameters that interpolate differentcan impose field two theories independent mixed living boundary on conditions among for the general mixedtion boundary condition become more obviou variation ( For scalar variation decays at least suppress the coordinate dependence of the variations for si to have a wellThen defined variational problem. This also renders where we abbreviate ˜ For the scalar mass slightly above the BF bound, we use ( 3.1.1 Scalar mass above the BF bound Now it is clear to see the possible quantizations. We emphasi effectively the same as the variational problem for the const For a different choice, it is required toexpectation fix values Note that is more clear in the following section JHEP11(2016)044 (3.9) (3.14) (3.13) (3.10) (3.15) (3.12) (3.11) n impose δα . 1) − = 0 φ . and is not fixed. c ( F φ F β , c = 0 Q δQ φ − d variational problem. F nt from the thermody- ) F µq = 0. For this case with β s Λ Q F δQ c φ ition + c eral. t. δβ e BF bound as we see below. r µq + β . α F or . c − µ log δµ 1) λ , q . + φ  α − = 1, the scalar expectation value is c ( µq − r } φ µ F δµ 1 1) = 0 − c = c 1) log ). i + − µq . Yet, before that, we need to take care µ − λ ] − for F F 2 2 1) αδβ ) + ( 3.2 φ φ c . ˜ κ φ ˜ hO κ c 2 d β c 3. c − . /  + ( + − F + ) has the following form for the scalar mass  c r + (2 1) – 13 – , F r F − = 0 in ( = const + ( + [1 βδα 2 log 2 2.22 βδα Q ) β φ δQ ˜ ˜ κ κ to be 1 φ α F α c + 1) Λ φ λ Q at the boundary. The corresponding field theory dual µq = 2] log c δα,δβ,δµ,δQ αδβ 2]( − F 1] i φ = µ φ c d/ = (2 c α d/ i c − ( φ 1) + + 1) φ hO =0 Λ and c = 0 or 2 ) − 2 α [ − µ ˜ κ δµ ˜ κ φ α F φ − 2.8 c c βδα hO c µq − (2 1) (2 λ 1) for , the variation ( − φ − . − F c − φ c φ φ c ) decays quickly compared to the other terms. In general we ca F − ( [Λ c [Λ φ { 3.1 − and +( +  (Λ = const φ x c d µ ∝ d M Z = δS There are various ways to find specific cases for the well define Before moving on, let us consider M = 0, it is required to fix δS F operators have the following expectation values for the scalar contribution.c This include the special case f If we choose For this to satisfy, we can impose two mixedIt conditions works, in for gen example, if one imposes the mixed boundary cond Thus the expectation value depends on both the boundary term two independent conditions among saturating the BF bound ( 3.1.2 Scalar massFor saturating general the BF bound This happens also for the scalar with the mass saturated at th Then namic first law fixes the parameter As we see below with a particular EMD background, a requireme Again the term ( of the divergent parts. We find the following choice works bes JHEP11(2016)044 . F δµ hic and , . (3.21) (3.19) (3.17) (3.22) (3.20) (3.18) δα . = const φ . c α d δα,δβ,δµ,δQ = 0 r δβ  = 0 F the parameters. If F . W Q δQ δβ F F µqc µq Q . δα δQ F F c  c in an example below. & µq ntrinsically ambiguous when W + + = 0 different field theories at the = β F µ ) F µqc i q . δµ d ) gives for general F F , Q − δQ c Q 1) µq = . + 2 (3.16) 6= 0 gives us new possibilities. From 1) µq − 2.22 hO i = 0 F 2 φ β µ d/ F −  c are finite at the boundary. One of the .( = 0 to find  ) c ˜ κ c d d F 1) F F F hO F + φ (2 F c c r 1) , = Q − Q δQ ∂ − δQ µ r δµ φ − + + ( W φ F c 2 for λ µq c φ c , ˜ κ . µq – 14 – c F F (Λ δβ ( γφn c + β 2 1) Q α = (2 φ − + i 2 ˜ κ − φ + c µ − ˜ κ F φ δµ √ φ α dc Q Λ ) F µ = δµ 1) c φ = const (Λ = 0 and d i − hO − = 0, it is natural to treat the variational problem between α − µ dc and µq − + + ( r + λ F δα F 2 2 1) d − hO c c = 1, it is natural to impose a mixed condition among ˜ 2 κ  α − φ δα and − ˜ κ γφ φ dβ r φ = 1, the variational problem is well defined for W β 2 . 2 + ( c Λ F − (Λ φ ˜ κ ˜ κ F = c c  √ c µqc i δα can not be fixed from the variational problem and the holograp = 0. Then the expression simplifies to +( α +  F W c φ φ x c hO d ), it is reasonable to fix two mixed conditions among = const F + ) actually contributes and provides interesting options. d c α  3.1 Z dβ = 0 and 3.20 and Λ ) µq φ φ = = 0, we require to set 2 φ c c , and the corresponding expectation values are c ˜ κ . α M − as δS (1 F ), then the variation becomes  δQ Let us impose Λ Let us mention some particular cases that can be done by fixing When = const 3.2 F the gauge and scalar fields separately. The case one chooses The new term ( Thus we impose Q parameters the condition ( and For the grand canonical case, In this case, the terms For example, if we choose in ( For a massless scalar, we have renormalization. Different parametersboundary. are This associated indicates that with fixingthe the scalar boundary mass terms saturates are the i BF bound.3.1.3 This is also Massless reflected scalar JHEP11(2016)044 φ Λ , = 1. φ (3.28) (3.24) (3.27) (3.26) (3.23) (3.25) c φ c particular . . . The coefficient F = 0 and − µq . Once again the = 0 Q φ λ d β r = 0. − F = F r α i . = Q δQ or Q . = 1. The corresponding F φ + q F µq F hO λ alar depends on the unde- . Q c ff y for F µq − 1 Q . c ). From the conserved quan- if r − ) d nction of the expectation value . r = + = 1, provides a rather different and the gauge field L 3.3 has the full variational property , nd gauge fields at the boundary, i δφ 2 ( F k µ F δµ κ ν c 1 φ W Q δφ 0 c A − F is more subtle. The boundary term µq F Q = 2 hO W rν = const Q λ as in ( k 1) 4 1 Q k = 0 after choosing Λ r F i φ F ) r ). Here we consider three different cases rt − F 4. This happens due to the presence of the Q between the gauge field and the coupling α Q F W / ∼ Q λ = 1 and F ) = , c q 2.3 1 c ( ) φ φ − W n hO ( . One might try to understand the variational c φ W = 1 s 1 ( c k + k 0 + ( + φ W – 15 – λ φ F are finite at the boundary. The coefficients φ r W + W γ = 0 for the well defined variational problem. Then , β Q δα φ ) − i µ ∼ α r 1 =  d F ∂ − √ ) s = r Q W 1) Q k x . φ φ t λ d ( − − d hO A d 2 or 3.1 µqc . This is consistent with and yet more general than the γφn φ ˜ κ − W φ F c = Z − − 0 r c ( c λ i φ 2 r √ 2 W − κ + c =0 φ = α β and = ) s and (Λ d φ s hO φ 2 = 0. φ 1) = α γφ i − 2 − , while the rest of the term are evaluated according to a given φ =0 ˜ κ c + √ α ( λ δβ r − hO + φ ), where we have been forced to set . One simple choice is to set − λ (Λ δα r cancels out once we use the relation ( µq ), the scalar field behaves as  3.23 determines the way to split the charge has the form 0 W = c 2.3 ) k W We consider the polynomial coupling Let us consider ) φ ( δφ . ∂φ W c ∂W W ( of the boundary terms are not fixed3.2 by the variational problem Polynomial coupling: Note that the radial dependence of the scalar should be eithe solution. For example, The case with the polynomial coupling boundary terms This happens for the massless scalar with the boundary fallo for the scalar mass as in section tity ( expectation values are Thus the variational problem is well defined for problem with full non-linear propertieswhich of is coupled beyond scalar the a scope of this paper. at the boundary. This gives the same form for of the dual current operator. Another case, Thus the expectation value of the dual scalar operator is a fu Here we treat this term semi-classically, meaning that as Note that the expectationtermined value parameters of Λ the operator dual to the sc Now we check this is a particular case we consider momentaril result ( situation in contrast to the Maxwell case, term JHEP11(2016)044 ) ) ), and onal 3.3 . 3.29 3.26 (3.31) (3.33) (3.35) (3.30) (3.36) (3.34) (3.29) (3.32) α F αδβ . The ) c F − λ . δQ and ) and ( φ 1] . i c . φ − F c 2.6 Q φ and c [ = 0 . hO − δα F F F F + µq Q Q and a mixed condition Q λ δQ . . φ for given = . c µq i − F = 0, then the term ( = 0 F φ + ion for general q . = 0 Q F e on-shell action to be finite. = 0, it is natural to fix c − δβ  = const φ hO = 1 for simplicity. Then ( αδβ , but also +(Λ 2 c F = φ α β β F ˜ κ at the boundary. Then c F i ) . δα , φ c Q µ δQ α c } + ) 2 2 , λ ˜ + ˜ hO + κ κ i = 0 = 0 λ − F δα,δβ,δµ,δQ µq F − δα Q − − F λ Q W ] λ . W k c − kc hO φ c λ c F β , +  log + ( 2  ) Q β δ F + ( – 16 – F − ) − µq W k λ + + Q c δQ βδα 2 λ δα + − ˜ κ ) has the following form, after using ( 1]  at the boundary. The scalar and vector variations δα − 1] + [1 µq αδα + ) φ − F − β W λ k = 0 depends not only µ µq c − ) ( c φ . The scalar solution is realized by its slower falloff. For αδβ 2.22 Λ φ φ ) − c λ c − − W k [ ] c + λ + c λ = λ )[ φ 2 − λ − c 2 = 0, it is required to fix i ˜ +2 + κ µ r + − 2 6= 1 is different. Consider d − ˜ δµ κ for α λ φ − λ − 2 c W + i F − β r φ − µq ˜ c − κ hO c α λ 2 c ) provides a finite contribution. Here we impose ( λ ) gives is no longer an option due to the presence of the term proporti ˜ − κ 1) 1] ( = − hO λ φ = β + [1 s ( − − 3.26 φ i φ 3.30 φ F (Λ α c c { , the variation ( (Λ )[ φ hO +( + +  c λ x d − d and = 1, fixing − φ Z φ λ c ( =  . Actually there is a more general mixed condition involving M For Instead, we consider some special cases. If one choose In contrast, the case at the boundary. Then µq F δS If we choose a special case Let us consider this in detail with some special cases. For We need to impose two conditions. It is possible to choose separate. We can use the mixed condition on the scalar variat Q with the condition ( vanishes, and it is natural to fix to have a well definedAfter variational that, problem. one This can also impose yield two th conditions among general Λ The expectation value Let us first consider 3.2.1 Scalar mass above the BF bound Note that the term ( variation is to JHEP11(2016)044 ) = δα δβ s . } , the φ 2.22 W (3.39) (3.37) (3.38) (3.42) (3.40) (3.41) F W c ,( = 0 φ r/c /c c 2 F ˜ κ F d before in and log Q δQ µq φ and c δβ F φ first. Then the µqk µq because it is too W 2 kc . ˜ κ + = 0 to have a well α + F φ c δβ α c ) + µqk/c W − 2 β . We consider the case ˜ λ κ = const alar solution, either F 1) F 1] µq Q c F − . For general equire boundary condition between − 2 φ µqk/c Q + c 2 . For general Λ = φ = 1. Of course, we can choose d/ , ( ue is given by κ c α + − [ 2 i λ = 1. We impose the condition − F r − ˜ 1) κ F + ˜ ) c − − λ F r Q W ] β δα . α c − 6= 1. c i φ + } β . W + c 2 φ F λ 1) hO ) F c r c 2 W = φ = 1 and c Q + d/ 2 c 2 − s is = /c λ − ˜ κ ˜ κ F φ 2 φ hO 1) log 2 s − c c ˜ F κ ) + ( + ˜ F κ φ ( α φ 2 − β φ Q Q ˜ κ . + [1 µq φ + (Λ 1)( c + F φ (Λ { – 17 –  r − Λ . Of course, we should not fix kc δα − + φ F = 2 λ F α r c . = (2 ˜ i W κ + log ) Q 6= 0, specifically c +2 α k δQ φ d α α +2  log βδα ) F Λ − − 2.8 r ) c hO } µq F − r F 2)( + 2 F = λ W ] c λ Q ˜ κ δQ d/ φ α . Let us focus on ] 1] c + r 2] log φ 2 µq c − F 2 d/ µ µqk/c − δµ φ c ) log 2 c φ − 2 [ κ c + µq d/ 2 − 2 + ˜ + [1 1) − µ − ˜ κ δµ r + [1 − φ α − λ φ W 1) µq = 0, the variational problem reduces to the case we considere φ (Λ c F +(1 c { c and (Λ 1) F − φ = . c ) gives, after using (  − φ +( + − s c x φ . ([Λ φ d F ) gives (2  d 2.22 c (Λ variation is tied with that of { x or d Z +( + 3.1.2 F + 2 d = const 3.40 Q Z d/ i = β 6= 0, the variational problem is too restrictive for = φ Now we come to the other scalar solution, Let us focus on the case with − F φ c r M Q M c W δS δS variation ( c 3.2.2 Scalar massHere saturating we the again BF consider bound two separate cases depending on the sc expectation value for the operator dual to If To make things a little more clear, let us fix restrictive. This signifiesthe a scalar new and possibility gauge to variations. have a mixed more general mixed boundary condition including gives hO section which is constant. The on-shell action is finite. We further r Thus If one chooses defined variational problem. The resulting expectation val Then ( JHEP11(2016)044 ) = 1  = 1. 3.40 F F . The (3.50) (3.47) (3.44) (3.43) (3.46) (3.45) (3.48) (3.49) F φ c c β Q δQ . µq F F c ) satisfied. δQ + . µ δµ 3.48 and = 0, the variation is µq = 0 . δα . φ β 1) c = 1 can work for F F φ ]. c φ − = 0 , = 0. The variation ( δα, δβ Q = 0 c = 0. If we consider δQ ar is ndition ( 18 F − F α F F c F r µq = 0 Q Q δQ . δQ δβ . + + ( log µq log = 0 or µq W k } 2  δβ δβ c δ , which is not fixed at the boundary. ˜ φ κ + F . W c F F β k − W W k β c ) c Q β . F δQ W . = 2 + c φ c δβ µq ˜ c 2 κ . If one chooses F µq µ β µqk/c ˜ κ + + µqk/c 2 + F − Q δβ 2 r c ˜ κ β log = ˜ κ = F δα + δ 2 2) i log F c i log r – 18 – ˜ c α κ and 1) d/ µ + =0 ] δµ + δα hO α φ − log β ( and impose the variational condition c δα β 2 µq } } F 2 2 . 2) hO ˜ κ , δ c α 2) are adjusted to satisfy the relation for general ˜ κ 1) W ( − d/ ] − d/ − ] φ µq and impose the condition = 0 φ c F W c 2 . 2 c + [1 , 2 µqk/c ˜ κ = const . Similar case was considered in [ δα 2 φ − = 0 and requires the mixed condition − κ + ( δβ µq (Λ α is directly related to that of = 0 µqk/c + ˜ k 2 + 2 δα W + [1 F c ˜ κ α ˜ κ = const + [1 r β δα φ { { Q 2 φ φ − c ˜ κ µq + (Λ log (Λ k = { − 2 β W c ˜ κ +   α δα x x = d d d d α Z Z = = M Once we consider special cases, we see that only Before moving on, it is interesting to examine the case δS proportional to This case is actually realized in an example below with the co In the last line, we use the relation Now we can fix due to the non-trivial coefficients of The expectation value of the scalar is also given by We can fix gives This brings the variation to the form Thus the variation of which is to be understood that In general, we need to impose two boundary conditions among The vacuum expectation value of the operator dual to the scal variational problem fixes JHEP11(2016)044 . W c = 1, = 0, ) has δα (3.56) (3.53) (3.57) (3.55) (3.51) (3.52) (3.54) φ = F c c W s 2.22 φ αδβ ) µqk/c = 1 below. d 2 φ ˜ κ F c c F . Another simple c − . + φ δβ . β F . = 0 1) and one mixed condition + (Λ µq F − . F QQ , then φ = 0 . e consider c Q δβ δα δQ ] . ( = pends on that of the charge . Let us start with . F α F d W i − F µq ) 1) = 1, we have − Q F δQ µq /c Q β . Again, the variation ( = const Q + φ = 0 2 , r − r c 0 = const ˜ + 2 κ µq α = φ F r hO r ˜ κ i F c i + log F . Then Q F ∼ . µqk δQ 2 Q dαδβ log Q to the variation of 2 s F δβ φ d/ ) + ( c α ˜ κ c φ µq , 2 hO − F 2 hO β r . + ˜ . κ − + Q 1)( α/ W k + W 6= 0 provides various possibilities. For β c −  . We consider some specific cases. For c  ) = const r δα − W F φ F φ ] d F , F c – 19 – c c F kδα c F F = Q W 1) log 2 . Similarly, for 2 δα i . Q Q Q s δQ 2 δQ ˜ κ µq κ α F +2 α /c φ − W 2˜ 2 2 r Q )( + φ µq µq ˜ ˜ κ κ − c φ and 3.1.2 ) simplifies to F WF F ( hO c . c = ] log with condition 2 − µqk + µqk/c φ + i + 2 ˜ κ 3.54 2 φ F dαδβ β ˜ β κ + 2 + Λ κ 2 F µ µ − Q δµ δµ d c i ˜ hO κ d/ [(Λ 2+Λ ] dβ 2. Then F = const ) φ µq µq 2 + ˜ / Q = + φ c d/ α c ) 1) and 2 1) i β/ d φ hO α = 1 φ c − − − − − k c 2 W r αδα φ 2 c hO F F 2 c φ − c [(1 c ([1 ˜ κ − Λ by coupling the variation of +( +( +  = ([(1 F F x  β d Q δQ = 0, the variation ( x d d = 0 and φ µq d Z φ Z = 0, one can refer to section = = = F M Let us be brief on the other case For Λ c M δβ 2 δS 2 δS ˜ κ α/ Then It is interesting tooperator due see to that the the coupling term scalar expectation value de For the massless scalar, there are two solutions 3.2.3 Massless scalar we set Λ the form for general a simple choice is to fix One can work out a variation for general In general, we can impose two conditions. For example We can impose two general mixed condition. For simplicity, w For case is to fix JHEP11(2016)044 . (3.59) (3.62) (3.60) (3.63) (3.64) (3.58) (3.61) = const α δβ ] W /c 2 to find ˜ . κ . µqk amples = 0 . fit together nicely. . F are functions of the unfixed . c F F F y variational problem, the on- F MD solutions with asymptotic = const µq + Q Q µq = 0 δQ Q F α ) = F . Q µq = d .. i i Q φ i F F + c F Q Q . and Q log − 2 . δ hO hO φ ˜ κ  hO F W dαδβ = const F = 0, it is straightforward to fix c F φ Q . c φ F Q d + δQ c + [(Λ W 2 δα µq − Q − ] c ˜ κ = const r – 20 – β , φ µq k W α , = d β W F Λ d βδα c c i /c βδα ) 2 φ 2 + − 2 ) F 2 c d ˜ ˜ κ κ + = ˜ κ d d ˜ φ κ Q = − s 1) − Λ , kδα 1) µ r φ δµ hO α and consider the expectation value for the operator dual − 2 = µqk − = ˜ κ . F i φ µq i φ c = 0 c β c ( α 1) ( , we can fix + hO δβ − d ) simplifies to hO − − α φ φ 1) φ F = const c (Λ 3.59 [Λ − (Λ F +( + φ  c Q x d to find d = ( . φ Z = M = const = 1, we fix is constant δS F F F Q Finally, we consider the case c Q and to For Let’s consider some specific examples. For We impose 4 EMD solutions In this section, we apply theshell general action programs and of the the stress boundar AdS energy boundary. tensors to It some is analytic pleasant E to check that all these programs Note that both theparameters. cases the expectation values of the scalar If we choose Λ One can impose a mixed condition as familiar from previous ex Then, the expression ( JHEP11(2016)044 ]. 7 (4.5) (4.4) (4.1) (4.2) (4.3) . is a horizon  . Depending h 3 3 r Q r / / ], this solution ]. . 8 2 7 ω L 3 49 . 2 / . There have been 1 + Q 1 κ 3 2 ) F 2  where ω − Q 2 2 = ]. This solution can be log L = 6= d 48 3 ωL µq al potential or the charge. 2 , Q =0 + ( , √ 2 bute to the mass. Thus the , 9 . = φ

3 nalyzed in [ ) Q 3 ular solution is supported by  = ) ]–[ Lκ 2 ) )  r − 2 h − 2 φ 45 grees of freedom as noted in [ r r ( + q . Thus , and the coefficient of the second F ( . Note ωL 2 + 6 3 2 V . / φ / Q O √ 2 1 1 Qω ( 3 Q ) − 3 e L / /L 2 dt , φ + 2 3 2 2 , √ Qω πµ − − L 4 ) 3 ! L r 2 2 Q ( 1 κ √ 6 3) Qω 2 / ∂φ r − = = O 1 ( = 1 3 √ using ( = 3 2 1 = ω  / √ + ) are satisfied. As noted in [ φ/ 2 µ 3 q − Q ω φ L r − Q , ). They are slightly above the BF bound. Both √ 3 – 21 – Q, ω 2 2.2 6 3 2  , s √ / F 2 , h 2.6 cosh( 1 ) √ 3 dr 3 φ 2 / + /  ( 6 2 C 1 = L Qω 2 ) r r L ] has the following action and metric h Q 3 − W φ 6 ω − 7 6 cosh Qω e / 2 + √ 3 5  L = 2 in ( ) Q 1 + 2 √ ) = = 2 ω R − ) + +  2 φ t ∂  2 − + ( λ ∂φ ( , we can see L A g

( Q d~x − A log − π − = 3 4 + 4 = √ 2 q x 2 + 4 µ ,V L d  3 = 1 and 2 φ 3 hdt r L √ m Z − − = (  λ 2 φ/ C T e 1 κ 2 background 1 2 4 e dA , A 4 solution considered in [ = log = = = 4 ) = 2 S F C φ ( ds The temperature and entropy density can be readily evaluate From the gauge field The mass of the scalar can be evaluated W radius. We express physical quantities in terms of the slower falloff of the scalar field of the scalar falloffs are normalizable. Note that the partic uplifted to resolve theA singularity solution by with including similar stringy potential de in asymptotic AdS is also a has a naked singularity in the extremal limit term is proportional to the charge density, One can check the equations of motion ( The constant term is the chemical potential extensive literature works that have dealt with this issue [ The gauge field term decayscalar sufficiently field fast has and does not contri 4.1 AdS on the choice of the boundary term, we can either fix the chemic The AdS JHEP11(2016)044 1 − (4.6) (4.7) (4.9) (4.8) φ (4.11) (4.13) (4.12) (4.10) c .  = 2 t field theory φ A rt s nothing but . F  r 2 ω φ 2 , φ L L . b W n 4 Λ S F . ω c 2 + , + 2 + 16 ω . L  φ 2 3 − ω 2 i r / 2 L j 2 ∂ 4 the Balasubramanian-Kraus r L φ A L the pressure of the system. + 16 3 , φ WF ri L 3 / φn ing the condition Λ + 16 hell action (density), we identify 4 Λ / 2 F 4 φ − 4 3) r 2 3 + 32 c / L L − 3 √ 4 V  2 Qω 3 /  . From the Einstein equation, we get / φ κ 4 L F W n ij 2 r φ/ 3 c 2 2 3 4 L γ / ∂ . 32 3 κ 2.3 2 r = 1, that is used to make the variational 2 / g 48 4 − Qω 1 + 2 ) is κ 4 t L − − F φn 4 − cosh( Qω − c L λ A (3) ij φ √ L 3 32 2 2 F Qω 2 φ c rt c 2.35 48 κ c F Q 12 L dr 32 ) c F LG − ǫ 48 − r 32 φ r – 22 – h − c 48 = 2 3 r − − 2 3 (3) ), with Z ) φ Q ij 3 − W n ) R x − γ Λ 3 φ ij Q 4 3.4 ∂φ 2 L c ) 2 L γ d ( Q (1 3 φ 2 2 1 c − h 1) − Z 3 − = F = 2 2 L ij − c i − 1 κ (1 γ φ R yy 2 . + c Θ (1 T = G h Θ + (3 = − − −  = ij = γ = shell i = i shell − − G tt xx − P on √ = Θ T T s on h h x ij 4 S d T = = 2 are identified as energy density and pressure. The pressure i is a density, the on-shell action divided by the volume of the κ P E ∂M P ) so that the stress energy tensor is finite) Z 2 shell 1 4.9 κ − and − on s E = The corresponding stress energy tensor ( To have a finite on-shell action, we impose the condition b S which can be used to evaluate to find where the boundary terms include theterms, Gibbons-Hawking and term, the scalar and vector boundary terms Below we also check that this grand potential is identical to This is consistent with the condition ( the grand potential, Here problem well defined. Then we get given in ( Let us evaluate the on-shell action following section 4.1.1 On-shell action Explicit computation gives the following data (after impos Here coordinates including the compactified time. Fromthe this thermodynamic on-s potential (density) JHEP11(2016)044 ). We (4.18) (4.16) (4.17) (4.15) (4.14) (4.20) (4.19) 4.4 potential ) does not . . 4.17 ω 2 2 4 L L L Qω 2 2 3 κ modynamic relation κ 2 √ + 16 6= 0 tential ( 32 . nds on the parameters − 3 φ c Q ω = is nowhere found in tem- 2 q . = 3 L . The corresponding on-shell en though this is legitimate ) is satisfied, all the possible − i , Q em, it is not acceptable from F yy . = = 0 ω 4.9 Q + 32 T 2 i ). h ω F 2 3 µ L 6= 2 c / κ 4 = 4 L . One can easily check the relation, hO Q L 4.15 i = 0. Then we get the following data L 3 P + 16 / xx φ for 32 2 2 − c 3 qdµ . 4 + 16 T κ h L 3 Q 4 , − 2 ) is realized with the scalar field ( Qω as in ( L Q = − κ F ) φ c φ 4.1 32 c = 0 = 32 c sdT = 1. Note = 0 48 3 – 23 – µq , β − − 2 F − is fixed for this solution. − − 1 = 0), we examine possible quantizations with the shell c ˜ κ 3 , λ ,P φ − (1 F c Q Ω = T s = on c ω d − s 2 = 2 1) β − L − ) 4 = + − − L M φ 2 G λ , λ c κ 2 + 32 Ω = − ˜ κ (3 3 32 = 0 Ω = + Q = λ ( − . A priori, as far as the condition ( M Q, β = = 3 2 i i √ = 0. . In fact, it holds for any tt 3.1.1 T = F h are identified as energy and pressure. We check that the grand µq fixed c α = = + P Q ∼ M T s α . At this point, one can readily check that the following ther and F = − c hO E E M Let us start tentatively by examine the case Now there is a troublesome fact. It turns out that the grand po One can also explicitly compute the mass density Let us consider the general form of the grand potential with and φ for the dual field theory Here Ω = Where we use For the grand canonical ensemble ( results in section action and thus the grand potential Ω are is nothing but the negative of the pressure, Ω = satisfy the differential from of the first law and the field theory stress energy tensors are quantizations are legitimate. Theare solution going ( to see how the parameters 4.1.2 Grand canonical ensemble Thus we confirm thatc the mass and the grand potential both depe holds if we set If one think a little more, the reason is obvious. The term quantization from the pointthe of point view of of view the of variational the probl thermodynamic first law. perature, entropy, charge and chemical potential. Thus, ev JHEP11(2016)044 ssed (4.29) (4.21) (4.22) (4.23) (4.25) (4.26) (4.24) (4.28) (4.27) is uniquely fixed as stringent constraint φ , c . ω 2 ω transform from the grand L 2 ameters are fixed. One can tance of the general bound- s the same as that given in L stent physical quantities. e energy is a Legendre trans- + 16 . 3 + 16 / 2 4 , . 3 L / L 2 ω 3 1 4 3 κ T s . / L 2 − 2 2 = 0 , 3 κ − / µdq 2 4 = 2 2 = P κ L Qω . κ + 4 φ M 2 , then the parameter ω 3 1 L Qω Λ 32 48 + 2 L = ), goes as 2 32 48 − = sdT qdµ E − 3 µq – 24 – φ − − − − ,P from the thermodynamic data. If one uses the dif- 4.10 c = 1 , Q 3 ) = φ 2 = 1 3 φ F c Q L Ω = i c ) sdT c ω 2 µ 3 φ µ = dF − κ c = Ω + T 3 φ − h c F = − (1 Ω = E d (1 = theory is described by the grand potential Ω − 4 = shell ). Thus it is required to put a mixed boundary condition discu − F on 4.9 s . The on-shell action, from ( is fixed by ( φ 4.1.2 ). Thus the differential form of the first law actually put more Summarizing, the AdS 3.8 The necessary information forsection the semi canonical ensemble i Thus the semi canonical ensemblepotential. is well defined as a Legendre also check that the trace condition is satisfied. and the field theory energy and pressure By demanding the differential form of the first law, all the par than that of the exactary form. value This problem demonstrates along that with the the impor on-shell action to get consi where Λ in ( ferential form of the first law, Now we can actually fix this parameter Thus it is consistent withformation the from picture the grand that potential, the Helmholtz fre and the free energy 4.1.3 Semi canonical ensemble satisfies, again, for We further confirm that the differential form of the first law JHEP11(2016)044 (4.31) (4.32) (4.33) (4.30) . , µq .  − 6 . 4.1 √ E , φ/  mble evaluated 2 = 2 which is directly 2  − r Q 2 e F F 2 r E Q M ), are satisfied. The + 4 1 + 6 2.2  1 + . √ as the energy above the of the holographic renor-  3 φ/ y tensor can be obtained s, ( log / e 4 8 . 6 log emi canonical ensemble. For he negative of the free energy L 2  . 3 3 3 2 √ ), to find f the mass / ) / 2 be found in section , 4 2 1 2 4 , L − = . Again the system has a naked − L 3 L  3 / r 4.14 2 −  F Qω ) / 4 ( κ 2 r 3 L Q and the coefficient of the second term φ 4 L 2 O ( 3  2 L ) = − / 6= V 2 Qω + 2 √ φ L . The mass of the scalar comes from the 4 3 ω κ ( Q log ω 2 − ω 2 L Q 2 3 2 − 2 Qω 2 L dt , φ ]. Here we find that the energy of the fixed 2 − = L √ ) 3 κ r √ ω 2 Q ! 51 = 2 µ ,V = does not play the role of mass in the Helmholtz − Q 2 ∂φ 2 h , 1 i L 6 ( – 25 – κ r ω ω F 3. It has a further contribution from the bound- 2 2 1 2 √ − 2 50 yy / + . Note = M T 1 L √ φ/ 2 2 = − h 2 2 2 q − Q √ e 2 L F Q , /L = ,D 1 4 ] has the following action and metric = Q F 3 = E 2 i 7 ) + i Q φ  φ = = tt xx dr 2 ( 2 Λ 2 ) = r t ω T T , D = F r h h φ Q 2 h 2 W A 3 ( + e / M √ ω , we have = = 2 Q A 1 + F F Q R − not by the energy of the extremal black hole, ) + = 1 P  E  2 − ,W φ g

c µq 2 d~x log ) − 2 = 3 1 + r √ 2 2 are identified as energy and pressure for semi canonical ense x + + 4 ωL F d 2  hdt P background r Q L Z − ( ( 5  solution considered in [ ) by using 2 . One can also explicitly compute the mass, ( C − 5 and 1 κ 2 F 2 dA , A e F 4.13 − E = log = = 1 = = From the gauge field For the rest of this sub-section, we comments on some results = 2 h S F C F ds RNAdS black holes withground a state fixed (the charge, extremal the black energy hole) [ is identified computed through the holographic renormalization for the s free energy. It will be interesting to figure out the meaning o is proportional to the charge density singularity at the extremal limit. Further discussions can Now it is curious to find that this mass ary term compared to the grand canonical ensemble extremal limit is given by charge differs by the malization of thefrom semi ( canonical ensemble. The stress energ 4.2 AdS The AdS The equations of motion for the metric, gauge and scalar field The constant term is the chemical potential Here from the stress energy tensor.P We note the pressure is again t JHEP11(2016)044 . (4.40) (4.39) (4.37) (4.35) (4.38) (4.34) (4.36) ) with ωL 2.8 √ = . 2 ation holds ) Q g but the grand e boundary. ω 2 + . L . 2.41 2 h . 2 r . / ω . + 3 1 2 , nds on the parameters ω L ω L = 0 case. From this on-shell 2 ω 2 2 Lω ) and ( 2 L / 4 F L ith the radial fall-off of the 1 + 3 c + L . We do not need to impose ω 2 ressure of the system. 2 L + 3 2 ) + 9 2.37 2 κ Q + 9 / Q 2 L L 1 for 2 − F L L 2 5 / ω c 2.35 / 2 1 . 2 1 p / L ω 1 2 2 12 ) ω Q / 2 2 κ ω 2 1 F − 2 6 − 2 Q c Q r 3 κ F Q ( πω F 2 5 c µq , 12 5 F c Q 2 , we use the relation κ 5 O L ) c L 5 12 − ω − 6 , we evaluate the on-shell action. φ L 2 12 = L 4 + 2 12 3 κ − 6 κ − 6 2 T s 2 Q 3 − 6 + Λ ) 3 and r Q – 26 – 3 φ − Q φ Q ) 2 3 c ) Q Q φ 4 ) φ , s M r φ + Λ Λ 2 − Λ = φ / = + Λ 1 c − (2 − 4 φ P φ 2 c 2 Lω − 4 − = − π − i + 2 − φ (2 4. Thus the scalar field has the boundary behavior ( φ c 2 L yy ) = c (2 T − Q = (4 h (4 − − = = , which is different compared to the AdS = = 2 φ = p shell i T, V, µ c i L − M G 2 φ xx tt = Ω( on . or T T m s ), this scalar solution realizes with the faster falloff at th h h T G φ − = = 2.8 are identified as energy and pressure. The pressure is nothin . One can readily check that the following thermodynamic rel = P E P F c P and and E φ The corresponding stress energy tensor is given by ( The temperature and entropy density are One can also explicitly compute the mass to find from ( , c = 4. It saturates the so-called BF bound. This is consistent w φ Thus we confirm thatΛ the mass and the grand potential both depe Again we check that this grand potential is identical to the p To write all the expressions in terms of scalar field Here 4.2.1 On-shell action Following closely the previous section on AdS d Compared to ( Note that the scalar boundarya terms condition are on finite Λ at the boundary potential term as action, we identify the thermodynamic potential (density) potential, JHEP11(2016)044 l . − φ , c φ (4.41) (4.44) (4.45) (4.46) (4.42) (4.43) sdT − ) the grand Below this Ω = 3.18 d . 5 for general Λ ω L e only fix the com- 2 2 e falloff of the scalar µq L . κ 3 . 6 ynamics ). Note this condition are independent of the − ω , ω 2 q . his happens because the = eld theory ( 2 T s L − i 4.41 ω L 2 − xpectation value of the dual yy = L + 3 m of the thermodynamic first T i + 3 6= 0, we are required to impose h M 3 µ s, T, q, µ 3 α 5 Q + 9 = = 2 ) Q L hO 3 ) i φ κ 2 P φ 5 5 Q yet. The stress energy tensor is κ xx ) − L L 6 , T , φ φ 2 6 + Λ h , while c + Λ Λ , = κ 2 2 φ φ ω 6 κ φ c = 2 2 − or − 5 G , c c 4 2 L 4 φ Q 2 L φ φ ˜ κ 3 φ − c 6 c − − − = 0 2 3 (2 φ – 27 – (2 = 4 c F = ,P r = φ − c (4 = 0), we examine possible quantizations as we have 5 i Λ G ω = = L ). One parameter remains unfixed. Even though the 2 = F yy 2 c β 2 L T i κ φ h 9 ˜ κ tt 6 c shell 4.41 T = . All the possible quantizations are legitimate with genera − h = = i on i = 4 to have a consistent variational problem. Here the scalar i = 3.1 s xx tt . Then d − T 4 T =0 h M h α = = = = through ( hO ) for G φ P E M c ) or special cases of that. ) with the faster falloff. This has an important implication. 3.16 = ) for mass saturating the BF bound. If are energy density and pressure. The pressure is nothing but and E 3.12 4.34 2.8 φ P = 0, we impose the condition . One crucial information on the possible quantization is th φ and α c E This quantization gives the expectation value for the dual fi Upon imposing the differential from of the first law of thermod For We mention an important implication of the boundary terms. W , we are required to impose the same condition given in ( and φ done in the previous section field given in ( potential. The grand potential has the relation that comes from ( For the grand canonical ensemble ( is realized as ( the condition ( qdµ parameters similar to AdS Here and grand potential and stress energy tensors are all fixed, the e grand potential depends on the parameters Λ is the one we have from the consistent variational problem. T condition is shown tolaw be and consistent also with to the fix differential the for mass to ADM mass. Λ bination of Λ The on-shell action and the grand potential Ω are 4.2.2 Grand canonical ensemble Note that we do not impose a condition on Λ JHEP11(2016)044 . F w is − (4.47) (4.50) (4.49) (4.48) = F P e the similar ] ), are , 55 2 4.41 dr . . ndetermined coefficients ) ion of the semi canonical ]. The action is the same r . 3 ( L 56 ) 2 g ed variational problem. The L n from the grand potential, 3 / , F 2 . 2 ition ( ) 2 1 entioned there. This is similar ]). / r e of the free energy L F 55 ω 1 L o be independent of the unfixed ]. For a certain value of dilaton 2 r/r 54 h scaling solution in IR and the 2 5 ω ( + , / 2 , Q 0 37 r/r L 2 1 2 ( k 2 Q κ L ω 0 53 12 κ 5 2 2 dy ). We further comment on this below k 5 / 6 12 L 2 1 − 2 Q L 1 + 6 r L ω 2 − ω 2 12 4.42 κ 2 5 ω ), to find + 6 Q L 2 L − ) = 2 2 3 r L 12 ω ( κ 3 2 4.14 dx 6 = ] (see also [ − 2 − L 2 i – 28 – 9 r L ω = 1 52 = 2 yy . = T + F L , f h 9 c 2 F 1 2 4.1 shell = dt E =  − ) i 4 i r 4 F on = ( r r xx tt s f T T F − 2 h h 2 r M L 1 + . = = =  − F F F µdq = P E . We further confirmed that the differential form of the first la ) = 2 + r T s ( ds g − sdT − = 3, and we use the coordinates system considered in [ does not play the role of mass in the Helmholtz free energy. Se M d F = M Ω = d µq ) with in UV, which has attracted much attentions recently [ Previously, three counter terms (different from ours) with u We comments on some results of the holographic renormalizat 4 2.1 = Ω + The on-shell action and Free energy, after imposing the cond Thus this Helmholtz freeF energy is a Legendre transformatio This mass satisfied, as ( 4.3 Interpolating solution In this section, weAdS consider the interpolation solution wit coefficient, while the field theory expectationto value our was not observation m done in this section. 4.2.3 Semi canonical ensemble as field theory shares similar properties [ were considered in the contextcoupling, of one linear of dilaton the coefficient gravityresulting [ remains on-shell unfixed action for and a conserved well charges defin are shown t scalar field remains unfixed as can be checked in ( It is curious to notice that the pressure is again the negativ One can also explicitly compute the mass, ( discussion at the end of the section ensemble. Energy and pressure are given by JHEP11(2016)044 done (4.55) (4.52) (4.53) (4.51) (4.54) 5 ,
7 tial feature F r at the spatial 1 4 , this particular 2 / + 41 β 3 ∼ r 3 . We compute the 2 F W + r 3 = 0. We come back to 4 Qκ . r 4 F r ) r α L 4 3 0 ur notation, even though . 30 field alone. log k − ed to the fact that the mass nserved charge comes from 2 3 2 r fall-off of the scalar field explicitly, we get the density 5 ate for the zero temperature ess energy tensor. Similar to / ( 2 α F 3 r / r r φ is is similar to the AdS O 0 = 3 ) ) k . Λ F 4 , + µ ) → . Note that the coefficient of the c φ F 2 3 2 9 c r / + 2 φ κ r 3 5 4
2 −
F + − L 4 2 r κ r 0 φ . 4 3 F 4 ( , k 6 r r ). This is contrast to the EMD solutions 2 L 9 4 ( Qκ F ) O φ 0 has the following form 4 4 r 2 ( 2 − k ) = 9 + + 4Λ F / L 6 3 A 3 r 2 + 11 0 W = φ 2 2 F
k / / c 4 µq 4 r 2 3 3 F r . – 29 – . They have the property F r r L 9 4 12 + r + 11 2 φ ) 0 2 3 ) 4 − k 4.2 2 − F + 4 r m 3 r F √ r 0 2 4 F 6 r 3 (9 2 k √ r r r F . Thus ( 0 + 3 r Qκ 2 = + Q 2 k = 4 F F
L 4 r 3 r L φ 4 r + 7 = 3 3 0 F + 2 shell
r k r )( rt 4 4 0 − 2 3 r k F and section F + on F 2 r ) = r s (9 3 φ r t 6 − ( + 0 4.1 + 4 F A 4 k 3 r r 4 r ) is not directly related to the charge density. This is essen 6 gW r 0 r (3 3 k − 4 3 ( κ 4.52 8 F √ 2 2 r r 2 r 2 2 0 2 κ 3 Q . The variational problem at hand is the case with k 6 − 8 s L 4.2 2 − − = q ) = ) = ) = , the scalar boundary contributions are finite. This is relat r r r The mass of the scalar field can be shown to be The solution reveals that the gauge field 5 ( ( ( ′ V φ W we consider in section second term in ( We follow the notationsthe of concept of the the previous thermodynamic sections relations by are not abusing appropri o 4.3.1 On-shell action Let us compute the on-shellAdS action and the corresponding str details below. boundary, and thus effectively the charge come from the gauge As usual, the constantcharge term is the chemical potential, in section solution is realized with the faster falloff. Thus the analys It saturates the BF bound. This is consistent with the radial due to a nontrivialthe dilaton scalar field coupling. through the Some dilaton fraction coupling of the co The solution is specified by Compared to the general falloff behavior of the scalar, of the scalar field saturates the BF bound. By keeping JHEP11(2016)044 = φ c (4.59) (4.60) (4.56) (4.61) (4.57) (4.62) (4.58) 2 given se / 3 − φ theory side. c ot the case. Let = 3 = 0, the variational φ = 0), instruct us to . α 3 F q . c F . r − ) 2 F ns through the parameters for = κ . ion of the well-defined vari- he dual operators are given epends on the undetermined c 4 3 i ) 9 3 egative of the on-shell action µ F n geometry. L . and infinite. It is a ‘density’ in . We also note that the energy F r 0 2 − nformal invariance even though 2 r k hO κ 3.1.2 ) / 2.35 , φ 2 T s . 4 3 F 3 L c − 0 − F 9 r k = = 0 φ + 4Λ , ) µq . 6 2 − = 0. After using Λ c 2 φ F κ P P φ − / c c 4 F 3 F 9 − satisfies a relation even at zero temperature c 2 = 3 L r E 12 κ 0 + 2 0 − = G 2 4 2 d k k − = + 4Λ φ 6 E L 3 √ 1) 0 φ – 30 – 2 P − (9 c √ k 4Λ (and in section − 6 − 2 φ = 12 = ˜ c φ − κ i c = i ,G φ − µ µ c = yy 2 G 3.2.2 T (9 T κ β h 3 2 h = (2 4 (12 φ − ˜ κ F c φ L r 6= 0 cases separately. = 0 = = Λ i k = F i c G i tt xx = T T h h =0 E = 0 α = = F hO c P E = 0 and = 0, one works with fixed chemical potential. For F c F c . Note that the energy would vanish if one would be able to choo . To fix this, it is required to have further input from the field F ) φ c c = 0, while the pressure would be finite. We see below that it is n 3.16 ), we get ) and F c φ Let us go back to energy and pressure for The corresponding stress energy tensor is given by ( 4.58 3.18 = Λ , φ φ problem we considered in section us consider the 4.3.2 Case with If one chooses Λ Thus the expectation value ofparameter the scalar in dual field theory d It is interesting to observe that the ‘potential’ and pressure satisfy the traceless condition c The physical quantities depend on the boundary contributio It resembles that of thermodynamics without the term choose ( time coordinate as well. We identify the ‘potential’ as the n solutions. In particular, the time direction is not compact Apparently the interpolating solutionthe still interior is respect much the modified co with the hyperscaling violatio in ( The parameters are partiallyational fixed problem. by the Thein consistency corresponding ( condit expectation values of t JHEP11(2016)044 . φ c (4.64) (4.66) (4.67) (4.65) (4.68) (4.63) ). The and ), to φ 2.17 . Moreover, φ 4.66 s Λ c . In particular, , 3.2.2 ). ry term ( = 0 3.48 .
. 3 ) amounts to F orresponding stress energy c F 2 q , r 1 κ ) 2 . 4 = 0 4.65 − / ng the condition ( F emselves throughout the paper. om fixing the conserved charge 3 F L F 3 c iven in ( c 0 r µq F 3 2 k 0 − 2 r 2 , µ κ 2 k ˜ κ 4 3 1) √ (1 F k L 2 c √ 0 − F W 2 3 log = 2 k c φ c δ F 2 ˜ c κ c + . Note also this condition reduces to the ) ( + 2 − = 2 3 / 3 β F F = = r − 3.49  0 . The corresponding vacuum expectation value Q 2 ,P β k φ – 31 – 2 2 d = 1 does not correspond to fixing charge, the 3 √ 2 c φ ] 3 κ shell 2 ˜ c √ κ φ 4 log F − F c 2 r δ − c 2 = 3 ) on κ = µL F s 4 ) φ F c = − c L − Λ W i 5 0 + µq c ), satisfying the condition ( k = =0 µ − 2 + [1 α 3, F / φ = 3( 6= 0. This case requires the sub-leading part of the gauge field (2 4.50 log 4 hO Λ = 0. . Setting F F δ . The expectation value depends on the parameter − =  c F 2 F Q 1) / c c 6= 0 E 3 = F r − F 0 2 k c , k F 3 √ c ) for 2 ( √ = 3 = 4.58 d can be considered on equal footing as the other two coefficient β F c = 0, the general variational problem is analyzed in section α To consider the ‘semi canonical’ ensemble, we add the bounda For the variational problem requires the following condition g on-shell action and ‘Free energy’ are modified, after imposi the mixed boundary condition is given by ( previous case ( where we use where we use Here again, we abusetensor our is notation given for by the ‘free energy.’ The c For the particular solution ( 4.3.3 Case with Let us briefly mention on These are functions of where the solution gives coefficient of the dual scalar operator is in the analysis ofbecause the the variational problem. scalar coupling It also is contributes different to fr the charge. 5 Two physical implications Here we consider two physical implications that manifest th JHEP11(2016)044 to out γ ], the ] (see − 52 52 √ R In [ ℓ 1 − ]. = 57 ct . This is true even S ) d ergent physical quan- ( en though constructed R n-shell action and stress d theory. y Roman Jackiw [ ding on which symmetry, γ, y well defined variational laim to have finite counter ass in the chiral Schwinger cancel the divergences, and theories are undetermined, dary geometry, such as the field theory side. ive correction, when inserted ounter terms we encountered sidered in [ count a few examples. heories we considered above, ction to field theory radiative al’ inputs are necessary to fix iangular graph of axial vector result. Three different classes uantities including anomalies. by subtracting the divergence l, one can add the terms such etermined because, due to chiral or indefinite result? 2 Pauli term in QED or photon and retains the symmetries of the − g ]. We examine the available examples in requires only a term 52 3 Can one formulate a criterion that will settle a – 32 – are not determined because these terms vanish too fast n ]: (a) radiative corrections are uniquely determined by c 52 ], holographic renormalization for the AdS has been carried 20 , and the Ricci scalar and tensor built from 19 ij γ In [ 1. The coefficients ≥ ]). n 54 , , n 53 R Jackiw asked the following question. Renormalization in Quantum field theory is a way to render div n c for the simplest geometry. For example, AdS to match the field theoryTrace expectations anomaly for for dilaton various physical coupled q theories also has been con from the localinduced and metric covariant functions of the intrinsic boun Here we consider severalterms known as examples well in as holography somecorrections. that relevant c cases that The provide closeproblems. holographic conne boundary The terms counterenergy are terms tensor. are Sometimes required the required b boundarythus to terms we are yield put also the them used to finite in equal o footing. They are far from unique, ev holography and compare them with those of the examples in5.1.1 fiel Examples in holography basic rule of thumb has beeninto the provided. bare If Lagrangian, the preserves computed thetheory, radiat renormalizability the radiative correction doesof not examples produce are a presentedspoiling definite in renormalizability or [ gauge invariance such as also [ priori whether the radiative correction produces a definite cancel the divergence inas the stress energy tensor. In genera to contribute to the finite part of the stress tensor.Anomalies. Let us re In this section we have aabove. fresh In look for particular, the we finite compare boundary the or situation c to that of the 5.1 Finite boundary terms the radiative corrections. See more details in [ mass in Schwinger model, (b) radiative correctionsanomaly, are there not d is nomodel, symmetry prohibiting and to (c) insertvector Radiative photon or m correction axial vector, can weanomaly. be choose Thus to determined for preserve depen the in cases the (b) case and of (c), tr further ‘experiment there have been casesnot when to the mention finite. radiative corrections This of phenomena field has been investigated b tities, such as mass andusing coupling the constants, radiative into corrections. finite ones Analogous to the gravity t JHEP11(2016)044 (5.7) (5.3) (5.4) (5.2) (5.5) (5.8) (5.6) (5.1) e holo- , . including the R ··· +1 d + , ℓ . µν 2 3 πG
, we get Ω (2) al invariance in the pro- 16 2 3 d γ 2 . − 4 r dS r after the renormalization. − regularized in a way to pre- ij θdψ = dary counter terms. r scribe here are similar to the + γ 2 plicitly broken, and the trace gy in global AdS. We consider r rgences of the effective action 2 + ∂ ) (0) µν → ∞ nce of the quantum field theory. ntation of the conformal class of dr , ij γ µν r 1 γ j 2.3 + cos µν ℓ − (0) r . 2 . 2 . dx  γ (2) i 2 2 R − γ L 2 2 − 2 r 2 θdφ L ··· dx 3 r r 2 1 κ 2 2 ij ℓr ℓ 8 + γ = Θ = 2 1 + ]. The holographic renormalization process = +1 = + r d  µν L 2 i + sin 2 1 52 3 ) κ tt (0) µν 2 + – 33 – 8 dr πG , T γ 2 h 8 2 dθ 2 ij µν = r L dt − γ = 2 , γ r tt (2)  L = = ∂ T 3 γ E 2 2 ℓ 2 r r ··· + L  2 t, θ, φ, ψ, r 2 ℓ 2 ds + − dt 1 + = (2) µν −  Θ + γ ] gives ij = −  Θ + 20 3 = 2 FT (0) µν 1 2 πG γ Similar to the anomalies computed and determined above, som ds 2 8 ds r − = = i µν i γ T is the metric for 3 dimensional sphere. The energy density by 2 3 Ω d in global coordinate with ( For the metric of the pure AdS This conformal anomaly is a consequence of breaking conform 5 The field theory energy density is given by (see section Then by taking the trace of the stress energy tensor for pure A standard counter terms [ where boundary limit is taken and To evaluate this we use the expansion of the metric where spoils the general covariance to fix theCasimir coefficient energy. of the boun Compared to the field theorycase expectations, (a) the features rather we than de (b) described above [ cess of regularization andand renormalization. of the Due boundary to extrinsicserve the the curvature dive general term, covariance, they analogue shouldThe of be regularization the procedure gauge picks invaria upthe a boundary particular theory. represe Inof this the way, conformal stress invariance energy is tensor ex is expected to have a unique answe graphic examples provide a definiteAdS answers for Casimir ener We use the following metric of the field theory living at one can compute the extrinsic curvature to be JHEP11(2016)044 ]. ], 52 61 ssed have (5.9) (5.10) graphic perties of ). This boundary 2.3 scribe by the term. olographic studies of e ( er term in holographic ated to the probe brane t the on-shell action to field theory finite radia- ]. Similar computations ot decide the field theory . 20 t e boundary term associated f space-time Sch¨odinger [ ) that bring out the mixing st law of thermodynamics or until ‘experimental’ data are he finite term can be fixed by A Here we briefly mention some ]. It is noted that a new type nical ensemble, the parameter φ ation of the EMD theories and ( rt ounter terms. 64 tive corrections discussed in [ es with field theory expectations. F W , was introduced to match the ex- r 2 is required to satisfy the first law. . φ 2 ]. The black hole solutions have gauge W n φ L F 1 2 61 c κ , 2 − 60 γ = − i √ tt – 34 – x T d ]. It has been argued that finite counter terms are h d 63 = , ∂M E Z 62 in global coordinate 2 1 κ 3 in the D3-D7 brane system [ = 3 F b S S ]. In other wards, the first law of thermodynamics requires to × boundary or counter terms in holography that have been discu 5 60 [ , which is actually finite. Similarly, the thermodynamic pro ] for more general discussion of Casimir energies in the holo 5 2 AdS φ 59 finite , 58 , 41 is always finite because it is nothing but the conserved charg F c Another class of finite counter terms has been considered in h We start with the boundary term for the gauge field, which is de The third class of examples of finite counter term has been rel These two classes of holographic examples are similar to the One class of the examples that has introduced the finite count remains unfixed. This is even more clear with the F physical systems, say grand canonicalc ensemble or semi cano This is actually the best known finite boundary term. If we do n thermal and electric responses in [ See also [ pected mass in AdS between the gauge and scalarwith variations. As one can check, th finite counter term is not directly related to any finite radia compare them with the situations of the field theory. vanish for super-symmetric theory. Thisdemanding is the an symmetry example discussed that in t the case5.1.2 (c). EMD theories Here we summarize the results of the holographic renormailz provided. Here the expectationstransport from properties, field are theory, required such as to fir determine the finitewrapping c around of finite term is possible. The finite counter term is used to se necessary to yield a consistent physical picture that match tive corrections. The finite counter terms are undetermined fields and scalar fields. There the finite counter term, the counter term where again the finite counter term proportional to give the energy density for AdS the R-charged black holes have been analyzed in the context o This is identified as a Casimir energy in the field theory side [ setup is in the context of R-charged black holes [ Holographic examples with finiteexamples boundary of terms. the context. in the literature. JHEP11(2016)044 = 0, (5.11) . For F asymp- c F heory in c 4 example. 2 / 5 1 for − tribution of the rameters, while φ 2 + 3 c / ivergent contribu- 3 f thermodynamics, ge comes from the y with AdS = 2 . Thus the boundary − dary provides certain ne. l, energy and pressure φ = 0. Surprisingly, this φ c φ . t requires further input aturates the BF bound. arge different from the ue of the operator dual c F r to the AdS the thermodynamic first c parameters of the theory  fined. It turns out that until we are forced to fix tic AdS boundary because n, Λ 2 isfy the differential form of ions. They are in general ition is necessary to have a φ φ = 3 c tional problem. φ ermodynamics. We encounter heses two conditions coincide L that we can get by requiring 2 for φ 4 Λ − + and φ . Thus the physical quantities are c φ φ φ r c ∂ r = 4 φ φn φ 2 c ). And then one of the remaining parameter  γ – 35 – − 5.11 √ x d d ∂M Z 2 1 κ = . We consider the general Λ φ r b S = 0, for example, from the beginning could lead a consistent t φ c : the boundary contribution of the scalar kinetic term has a d : the scalar mass saturates the BF bound, and the boundary con 4 5 = 0, the energy and pressure are fixed and independent of the pa F tion. It is required to cancel the divergence with a conditio the expectation value of the scalar depends on the parameter AdS which we consider as an independent input. Thus the EMD theor can be fixed by requiring the differential form of the first law o from the boundary counter terms ( charge provided by thedilaton gauge coupling, and field. thus fromThe Part the scalar. features of The of the scalar the conserved mass s holographic char renormalization are simila terms are not completely determinedor by experimental the data gravity from theory. the I boundary field theory. c totics itself can completelylaw. determine the parameters with well defined boundary variational problem Λ scalar kinetic term is actually finite. Nevertheless, a cond General variational problem provides a condition Λ particular condition coincides with thethe requirement first to law sat of thermodynamics!are The determined thermodynamic with potentia theto condition. the Yet, scalar the is expectation a val function of a parameter, say AdS not completely fixed even after imposing the first law. Interpolating solution: the example gives the conserved ch is independent of Let us consider the three examples we have considered one by o As we demonstrate above, the variational problem at the boun There are two additional boundary terms for the scalar fields r • • • ∂ r if the boundarydifferent terms when are the used boundaryare to terms not completely cancel are fixed finite. divergent even after contribut numerous imposing In examples the that first from law case, the of th the consistency of the general varia conditions for thethis parameters condition such is, that infiniteness the of general, theory the different on-shell is from action well the and conditions de stress energy tensor. T These two terms have the samen boundary falloffs in the asympto the end, but we do not take that approach. them. Choosing JHEP11(2016)044 4 and (5.14) (5.13) (5.15) (5.16) (5.12) (5.17) served rt F = 4) is d , is proportional to 4 . There we see that 1 r s .
ominant profile of the 4.3 t physics. 2 (and thus ) φ 5 e charge is hidden in the les the Fermi liquid state ( of the geometry from the Qκ gauge field point of view. 6 F ion 4 r s means that the conserved W ntirely come from the gauge L d 27 2 0 ial behavior at the boundary. − k y examine whether this charge ty , F d to see the same for the AdS 0 at the boundary.) It actually +1 4 Q Q λ ωQ 3 = → , it is easy to see that the coupling r 2 r 1 e rt √ − 4.2 . 2 , d at constant charge density and constant 4 L T. 3 Q = F ) 2 C λ L ωQ κ Q r ∼ rt 2 2 2 ,F 3.3 κ √ 3 C = / = – 36 – 4 ∼ rt = − rt . This example is very different from the previous F s q φ , behaves as ,F Q 2 1 e 1 4 ∝ ]. For example, entropy density used in ( = 7 2 gW F [ = r ∼ q − ∞ 4 T example in section κ √ W = 2 W 2 5 6 F . Thus the conserved charge for AdS r 4 κ Q r 6 2 in L 27 ωQ − e 2 0 2 k = is conserved, there are tension between the field strength 2 √ . q q depend on the details of the solution. The corresponding con = 4.1 = 2 F F W Q ,Q ) at the boundary can be expressed in a slightly different form and = 3 Q 2.3 Q λ λ ). We go back to our examples to see the qualitatively differen ) and gauge field behave, at the boundary, as Let us consider the AdS These examples provide the physical properties that resemb Let us turn to the interpolating solution we consider in sect φ φ ( ( W where of matter at low temperature Because the charge charge ( Thus we check that thefield contribution to in the this conserved charge particularsolution e solution. in section It is also straightforwar where W which give the conserved charge (The physical coupling, examples. In particular, the coupling has a rather non-triv the temperature. Similarly, the specific heats chemical potential are also coincide with the entropy densi In this last sectionWe we focus examine on the thegauge conserved case field charge with at from asymptotic the the AdS boundary boundary. Using the d 5.2 Non-Fermi liquids & charge splitting dominates at large radiuscharge compared does to not the entirely fieldgeometry come strength. because from Thi the the gauge dilatonstring field. field theory can In point be a of considered way, view. th as It a will part be interesting to closel JHEP11(2016)044 (6.2) (6.1) (5.18) ) 3.25 . ) ]. For certain = 0 68 3.24 – ] are equivalently F F 66 39 Q ). δQ ). We also check that . µq ). There the parameters 2.41 = 0 as ( that the mixed boundary F alar fields in contributing F potential in general EMD advertised to indicate the c 2.35 α ers (dynamical and Hyper- on the role of the dilaton rmula [ ible. Let us illustrate this Q i matter. Physical properties or ( 2.22 ting solution we consider in + ]. They are readily different ( current operator ( F iational problem that includes on exponents, the holographic c violation and extensive vol- µ Q 55 δµ directly responsible for various vial and interesting physics. For ce of the exotic phases of mat- ctionalization of charge that has hO ) and ( µq W 1) ]. c 2.33 − 65 , + ]). , ) in the context of exponential coupling 7 F β / ). Now the expectation value of the dual c 69 ) 4 56 3.3 d , T = 1 that gives the expectation value for the + ( 1) 3.25 49 ∼ F for the gauge field. The corresponding on-shell = 0 in the boundary expansion of the massless − c δα – 37 – C F 2 φ  α c c ˜ κ ∼ ( for W as in ( . s . ] (see also [ − F T µqc φ µq 56 Q F , ∼ c (Λ = 55 C + i = = const F ∼ i β Q . We have the general variation for F ) s d =0 Q hO α 3.1 1) hO − 2 φ ˜ κ c ( , along with the gauge field ( d β − r are the entropy density and specific heat [ φ + C (Λ α reveals  ] can be signified by the holographic entanglement entropy [ and → given in section 43 4.3 s are introduced for the scalar, and φ , φ e φ In closing, we mention that the dilaton coupling and a scalar From the analysis of the boundary value problem, we conclude Due to this interesting competition between the gauge and sc 42 ∼ , c φ action and stress energy tensor are determined in ( where scalar operator is given by the expectation value of the dual scalar W the mass density andgiven pressure by evaluated the by components the of Brown-York fo the field theory stresscondition energy between tens the gaugewith and a scalar simple example fields for are a indeed scalar poss with We have examined thecoupling. holographic Due to renormalization this coupling, focusing the we boundary consider terms the for boundary the var gaugeΛ and scalar fields together in entanglement entropy interpolates betweenume dependence the of logarithmi entanglement entropy.presence The of former the has Fermi been surfaces [ 6 Conclusion parameter ranges of the dynamical and Hyperscaling violati section from the Fermi liquid case to the conserved charge, the modelexample, provides a it highly non-tri provides anare example much of deviated non-Fermi from liquid the state conventional of ones. The interpola theories can be used toscaling accommodate violation two independent exponents), paramet whichinteresting are physical directly properties. and/orter in For [ example, the existen We impose the condition been considered in the holographic context [ splitting has some direct or indirect connections to the fra dual current operator as JHEP11(2016)044 . 2 in β r the 5 and (6.5) (6.3) (6.4) (6.6) + φ r to satisfy 4.2.2 log 2 α r qdµ is not coincide , but also the − → and the charge β tic AdS bound- q φ 2 ) required by the amined along with sdT √ L ent for the theories Q − n of 3.16 . the parameters Λ ted in section semble = = 0 in ω Ω = 2 µ α d L e advertised throughout the the consistency between the ics s take the example of AdS + 3 . i . This is contrasted to the other 3 r with F 2 Q Q 5.2 . ) κ . φ 5 ω hO L 2 5.1 4 at the boundary. The same property , 2 6 . / . These coefficients are partially fixed by r + Λ √ 2 1 φ 2 2 φ Q r − c ). The scalar field saturates the BF bound. Q ∼ 4 φ 2 3 − at the boundary is given by c as well. The corresponding physical properties − W 2 and Λ 5.17 r A – 38 – φ √ L (2 = 4 c 4.1 = Q φ − φ Λ = = = 0. Then the mass evaluated by stress energy tensor t α A given in ( shell − W , which is conserved charge. This happens because of the on q s − ensemble. that demonstrates non-Fermi liquid states. It is related to = F G Q 4.3 . This happens because ω 2 3 L √ Q 2 1 κ to summarize. The gauge field ). This condition turns out to be the same condition ( is not identical to = . This demonstrates the finite boundary terms that are preval q 4.2 F 4.41 φ dilaton coupling. This demonstrates that the fixed charge en with the fixed expectation value of the dual current operator c previous examples in the literature in section with a scalar field. The finite boundary or counter terms are ex Q The expectation value of the dual scalar actually depends on First, the expectation value of a scalar is not only a functio The on-shell action and the Grand potential have been evalua We examine the program with three EMD solutions with asympto • • • imposing the differential form of the first law of thermodynam agrees with the ADM mass. They are functions of the two parameters consistent variational problem for are similar to those ofexample the Fermi in liquid states section in section ary. Depending on theboundary particular variational problem solutions, and we the demonstrate on-shell action. Let u section The two terms are directly related to the chemical potential density is shared by the example given in section This solution is realized with the faster falloff of the scala paper. This example demonstrates several important results we hav as in ( non-trivial boundary profile of JHEP11(2016)044 , 5 (6.7) , ] in the ommons 9 ]. SPIRE ]. ) IN [ 4.42 , ndition, the expectation SPIRE ]. Limitations on the IN value, further information 4341. Robert Mann and Ioannis rved the expectation value backgrounds with different in ( lue of the current operator. redited. ne successfully in [ φ e a mixed boundary condition ]. , c SPIRE ave been considered previously er space with emphasis on the (1991) 2353 ]. ]. IN ilczek, . [ 2 nces and valuable comments on the 2 SPIRE Q A 6 ˜ κ φ IN SPIRE SPIRE c (1992) 3888] [ ][ IN IN 2 3 [ Charged black holes in string theory ][ r D 45 = ]. β 2 Dual dilaton dyons φ ˜ κ – 39 – c arXiv:0812.0530 (1988) 741 , SPIRE = Mod. Phys. Lett. . i IN , [ Black holes and membranes in higher dimensional theories arXiv:0911.2898 =0 Black holes as elementary particles 5.1 [ α hep-th/9202014 B 298 [ Erratum ibid. [ ), which permits any use, distribution and reproduction in Peculiar properties of a charged dilatonic black hole in AdS hO (1991) 2677 (1992) 447 Nucl. Phys. 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