Classical and Quantum Black Holes Le Studium, Tours, May 30-31, 2016 Black hole thermodynamics from a variational principle: Asymptotically conical background

Mirjam Cvetič

Ok Song An, M.C., Ioannis Papadimitriou, 1602.0150, JHEP03 (2016)086 Asymptotically flat or anti-deSitter (cosmological constant Λ < 0) spacetimes have been studied extensively, but new asymptotic backgrounds, asymptotically supported by matter fields, attracted attention only relatively recently.

Examples: ii. Flux vacua of string compactification, dual to certain supersymmetric quantum field theories (Klebanov-Strassler) ii. Lifshitz black holes, dual to certain non-relativistic quantum mechanical condensed systems iii. `Subtracted geometry’, proposed to addressed microscopics of certain non-extremal black holes.

Understanding the mesoscopic (geometric) properties of black holes with such exotic asymptotics à a first step towards their microscopic description. While black hole entropy & temperature is not sensitive to the asymptotics, as they are intrinsically connected with the horizon, conserved charges and the free energy depend on the spacetime asymptotics and receive contribution from asymptotic matter fields.

E.g., large distance divergences, plaguing conserved charges and free energy may not be always remedied by, say, Komar integral or background subtraction.

Main motivation: formulating well posed variational problem for new asymptotic geometries (following lessons from AdS geometries Heningson,Skenderis’98; Balasubramanian,Kraus’99; deBoer,Verlinde 2 ’99, … ) : to determine conserved charges & black hole thermodynamics.

Approach general (does not rely on the specific theory or its asymptotic solutions), but apply to concrete example of subtracted geometry. Summay: variational problem for new asymptotic geometries

Achieved through an algorithmic procedure:

• integration constants, parameterizing solutions of the eqs. of motion, separated into `normalizable’ - free to vary & ‘non-normalizable’ modes – fixed

• non-normalizable modes – fixed only up to transformations induced by local symmetries of the bulk theory (radial diffeomorphisms & gauge transformations)

• covariant boundary term, Sct, to the bulk action - determined by solving asymptotically the radial Hamilton-Jacobi eqn. à total action S+Sct independent of the radial coordinate

Skenderis,Papadimitriou’04,Papadimitriou’05 • first class constraints of Hamiltonian formal. lead to conserved charges associated with Killing vectors. [Sct ‘renormalizes’ the phase space: charges are independent of the radial coordinate.]

• Conserved charge automatically satisfy the first law of thermodynamics Outline:

I. Microscopics of Black Hole Entropy & String Theory

II. Motivation and review of subtracted geometry: Classical geometry insights Summary of quantum aspects

III. Thermodynamics: via variational principle Key Issue in Black Hole Physics:

How to relate Bekenstein-Hawking - thermodynamic entropy: Sthermo=¼ Ahor (Ahor= area of the black hole horizon; c=ħ=1;GN=1) to Statistical entropy: Sstat = log Ni ?

Where do black hole microscopic degrees Ni come from?

Horizon

Space-time singularity In String theory, main progress on microscopic origin of entropy primarily for extreme black holes w/

M = Σ |Q | + Σ | P | (schematic) i i i i M-mass, Qi-electric charges, Pi-magnetic charges Extreme à primarily supersymmetric (BPS) multi-charged black holes, J=0 & w/ Λ=0. Maldacena’97

Systematic study of microscopic degrees quantified via: AdSD/CFTD-1 (Gravity/Field Theory) correspondence

[ A string theory on a specific Curved Space-Time (in D-dim.) related to specific Field Theory (in (D-1)-dim.) on its boundary]

Specific microscopic examples of black holes in string theory: relation to 2-dim. CFT via AdS3/CFT2 correspondence extensively explored:

- supersymmetric (BPS) multi-charged black holes w/ Λ=0:

M = Σ |Q | + Σ | P | J=0 Strominger, Vafa’96.. i i i i (prototype: four-charge BPS dyon M.C., Youm 0507090) - near-BPS l black holes: M ~ Σ |Q | + Σ | P | Maldacena, Strominger’97.. i i i i - near-BPS multi-charged rotating black holes M.C., Larsen’98

Further developments: - (near-)extreme rotating black holes 2 M > Σi |Qi | + Σi| Pi|, but M =J Kerr/CFT correspondence Guica,Hartman,Song, Strominger’08 - extreme AdS charged rotating black holes in diverse dim. ….M.C., Chow, Lü, Pope 0812.2918 How about Microscopics of Non-extreme Black Holes?

M > Σi |Qi | + Σi| Pi |

Motivation for Subtracting Geometry Subtracted Geometry - Motivation

Introduced to quantify `past observations’ (later) M.C., Larsen ‘97-’99 that not only extreme, but also non-extreme black holes might have microscopic explanation in terms of dual 2-dim. QFT: focus on non-extreme asymptotically flat (Λ=0), charged ,rotating black holes in four (and five) dimensions.

Should focus on the black hole `by itself’ à enclose the black hole in a box (à la Gibbons Hawking), thus creating an equilibrium system with manifest conformal symmetry (later)

The box leads to a `mildly’ modified geometry changing only the warp factor Δ0à Δ

Subtracted Geometry M.C., Larsen 1106.3341,1112.4856

Non-extreme black holes in string theory w/ Λ=0

M - mass, Qi, Pi - multi-charges, J - angular momentum w/ M > Σi |Qi| + Σi| Pi |

Prototype solutions of a sector of maximally supersymmetric D=4 Supergravity [effective action of toroidally compactified string theory] à so-called STU model

2TheSTUmodelanddualityframes 2TheSTUmodelanddualityframes In this section we review the bosonic sector of the 2-charge truncation of the STU model that is In this section we review the bosonic sector of the 2-charge truncation of the STU model that is relevant for describing the subtracted geometries. We will do so in the duality frame discussed in relevant for describing the subtracted geometries. We will do so in the2TheSTUmodelanddualityframes duality frame discussed in [14], where both charges are electric, as well as in the one used in [17],[14], where where there both is one charges electric are and electric, as well as in the one used in [17], where there is one electric and one magnetic charge. We will refer to these frames as ‘electric’ andone ‘magnetic’In magnetic this section respectively. charge. we review We As will it the refer bosonic to these sector frames of the 2-chargeas ‘electr tic’runcation and ‘magnetic’ of the STU respectively. model that As is it will become clear from the subsequent analysis, in order to comparewill therelevant become thermodynamics for clear describing from in the the twothe subsequent subtracted analysis, geometries. in order We will to cdoompare so in the the duality thermodynamics frame discussed in the in two frames, it is necessary to keep track of boundary terms introducedframes, by[14], the duality it where is necessary transformations. both charges to keep are electric, track of as boundary well as in the terms one intro usedduced in [17], by where the there duality is one transformations. electric and one magnetic charge. We will refer to these frames as ‘electric’ and ‘magnetic’ respectively. As it 2.1 Magnetic frame 2.1will Magnetic become clear frame from the subsequent analysis, in order to compare the thermodynamics in the two frames, it is necessary to keep track of boundary terms introduced by the duality transformations. The bosonic Lagrangian of the STU model in the duality frame usedThe in bosonic [17] is given Lagrangian by of the STU model in the duality frame used in [17] is given by 1 1 2 2ηa a a 2.1 Magnetic frame 1 1 2κ4 4 = R ⋆ 1 ⋆ dηa dηa e ⋆ dχ dχ 2 κ STU2 = RModel⋆ 1 Lagrangian⋆ dη dη e 2ηa ⋆ dχa dχa L − 2 ∧ − 2 ∧ 4L4 − 2 a ∧ a − 2 ∧ 1 η0 0 0 1 2ηa η0 a a 0 Thea bosonica 0 [A Lagrangian sector of toroidally of the STU compactified model in the effective duality string frame theory] used in [17] is given by e− ⋆ F F e − ⋆ (F + χ F ) (F + χ F ) 1 η0 0 0 1 2ηa η0 a a 0 a a 0 − 2 ∧ − 2 ∧ e− ⋆ F F e − ⋆ (F + χ F ) (F + χ F ) 2 − 2 1 ∧ − 2 1 2ηa a a ∧ 1 a b c 1 a b 0 c 1 a b c 0 2κ0 4 4 = R ⋆ 1 ⋆ dηa dηa e ⋆ dχ dχ + Cabcχ F F + Cabcχ χ F F + Cabcχ χ χ F F , L 1(2.1)− 2a b ∧c 1− 2 a b 0 ∧ c 1 a b c 0 0 2 ∧ 2 ∧ 6 ∧ + Cabcχ F F + Cabcχ χ F F + Cabcχ χ χ F F , (2.1) 1 η0 0 0 1 2ηa η0 a a 0 a a 0 3 2 e− ⋆ F ∧ F 2 e − ⋆ (F ∧+ χ F 6) (F + χ F ) ∧ where ηa (a =1, 2, 3) are dilaton fields and η0 = a=1 ηa.ThesymbolCabc is pairwise symmetric− 2 ∧ − 2 3 ∧ with C =1andzerootherwise.TheKaluza-Kleinansatzforobtaininwhere ηagthisactionfromthe(a =1, 2, 3) are dilaton1 fields and η10 = a=1 ηa.Thesymbol1 Cabc is pairwise symmetric 123 ! + C χaF b F c + C χaχbF 0 F c + C χaχbχcF 0 F 0, (2.1) 6-dimensional action (1.1) is given explicitly in [17]. Thisframepossessesanexplicittrialitysym-with C123 =1andzerootherwise.TheKaluza-Kleinansatzforobtaininabc abc abc gthisactionfromthe 2 ∧ 2 ! ∧ 6 ∧ metry, exchanging the three gauge fields Aa,thethreedilatonsηa 6-dimensionaland the three axions actionχa.Inthis (1.1) is given explicitly in [17]. Thisframepossessesanexplicittrialitysym- where η (a =1, 2, 3) are dilaton fields and η = 3 η .ThesymbolC is pairwise symmetric frame, the subtracted geometries source all three gauge fields Aa magnetically,metry, exchanginga whilew/ A 0 the &is electri- three gauge fields Aa,thethreedilatons0 a=1 a ηa and theabc three axions χa.Inthis with C =1andzerootherwise.TheKaluza-Kleinansatzforobtainingthisactionfromthe cally sourced. Moreover, holographic renormalization turns out toframe, be much the more123 subtracted straightforward geometries source all three! gauge fields Aa magnetically, while A0 is electri- 6-dimensional action (1.1) is given explicitly in [17]. Thisframepossessesanexplicittrialitysym- in this frame compared with the electric frame. cally sourced. Black Moreover, holes: holographic explicit solutions renormalization of equations turns of out motion to be for much the above more straightforward metry, exchanging Lagrangian the three w/ metric, gauge fields fourA gaugea,thethreedilatons potentials andηa threeand the axio-dilatons three axions χ a.Inthis In order to describe the subtracted geometries it suffices toin consider this frame a truncation compared of (2.1), with the electric frame. a 0 frame,1 the2 subtracted3 geometries source all three gauge fields A magnetically, while A is electri- corresponding to setting η1 = η2 = η3 η, χ1 = χ2 = χ3 χ,andA = A = A A.The ≡ ≡ Incally order sourced. to Prototype, describe Moreover,≡ the four-charge holographic subtracted renormalization rotating geometries black it tur su holensffices out, originally to to cbeonsider much obtained more a truncation straightforward via of (2.1), resulting action can be written in the σ-model form 1 2 3 correspondingin this frame solution to compared setting generatingη with1 = theη2 = electrictechniquesη3 frame.η, χ 1 = χ 2 = χ 3 χ ,and M.C.,A Youm= A 9603147= A A.The 1 1 ≡ Chong,≡ M.C., Lü, Pope 0411045 ≡ S = d4x√ g R[g] ∂ ϕI ∂µϕJ F Λ F Σµν resultingϵµνρσInF action orderΛ F Σ to can+ describeS be, written(2.2) the subtracted in the σ-model geometries form it suffices to consider a truncation of (2.1), 4 2 IJ µ ΛΣ µν ΛΣ µν ρσ Four- GHSO(1,1) transfs. cosh i sinh i 2κ4 − − 2G − Z − R H = , 1 2 (34)3 "M # corresponding1 $ to setting η1 = 1η2 = η3 sinhη, χ1cosh= χ2 = χ3 χ,andA = A = A A.The S = dtime-reduced4x√ g R [Kerrg] BH ∂ ✓ϕ≡I ∂µϕi J i ◆F Λ F Σ≡µν ϵµνρσF Λ F Σ +≡S , (2.2) where 4resulting2 action can be written inIJ theµσ-model formΛΣ µν ΛΣ µν ρσ GH 2correspondsκ4 to: − − 2G − Z − R 1 3 "M # $ SGH = d x√ γ 2K, Full four-electric(2.3) and four-magnetic charge solution only recently obtained 2 1 4 2 1 I2 µ J Λ Σµν µνρσ Λ Σ 2κ4 ∂ − where x p 1+ ↵ x 0 1 p 1+ ↵ x Chow, Compère 1310.1295;1404.2602 " M S4 = 2y 0 =d y,√ g e R[g] = ⇤IJe ∂µϕ ∂, ϕ ΛΣFµν F ΛΣϵ Fµν Fρσ + SGH, (2.2) 2κ − − 2G 1 − Z3 − R is the standard Gibbons-Hawking [36] term and we have defined the doublets 4 1 2p1+↵2x 2 2 x√ 2 2 2p1+↵2x 0"M= ⇤ [ +# (eSGH = (12 + ↵ ) d)] ; ⇤ =(γ2 K,+ 1) e . (35)$ (2.3) where p1+↵2 2κ4 ∂ − η A0 " M ϕI = ,AΛ = ,I=1, 2, isΛ the=1 standard, 2, Gibbons-Hawking(2.4) [36] term1 and we have3 defined the doublets χ A Note, this transformation can also beSGH determined= 2 as and analyticx√ γ continuation2K, of transformations (2.3) # $ # $ given in Section 2 of [19]. A Harrison transformation2κ4 ∂ in the limit− of an infinite boost corresponds η A" 0M as well as the 2 2matrices is theto standard 1 5. In Gibbons-Hawking theϕ caseI = of ↵ =,A1 [36], we term shallΛ = act and with we,I (33) have on defined the=1 Schwarzschild, 2,theΛ doublets=1 solution, 2, with (2.4) × ! χ p3 A 30 1 e 3η +3e ηχ2 3e ηχ e12U =1χ3 23mχ,2 = 0, = 0. The transformation (34) with =1resultsin⇤ = 2m , and the − − − 2r # $ # 0$ r IJ = 2η , ΛΣ = η η , ΛΣ =metric3 (6)2 with the. subtracted(2.5)I η geometry warpΛ factor:A G 03e Z 4 3e− χ 3e− asR well as4 theχ 2 3χ2matricesϕ = ,A= ,I=1, 2, Λ =1, 2, (2.4) # $ # $ # 2 × $ χ A # $ 3η4 # η $2 3 η 3 3 2 30 1 se0 = r +3es =(2χ m)3r,e χ 1 χ(36) χ As usual, ϵµνρσ = √ g εµνρσ denotes the totally antisymmetric Levi-Civita tensor, where εµνρσ = − ! − − 2 − asIJ well= as the2 2η ,2matricesΛΣ = η η , ΛΣ = 3 2 . (2.5) 1istheLevi-Civitasymbol.ThroughoutthispaperwechoosetheG orientation03e in × so thatZ 4 3e− χ 3e− R 4 χ 3χ ± and# the30 scalar$M field and the electric1# fielde strength3η +3e : ηχ2 3e ηχ $ 1 #χ23 3 χ2 $ εrtθφ =1.WenoteinpassingthattheLagrangian(2.2)isinvariantunder= the global symmetry, = − − − , = 2 . (2.5) As usual,IJ ϵµνρσ = √2η g εµνρσ ΛΣdenotes the totallyη antisymmetricη Levi-CivitaΛΣ tensor,3 2 where εµνρσ = transformation G 03e Z 24 2m3e− χ 2 3e−1 R 4 2 χ 3χ − p3 1istheLevi-Civitasymbol.Throughoutthispaperwechoose# $ e #= , Ftr = ,$ the orientation# (37) in$ so that η 2 η 2 0 3 0 2 2 r 3 2m e µ e , χ µ− χ,A µ A ,A µA,± As ds usual,ds ,ϵµνρσ = √ g(2.6)εµνρσ denotesr the totallyr antisymmetric Levi-Civita tensor, whereMεµνρσ = → → → → εrtθφ =1.WenoteinpassingthattheLagrangian(2.2)isinvariant→ − under the global symmetry 1istheLevi-Civitasymbol.Throughoutthispaperwechoosei.e., this is the static subtracted geometry of Subsection 2.1, with ⇧c =1, ⇧thes = orientation 0. in so that where µ is an arbitrary non-zero constant parameter. transformation± The subtracted geometry for the Kerr spacetime can be obtained by reducing the spacetime M εrtθφ =1.WenoteinpassingthattheLagrangian(2.2)isinvariantunder the global symmetry on the time-likeη Killing2 η vector and acting2 on the0 Kerr black3 0 hole with an infinite boost2 Harrison2 transformatione µ e , χ µ− χ,A µ A ,A µA, ds ds , (2.6) transformation→ for Lagrangian density→ (1), where→ we set 1 = 2 =→3 , '1 =→'2 = '3 5 η 2 η 2 0 3 0 ⌘ 2 2 ⌘ where µ2is, anF arbitrary= F e= non-zeroµ e , 2 Fχ constantandµ− χ=,A parameter.p2 , i.e.µ anA Einstein-Dilaton-Axion,A µA, ds gravityds , with (2.6) p3 ⇤ 1 2 →⇤F1 ⌘ 3 → F2 F → → → two U(1) gauge fields and respectiveq dilaton couplings ↵ = 1 and ↵ = p3. The subtracted where µ is an arbitrary non-zero constant parameter.1 p3 2 geometry of the multi-charged rotating black holes is expected to arise as a specific Harrison transformation on a rotating charged black solution5 of (1). We defer technical aspects of these calculations to follow-up work. 5

3 Asymptotically Conical Metrics 3.1 Lifshitz Scaling The scaling limit, or equivalently the subtraction process, alters the environment that our black holes find themselves in [12, 13]. In fact the asymptotic metrics take the form

R 2p ds2 = dt2 + B2dR2 + R2 d✓2 +sin2 ✓2d2) (38) R0 with B and R0 constants. In our case B = 4 and p = 3. In general metrics with asymptotic form (37) may be referred as Asymptotically Conical (AC). The spatial metric is conical because the radial distance BR is a non-trivial multiple of the area distance R. Restricted to the equatorial plane the spatial metric is that of a flat two-dimensional cone

2 2 2 2 2 dsequ = B dR + R d (39)

5One may verify that (33) with b 1intheEinstein-Maxwellgravity(↵ =0)takestheSchwarzschildmetric to the Robinson-Bertotti one. This! type of transformation was employed recently in [21]. For another work, 2 relating the Schwarzschild geometry to AdS2 S ,see[22]. ⇥

10 Metric of rotating four-charge black holes in String Theory: M.C. & Youm 9603147 Chong, M.C., Lü & Pope 0411045

= 0 outer & inner horizon

Special cases: Mass δI = δ Kerr-Newman & a = 0 Reisner-Nordström; Four charges δI = 0 Kerr & a = 0 Schwarzschild; Angular momentum δIà∞ mà0 w/m exp(2δI)-finite Or equivalently : m, a, δI (I=0,1,2,3) extremal (BPS) black hole Thermodynamics - suggestive of weakly interacting 2-dim. CFT w/ `left-’ & `right-moving’ excitations first noted, M.C., Youm’96 M.C., Larsen’97 Area of outer horizon S = S + S 2 2 + L R SL =S⇡L m= ⇡(⇧mc +(⇧⇧cs+)(1)⇧s)(1) [Area of inner horizon S = S – S ] S p⇡ m2pm22 a2 - L R SR = ⇡R m= m 2 a2 (⇧c (⇧⇧cs)(2)⇧s)(2) SL S=L ⇡=m⇡m(⇧(c⇧+c +⇧s⇧)(1)s)(1) L =2L⇡=2m (⇡⇧mc (⇧⇧cs)(3)⇧s)(3) 2 2 2 2 Surface gravity (inverse temperature) ofS S=R ⇡=m⇡pmmpm a a(⇧(⇧c ⇧ ⇧)(2)s)(2) R 2⇡ m22⇡ m2 c s = (⇧ + ⇧)(4) R R = 2 2 c (⇧cs+ ⇧s)(4) outer horizon βH = ½ (βL + βR ) Lp=2mL =2p⇡mm⇡a2(m⇧(ca⇧2 c ⇧s⇧)(3)s)(3) ⇧ = ⇧ =03 cosh3 ,Pi2 2 = ⇧ =03 sinh3 (5) c ⇧c I= ⇧I =0 cosh2I⇡2m⇡Im,Pis sI= ⇧I =0 sinhI I (5) [inner horizon β- = ½ (βL - βR ) ] R =R = (⇧(c⇧+c +⇧s⇧)(4)s)(4) HW 7, Phys 151 pmp2m2 a2 a2 HW 7, Phys 151 3 3 3 3 Similar structure for angular ⇧velocities=⇧c⇧= ⇧=0ΩI, =0 Ωcosh andcosh momenta,PiI ,Pi J=,s J=⇧. ⇧ =0I =0sinhsinh I (5)(5) Ch. 29: c I + - I s + - I I Ch. 29: Exercises:HW 7, 2,6,11,13,16,20,23,37 Phys 151 DependHW onlyExercises: 7,on four Phys parameters: 151 2,6,11,13,16,20,23,37 m, a, Problems: 2,7 Problems: 2,7 Ch. 29: Shown moreCh. recently, 29: all independent of the warp factor Δ ! HWExercises: due on Wednesday, 2,6,11,13,16,20,23,37 October 22nd. o Exercises:HW due 2,6,11,13,16,20,23,37 on Wednesday, October 22nd.M.C., Larsen’11 Problems: 2,7 Problems:Reminder: 2,7Next Midterm is on Thursday, October 23rd, 11 AM in A1. Reminder: Next Midterm is on Thursday, October 23rd, 11 AM in A1. HW due on Wednesday, October 22nd. HWHW due and on other Wednesday, information October available 22nd. on: http://www.physics.upenn.edu/phys151.htlmHW and other information available on: Reminder:(orReminder: equivalentlyhttp://www.physics.upenn.edu/phys151.htlmNextNext at Midterm http://www.physics.upenn.edu/ Midterm is on is on Thursday, Thursday, October October cvetic) 23rd, 23rd, 11 11 AM AM in in A1. A1. (or equivalently at http://www.physics.upenn.edu/ cvetic) HWHW and and other other information information available available on: on: http://www.physics.upenn.edu/phys151.htlmhttp://www.physics.upenn.edu/phys151.htlm (or(or equivalently equivalently at http://www.physics.upenn.edu/ at http://www.physics.upenn.edu/ cvetic) cvetic) Determination of new warp factor Δ0 à Δ via massless scalar field wave eq.: wave eq. separable & the radial part is solved by hypergeometric functions w/ SL(2,R)2 (manifest conformal symmetryà promoted to 2-dim. CFT)

(with warp factor Δ0 implicit):

w/

Δ à Δ such that wave eq. is separable: f(r)+g( θ ) (true for Δ and Δ) 0 0

& the radial part is solved by hypergeometric functions:

uniquely fixed f(r)+g(θ)-const.

M.C., Larsen 1112.4856 Subtracted geometry for rotating four-charge black holes

4 Comments: while Δ0 ~ r , Δ ~ r subtracted geometry depends only on four parameters: m, a, Remarks:

Asymptotic geometry of subtracted geometry is of a Lifshitz-type w/ a deficit angle:

p=3, B=4

à black hole in an ``asymptotically conical box’’ (w/ deficit angle) M.C., Gibbons 1201.0601

à the box is confining (``softer’’ than AdS) 2TheSTUmodelanddualityframes

In this section we review the bosonic sector of the 2-charge truncation of the STU model that is relevant for describing the subtracted geometries. We will do so in the duality frame discussed in [14], where both charges are electric, as well as in the one used in [17], where there is one electric and one magnetic charge. We will refer to these frames as ‘electric’ and ‘magnetic’ respectively. As it will become clear from the subsequent analysis, in order to compare the thermodynamics in the two 2TheSTUmodelanddualityframesframes, it is necessary to keep track of boundary terms introduced by the duality transformations.

In this section we review the bosonic sector of the 2-charge truncation of the STU model that is 2.1 Magnetic frame relevant for describing the subtracted geometries. We will do so in the duality frame discussed in [14], where both charges are electric,The as well bosonic as in the Lagrangian one used in of [17], the where STU there model is onein the electric duality and frame used in [17] is given by one magnetic charge. We will refer to these frames as ‘electric’ and ‘magnetic’ respectively. As it 2 1 1 2ηa a a will become clear from the subsequent analysis, in2κ order4 4 = toR compare⋆ 1 the⋆ d thermodynamicsηa dηa e in⋆ thedχ two dχ frames, it is necessary to keep track of boundary termsL introduced− 2 by the duality∧ − transformations.2 ∧ 1 η0 0 0 1 2ηa η0 a a 0 a a 0 e− ⋆ F F e − ⋆ (F + χ F ) (F + χ F ) 2.1 Magnetic frame − 2 ∧ − 2 ∧ 1 1 1 + C χaF b F c + C χaχbF 0 F c + C χaχbχcF 0 F 0, (2.1) The bosonic Lagrangian of the STU model in the duality frame2 uabcsed in [17]∧ is given2 byabc ∧ 6 abc ∧ 1 1 3 2 where ηa (a2=1ηa , 2,a3) area dilaton fields and η0 = ηa.ThesymbolCabc is pairwise symmetric 2κ4 4 = R ⋆ 1 ⋆ dηa dηa e ⋆ dχ dχ a=1 L − 2 with∧ C−1232 =1andzerootherwise.TheKaluza-Kleinansatzforobtainin∧ gthisactionfromthe 1 1 ! η0 0 0 2ηa η0 3Asymptoticallyconicalbackgroundsa a 0 a a 0 e− ⋆ F 6-dimensionalF e − action⋆ (F + (1.1)3Asymptoticallyconicalbackgroundsχ F is) given(F + explicitlyχ F ) in [17]. Thisframepossessesanexplicittrialitysym- − 2 metry,∧ − exchanging2 the three gauge∧ fields Aa,thethreedilatonsηa and the three axions χa.Inthis 1 a b c 1 a b 0The generalc 1 rotatinga b subtractedc 0 0 geometry backgroundsa are solutions of the equations0 of motion + Cabcχ F frame,F + theCabc subtractedχ χ F F geometries+ Cabcχ sourceχ χ F allF three, gauge(2.1) fields A magnetically, while A is electri- 2 ∧ 2 following∧ The from6 general the action rotating (2.2)∧ subtractedor (2.10) and take geometry the form backgrounds [14, 17]5 are solutions of the equations of motion cally sourced. Moreover, holographic renormalization turns out toM.C., be Gibbons much 1201.0601 more straightforward 3 following from the action (2.2) or (2.10) and take the form [14, 17]5 where ηa (a =1, 2, 3) are dilaton fields and η0 = Matterηa.Thesymbol fields (gaugeCabc is potentials pairwise symmetric and scalars) in this frame compareda=1 2 with√∆ the2 electricG frame.2 2 X 2 2 with C =1andzerootherwise.TheKaluza-Kleinansatzforobtaininds = dr¯ gthisactionfromthe(dt¯+ ) + √∆ dθ + sin θdφ¯ , 123 In order to3Asymptoticallyconicalbackgrounds describe theX subtracted− √∆ geometriesA it suffiGces to consider a truncation of (2.1), ! √∆ G ! " X 6-dimensional action (1.1) is given explicitly in [17]. Thisframepossessesanexplicittrialitysym-2 2 2 2 2 1 2 2 ¯2 3 corresponding to Scalars: settingη dsη(21 m==) η2 =drη¯3 a (Πηc, χ(Π1dst¯=) +χ2 )=+χ3√∆χ,anddθ +A =sinAθd=φ A , A.The metry, exchanging the three gauge fields Aa,thethreedilatonsThe generale = rotatingηa and, X the subtractedχ three= −≡ axions√ geometry∆− χa.Inthiscos backgroundsAθ, ≡ are solutionsG of the equations≡ of motion resulting action can be written√∆ in the σ-model2m form 5 frame, the subtracted geometries source all three gaugefollowing fields fromAa themagnetically, action (2.2) while or (2.10)A0 andis electri- take the form! [14, 17] " 4 (2m)2 a (Π 2 Π 2) 2 4 0 (2ηm) a (Πc Πs) 2 ¯ (2mac ) coss θ (Πc Πs) +(2m) ΠcΠs ¯ cally sourced. Moreover, holographic renormalization1 4 turnsA out=e to= be1 much− , moreIsin straightforwardµχθd=Jφ + −Λ Σcosµν 2 θ−, 2 µνρσ Λ Σdt, S4 = d x√ √g ∆R[g] G√∆ IJ∂µϕ ∂ ϕ ΛΣ2mFX F (Πc ΠsΛΣ) ∆ϵ F F + SGH, (2.2) in this frame compared with the electric frame.2 ds2 = dr¯2 (∆dt¯+ )2 + √∆ dθ2 + µνsin2 θdφ¯2− , µν ρσ 2κ4 − 2m cos− θ2G − Z − R "M X# − √∆ 4 A 2 2 G2 ¯ 2 2 $ 2 ¯ 4 In order to describe the subtracted geometries it suffiAces= to consider(2m∆) aa (2( truncationΠma) (Πc !) ofΠs (2.1),) sin θ dφ(2ma2"ma) (2cosmΠsθ+¯(rΠ(Πc ΠΠs))) d+(2t , m) Π Π where 20 ∆ − c s− 2 ¯ − c − s c s ¯ Gaugeη (2 potentials:mA) = 1 a (Πc 2 Π−s) 3 sin θdφ + − dt, corresponding to setting η1 = η2 = η3 η, χ1 = χ2e==χ3 χ,,and1 χA#$= = A∆(2−ma=1)A2(cosΠ θ,AΠ.The) % (2ma)2(Π Π )[2(mΠΠ2 +¯Πr(2Π) ∆Π& )] cos2 θ ≡ A√≡=∆ r¯ Sm =2m ≡c −d3sx√dt¯γ+ 2K, c − s c s s c − s (2.3)dt¯ resulting action can be written in the σ-model form 2m GH (2m)23(Π + Π ) 2m∆− −4 2m−cos−θ 2κ4 ∂c 2s 2− 2 4 (2m) a (Πc! Πs) "(2ma) cos2" θ (Πc Πs2) +(22 m) Π¯cΠs ¯ 1 1 0 A = 2 ∆¯ (2Mma) (Πc Π(2s)masin)2(Πθ dΠφ)2 cos2¯ma2 θ (2mΠs +¯r(Πc Πs)) dt , 4 I µ J A Λ= 'Σµν − µνρσsinΛ θdΣφ + 2 2 − 2 c s dt, ¯ S4 = d x√ g R[g] isIJ the∂µϕ standard∂ ϕ ΛΣ Gibbons-HawkingF F ∆ ΛΣϵ [36]∆ F termF+ a and(−Π+c S weGHΠs, have)sin(2.2)θ defined(Π−1+ Π the) ∆ doublets− − dφ, (3.1)− 2 µν µν ρσ − c − s ∆ 2κ4 − − 2G − Z − R 2 ! 2 " 2 "M # 2m cos θ 1 2#$ $ 0(22ma2) (Π¯c Πs) % (2ma) (Πc Πs)[2mΠs +¯r(Πc Π&s)] cos θ A = I A =η ∆ (2mar¯)Λ(Πmc AΠs) sin θ dφ 2ma (2mΠdt¯s ++¯r(Πc Πs)) dt¯ , dt¯ where whereϕ ∆= ,A= ,I3 =1−, 2, Λ =1, 2, − (2.4) − χ −−2m − −−A (2m) (Πc −+ Πs) − 2m∆ 1 3 ! 2 2 " 2 SGH = d x√ γ 2K,1 ##$$ (2ma) (Πc# Πs$) (2.3)% (2ma) (Πc Πs)[2mΠs +¯r(Πc 2 Π&s)] cos θ 2 2 2 A = 2r¯ m 2 − d2t¯+ 2 −2ma Magnetic(2 maframe)− (Π Π2 )dt¯¯cos θ as well2κ as4 the∂ 2 2matrices− −X2m=¯'r −2m−r¯ +(2am,G)3(Π +=ΠX) a sin θ, = 2 ((Π2mc ∆Πs)¯r +2mΠcs)sin sθdφ, ¯ " M × ! − c s −"+ a(Πc ΠsA)sinGθ 1+− − dφ, (3.1) is the standard Gibbons-Hawking [36] term and we have defined the3 doublets2 23η 4η 22 η 2−(2ma)2(Π2 Π2 )2 cos2 θ 3 3 2 ∆ 30' w/ ∆ =(2m) (Π1c eΠ−s)¯r +(2+3me−) Πχs 2 (23ema− )χ(Πc Πs) c cos sθ, 1 χ ¯ χ (3.2) − + a(Πc Πs)sin−θ 1+ − − ! dφ,2 (3.1) " IJ = 0 2η , ΛΣ = −η η , ∆ΛΣ = 3 2 . (2.5) I η G Λ A03e Z 4 3e χ 3e R 4 χ 3χ ϕ = ,A= ,Iand=1whereΠ ,, 2Π, , aΛand=1m, 2are, parameters− of the(2.4)! solution.− 2 " χ #A $ c s # $ # $ where In order to study the thermodynamics of these backgrounds it is necessary to identify which # $ As usual,#ϵµνρσ$ = √ g εµνρσ denotes the totally antisymmetric Levi-Civita tensor,2 whema re εµνρσ = as well as the 2 2matrices parameters− are fixed2 by the boundary2 conditions in the2 variat2 ional problem. A full analysis of the 2 ¯ 1istheLevi-Civitasymbol.Throughoutthispaperwechoose2 X =¯r 2 2mr¯ + a ,G2 2 = X a2masintheθ, orientation= in ((2Πsoc¯ thatΠs)¯r +2mΠs)sin θdφ, × Xvariational=¯r 2m problemr¯ + a ,G for the= actionsX a (2.2)sin θ or, (2.10)= require((sknowledgeofthegeneralasymptoticΠc Πs)¯r +2mΠGs)sin θdφ, ± 3η η 2 η − 3 3 2 − G A M − 30 1εrteθφ− =1.WenoteinpassingthattheLagrangian(2.2)isinvariant+3e− χ 3e− χ − 13 χ2 2−χ2 4 A2 under2− the global2 2 symmetry IJ = 2η , ΛΣ = η solutionsη , and3∆ =(22 isΛΣ beyond=m2 ) ( theΠ3 2 scope4 Π2 of)¯r the.+(2(2.5) present2m) Π paper.2 (2 Howevma2 ) er,(Π wec canΠs consider) cos theθ, variational (3.2) G 03e Z 4transformation3e− χ 3∆e−=(2m) (ΠRc Πs)¯r4+(2χcm) Π3sχs (2ma) (Πc Πs s) cos θ, (3.2) # $ # problem$ within− the class# of2 stationary− −$ solutions (3.1).− − To this end, it is− convenient to reparameterize η 2 η 2 0 3 0 2 2 As usual, ϵµνρσ = √ g εµνρσ denotes the totally antisymmetricand Πthese, Π backgrounds, a and Levi-Civitam are by parameters means tensor, of whe a of suitablere theεµ solution.νρσ coordinate= transformation, accompanied by a relabeling − e c andµs e Π, c, χΠs, aµ−andχ,Am are parametersµ A ,A of theµA, solution. ds ds , (2.6) 1istheLevi-Civitasymbol.ThroughoutthispaperwechooseInof→ order the free to study parameters.the→ the orientation thermodynamics In particular, in → weso of introduce thatthese backgrounds→ the new coordinates it is→ necessary to identify which ± where µ is an arbitrary non-zeroIn order constant to study parameter. theM thermodynamics of these backgrounds it is necessary to identify which εrtθφ =1.WenoteinpassingthattheLagrangian(2.2)isinvariantparameters are fixed byunder the4 boundary the global3 conditions symmetry2 2 in the variat4 2 ional problem.2 A2 full analysis of the parameters areℓ r fixed=(2m by) (Π thec boundaryΠs)¯r +(2m) conditionsΠs (2ma) ( inΠc theΠs variat) , ional problem. A full analysis of the transformation variational problem for the actions (2.2)− or (2.10) requiresknowledgeofthegeneralasymptotic− − k 1 2ma(Π Π ) solutionsvariational and is beyond problem the scope for of the the present actions¯ paper. (2.2) Howev¯ orer, (2.10) we canc require considers ¯ sknowledgeofthegeneralasymptotic the variational η 2 η 2 0 3 0 2 3 t =2 3 2 2 t, φ = φ 3 2− 2 t, (3.3) e µ e , χ µ− χ,A µ A ,A µA, ds ℓ ds ,(2m) (Π5c Π(2.6)s) − (2m) (Πc Πs) → → → problem withinsolutions→ the class and of→ is stationary beyond solutions the− scope (3.1). of To the this present end, it is paper.convenient− Howev to reparameterizeer, we can consider the variational where µ is an arbitrary non-zero constant parameter.thesewhere backgroundsproblemℓ and k byare within means additional the of a class suitablenon-zero of stationary coordinate parameters, transfo whosesolutionsrmation, role will (3.1). accompanied become To tclearhis byend, shortly. a relabeling it Moreover,is convenient to reparameterize of thewe free definethese parameters. the backgrounds new parameters In particular, by means we introduce of a suitable the new c coordinateoordinates transformation, accompanied by a relabeling 4 3 2 2 4 2 2 2 of the free4 ℓ parameters.r =(23m) (2Π In2Π particular,)¯r +(2m2 ) Π we(2 introduce2ma) (Πc2 theΠ2s) , new3 c2oordinates2 5 ℓ r =(2m) m(Πc +c Πs) s (2ma) (Πc s Πs) m a (2m) (Πc Πs), ± − − −− ± −− − 3 k 1 2ma(Πc Πs) ℓ ω =2t ma= (Πc Πs4),Bt,¯ =23m,φ =2 φ¯ 2 ( 4 t,¯2 2 (3.3)(3.4)2 3 −3 2 2 3 2− 2 ℓ (2m) (Πℓc r =(2Πs) m) (Πc −Π(2s)¯mr)+(2(Πc mΠ)s)Πs (2ma) (Πc Πs) , 5 − − − − − In order to compare this backgroundk with the expressions1 given in eqs. (24) and (25) of [14],2ma one( shouldΠ takeΠ into) whereaccountℓ and thek are field additional redefinition non-zeroA →−A, χ parameters,→−χ,beforethedualizationof whose role will¯ becomeA, as mentionedclear¯ shortly. in footnote Moreover,c 4, and adds a¯ 3 t = 3 2 2 t, φ = φ 2 2 ¯ 3 2− 2 t, (3.3) we defineconstant the pure new gauge parameters term. Moreover,ℓ there is(2 am typo) in(Π eq. (25Π)of[14]:theterm2) mΠs−cos(2θdmt should) (Π be replacedΠ ) − 2 ¯ c s c s by 2mΠs(Πs Πc)cos θdt. − − 4 3 2 2 2 2 2 2 3 2 2 whereℓ r =(2ℓ andm) km(areΠc + additionalΠs) (2ma) non-zero(Πc Πs) parameters,m a (2 whosem) (Πc roleΠs will), become clear shortly. Moreover, ± − − ± − − 3 weℓ defineω =2ma the(Πc newΠs parameters),B=2m, ( (3.4) − 5In order to compare this background with the expressions given in eqs. (24) and (25) of [14], one should take into 4 3 2 27 2 2 2 2 3 2 2 account the field redefinitionℓ Ar →−=(2A, χm→−) mχ,beforethedualizationof(Πc + Πs) (2maA), as(Π mentionedc Πs) in footnotem 4, anda add(2 am) (Πc Πs), ± 2 2 constant pure gauge term. Moreover, there is a typo in eq. (25)of[14]:theterm2− m−Πs cos θd±t¯ should be− replaced − 2 3 by 2mΠs(Πs − Πc)cos θdt¯. ℓ ω =2ma(Πc Πs),B=2m, ( (3.4) − 5In order to compare this background with the expressions given in eqs. (24) and (25) of [14], one should take into account the field redefinition A →−A, χ →−χ,beforethedualizationofA, as mentioned in footnote 4, and add a 2 2 constant pure gauge term. Moreover, there is a typo in eq. (25)of[14]:theterm2mΠs cos θdt¯ should be replaced 2 by 2mΠs(Πs − Πc)cos θdt¯. 7

7 The quantities β , β and β are removed by a gauge transformation φ φ+(β +β +β )t. 1 2 3 −→ 1 2 3 We shall assume from now on that this transformation has been performed. The final metric The quantities β , β and β are removed by a gauge transformation φ φ+(β +β +β )t. can be cast in the1 following2 3 form: −→ 1 2 3 We shall assume from now on that this transformation has been performed. The final metric X F 2dr2 F 2 sin2 θ 2 √ 2 √ 2 2 can be castds in= the∆ following2 dt form:+ 2 + ∆dθ + (dφ + Wdt) , (6.22) F !− X " √∆ X F 2dr2 F 2 sin2 θ where 2 √ 2 √ 2 2 ds = ∆ 2 dt + 2 + ∆dθ + (dφ + Wdt) , (6.22) F !− X " √∆ where X = r2 2mr , − 2 2 3 2 2 2 F X=(2= rm) 2mr(Πc , Πs)r +(2m)Πs , − − 4 2 16m3#ΠsΠ2 cβ4 2 2 $ WF =(2m) (Πc Πs,)r +(2m)Πs , = 2 − − 16m#4FΠ Π β $ W 2 s 6 c 24 , 2 2 ∆ ==F +(2m2) β (Πc Πs) sin θ . (6.23) − F 4 − 2 6 2 2 2 ∆ = F +(2m) β4 (Πc Πs) sin θ . (6.23) The dilation fields are of the form: − The dilation fields are of the form: Q eφ1 eϕ2 eϕ3 , = = = Q (6.24) eφ1 = eϕ2 = eϕ3 = √∆, (6.24) √∆ and the axion fields vanish. The Kaluza-Klein U(1) gauge field becomes and the axion fields vanish. The Kaluza-Klein U(1) gauge field becomes 3 3 2 2 2 2 Q 2mΠcΠs Q (2m) β (Π Π )sin θ Q3 m Q3 m 2β 4 2c 2s 2 θ 2 = 2 2 Π2 cΠ2s dt (2 ) 4(Πc −Πs)sin dφ+ , (6.25) A 2 =(Πc 2 Πs)2F 2 dt− ∆− dφ+ , (6.25) A (Π−c Πs)F − ∆ − −1 where Q is defined in (6.5) and φ+ = φ + Ω+t,withΩ+ = β4ΠcΠ−1s .Theremainingthree where Q is defined in (6.5) and φ+ = φ + Ω+t,withΩ+ = β4ΠcΠs .Theremainingthree gaugegauge potentials potentials become become identified identified and and are are of of the the form form (6.3) (6.3) b bysettingysettinga a=0.=0. OneOne can can of of course course remove removeQQinin the the scalar scalar and and gauge gauge fields fields via via a a gauge gauge transformation. transformation. However,However, it is it useful is useful to to keep keep it it in in the the discussion discussion of the the lift. lift.

6.36.3 Subtracted Subtracted Geometry Geometry Lifted Lifted to to Five Five Dimensions Dimensions We now provide a lift of the subtracted rotating geometry to five-dimensions5.Thefive- We now provide a lift of the subtracted rotating geometry to five-dimensions5.Thefive- dimensional metric for the scaling limit takes the form: dimensional metric for the scaling limit takes the form: M.C., Larsen 1112.4856 Lift of subtracted geometry on circle S1 to five-dimension turns out to 2 ϕ1 2 −2ϕ1 2 locallyds5 = e factorizeds4 + e AdS(dz x+ S2 2 )([SL(2,R), 2 x SO(3)]/Z symmetry)(6.26) 2 ϕ1 2 −2ϕ1 3 A 2 2 ds 5 = e ds4 + e (dz + 2) , (6.26) where we have to implement the scaling z zϵ−1.Thismetrictakestheform:A [globally S2 fiberedwith over Bañados-Teitelboim-Zanelli (BTZ) black hole → −1 where we have to implement the scaling z zϵ .Thismetrictakestheform:16ma(Πc Πs) -3 w/ mass M−32, angular momentum J3 &φ¯ 3dφ cosmol. const. Λz =lt ], 2 3 → 2 2 = + 3 − ( + ) (6.29) ds5 = ϵ (dsS2 + dsBTZ) , ℓ (6.27) with − 2 2 3 and2 2 ds5 = ϵ (dsS2 + dsBTZ) , 16ma(Πc Πs) (6.27) φ¯ =2 φ +2 2 2 − (z + t) , 2 2 (6.29) where (r r )(r r 3−) ℓ r r r − ds2 3 −with3+ 3 − ℓ3 dt2 3 dr2 r2 dφ 3+ 3 dt 2 , BTZ = 2 2 3 + 2 2 2 2 3 + 3( 3 + 2 3) 2 1 2 2 2− ¯2 ℓ r (r r )(r 16rma−()Πc Πs) ℓr where anddsS2 = 4 ℓ dθ +sin θdφ , 3 3 φ¯3+=(6.28)φ +3 3 − (z + t) , 3 (6.29) − − ℓ3 (6.30) 5 2% 2 2 2 & 2 2 Partial results were provided in2 [5, 9].1 Here(2r wer2 take)(r particu2 r −¯)2lar care of theℓ dimensionsr and of the r r − ds2 2 = ℓ 3 dwhereθ 3++sin3 θ3dφ and2, 3 2 (6.28)2 3+ 3 2 dsBTZS = 4 − 2 − dt3 + 2 2 2 2 dr3 + r3(dφ3 + 2 dt3) , ℓ2 r r r 2 r2 r2 2 2 ℓ2r periodicities of metric coordinates. − 3 ( 3 (r3+)(r3 )(r 3−)r −) ℓ r 3 r r − 5 % &dsz2 − 3 − 3+− 3 − 3 dt2 3 dr2 r2 dφ 3+ 3 dt 2 , Partial results were provided in [5, 9]. Here we take particularBTZ care= of the dimensions2 2 and3 + of the2 2 2 2 (6.30)3 + 3( 3 + 2 3) φ3 = , − ℓ r (r r )(r r −) ℓr R 3 3 − 3+ 3 − 3 3 periodicities of metric coordinates.where 23 ℓ (6.30) t3 = t, z whereR 2 φ3 = , 2 16(2mR) z2 2 2 2 2 2 23R r3 = φ2m=(Πc , Πs)r +(2m) Πs a (Πc Πs) . (6.31) ℓ4 3 − − − ℓ ! R " Conformal t symmetryt, of AdS can be promotedℓ to Virasoro algebra1 3 = 3 1 2 2 3 Here,R R is the size of the circlet3 S= andt, ℓ =4m(Πc Πs) is the radius of the AdS3. of dual two-dimensional2 CFT R à la Brown-− Hennaux 2 Furthermore16(2mR) 2 2 16(22 mR2 )2 2 2 Standard statisticalr = entropy2m ( (viaΠ AdSΠ )r3+(2/CFT2 m2)) Π a ( Π c 2 Π às )la2 Cardy. 2 2(6.31)2 2 3 4 c smRr3 = s 2m(Πc Πs)r +(2m) Πs a (Πc Πs) . (6.31) ℓ − 8 ℓ4 − 2−− 2 − − à Reproduces entropy! ofr3± 4D= blackm(Π holesc + Πs) ! m a (Π"c Πs) . (6.32)" ℓ2 1 1 1 # 2 2 ±3 1 − − 2% 2 Here, R is the size of the circle Here,S andR isℓ the=4 sizem(Π ofc theΠ circles) $isS theand radiusℓ =4m of(Π thec Π AdSs) 3 3is. the radius of the AdS3. The periodicity of z coordinate is 2πR,andthustheangularcoordinate− − φ3 has the correct Furthermore Furthermore periodicity of 2π.Notealsothatthe2π periodicitymR of φ¯ is ensured if 16ma(Π Π )ℓ−3 is mR 8 2 2 c s 8 r23± = 2 m(Πc + Πs) m a (Πc Π−s) . (6.32) r3± = m(Πc −+ Πs) m a (ℓΠ2 c Πs) . (6.32) quantized inℓ2 units of R 1. ± − −# ± $ − − % # $z πR % φ The periodicity of coordinate is 22 ,andthustheangularcoordinate 3 has the correct The periodicity of z Thecoordinate lifted geometry is 2πR,andthustheangularcoordinate is indeed locally AdS3 S with theφ3 radiushas the of correct AdS3 equal to ℓ and −3 periodicity of 2π.Notealsothatthe2× π periodicity of φ¯ is ensured if 16ma(Πc Πs)ℓ is 2 ℓ −3 periodicity of 2π.Notealsothatthe2the radius of S equalπ toperiodicity. of φ¯ is ensured if 16ma(Πc Πs)ℓ is − quantized2 in units of R−1. − −1 quantized in units of R . 2 Subtracted MagnetisedThe lifted Geometry geometry is indeed locally AdS3 S with the radius of AdS3 equal to ℓ and 2 × The lifted geometry is indeed locally AdS3 2 S withℓ the radius of AdS3 equal to ℓ and the radius of S equal to 2 . This geometry also lifts to (6.27)× where now φ¯ in (6.28) is defined as6 the radius of S2 equal to ℓ . 2 Subtracted Magnetised Geometry φ¯ = φ + β4 z, (6.33) Subtracted Magnetised GeometryThis geometry also lifts to (6.27) where now φ¯ in (6.28) is defined as6 and we set in all expressions above a =0,i.e.theBTZcoordinatesarerelatedto t, r, z 6 { } This geometry also lifts to (6.27) where now φ¯ in (6.28) is defined asφ¯ = φ + β4 z, (6.33) as in (6.31) with a =0.(Obviously,β4 =0correspondstotheliftofthestaticsubtracted −1 geometry.) Note thatand thewe¯ set shift in requiresall expressions that β above4 be quantizeda =0,i.e.theBTZcoordinatesarerelatedto in units of R ,inorderforφ¯ t, r, z φ = φ + β4 z, (6.33) { } to have the correctas periodicity in (6.31) with of 2aπ=0.(Obviously,. β4 =0correspondstotheliftofthestaticsubtracted and we set in all expressions above a =0,i.e.theBTZcoordinatesarerelatedto t, r, z −1 ¯ geometry.) Note that the shift requires that β4 be quantized{ in} units of R ,inorderforφ as in (6.31) with 6.4a =0.(Obviously, Relation ofto havetheβ4 =0correspondstotheliftofthestaticsubtracted BTZ the correct Black periodicity Hole Coordinates of 2π. to the AdS3 Coordinates −1 geometry.) Note that the shift requires that β4 be quantized in units of R ,inorderforφ¯ According to [14, 15] AdS3 is the quadric 6.4 Relation of the BTZ Black Hole Coordinates to the AdS3 Coordinates to have the correct periodicity of 2π. u2 + v2 x2 y2 = ℓ2 , (6.34) According to [14, 15] AdS−3 is the− quadric 6It was observed in [25] that such a shift produces a magnetic field for the Kaluza-Klein U(1) gauge 6.4 Relation of the BTZ Black Hole Coordinates to the AdS3 Coordinates u2 + v2 x2 y2 = ℓ2 , (6.34) potential and thus a four-dimensional geometry in a Kaluza-Klein− magnetic− field. 6It was observed in [25] that such a shift produces a magnetic field for the Kaluza-Klein U(1) gauge According to [14, 15] AdS3 is the quadric potential and thus a four-dimensional24 geometry in a Kaluza-Klein magnetic field. u2 + v2 x2 y2 = ℓ2 , (6.34) − − 24 6It was observed in [25] that such a shift produces a magnetic field for the Kaluza-Klein U(1) gauge potential and thus a four-dimensional geometry in a Kaluza-Klein magnetic field.

24 order to demonstrate the procedure we shall present the details for the Schwarzschild black hole, only. In this case it is sucient to employ the Einstein-Dilaton-Maxwell Lagrangian density, with the dilation coupling ↵ = 1 , which is a consistent truncation of the Lagrangian density p3 (1) with = 0, ' = ' = ' 2 , F = F = 2 F and = 0. [Of course for i i 2 3 ⌘p3 ⇤ 1 2 ⇤F1 ⌘ 3 F2 the multi-charged rotating black holes one has to employ the fullq N=2 supergravity Lagrangian density (1).] We begin by considering static solutions to general Einstein-Dilaton-Maxwell equations with the general dilation coupling ↵. The Lagrangian density is 4.:

1 1 2 1 2↵ 2 p g R (@) e F . (24) 4 2 4 ⇣ Making the Ansatz 2 2U 2 2U i j ds = e dt + e dx dx ,F= @ (25) ij i0 i we obtain an e↵ective action density in three dimensions of the form

ij 2U 2↵ p R( ) 2 @ U@ U + @ @ e e @ @ (26) ij i j i j i j ⇣ ⇣ ⌘⌘ Defining U + ↵ ↵U + x ,y , (27) ⌘ p1+↵2 ⌘ p1+↵2 the e↵ective action density becomes

ij 2p1+↵2x p R( ) 2 @ x@ x + @ y@ y e @ @ . (28) ij i j i j i j ⇣ ⇣ ⌘⌘ Evidently we can consistently set y = 0 and we obtain a sigma model, whose fields x, map into the target SL(2, R)/SO(1, 1), coupled to three dimensional Einstein gravity. The non-trivial action of an SO(1, 1) subgroup of SL(2, R) is called a Harrison transformation. More concretely, and following [19] but making some changes necessitated by considering a reduction on time-like, rather than a space-like Killing vector we define a matrix (See also, e.g., [20] and references therein.):

2p1+↵2x 2 2 2 p1+↵2(x+y) e (1 + ↵ ) p1+↵ P = e , (29) p1+ ↵2 1 ✓ It is straightforward to◆ show that a Harrison transformation: so that T 2p1+↵2y 10 P = P , det P = e . (30) H = , (34) 1 Taking H SO(1, 1) which acts on P as ✓ ◆ 2 P HPHT corresponds, to: (31) ! p1+↵2x0 1 p1+↵2x it preserves not only the properties (30) but also the Lagrangiany0 = densityy, (28) e which can= be⇤ caste , in the form: Further insights into geometry of subtracted1 geometry 2p1+ ↵2x 2 2 2 2 2p1+↵2x 1 ij 0 1 = ⇤ [ + (e (1 + ↵ ) )] ; ⇤ =( + 1) e . (35) p R( )+ Tr(@ P @ P ) . p1+↵2 (32) ij 1+↵2 i j It is straightforward i. Subtracted to show geometry that⇣ a Harrison as an transformation: infiniteNote, boost this Harrison transformation⌘ transformations can also be determined as an analytic continuation of transformations on the original BH,SO(1,1) transformationsgiven in Section [change 2 of aysmptotics [20]. A Harrison] transformation in the limit of an infinite boost corresponds 10 M.C. & Gibbons 1201.0601 H to, 1. One may verify that (34) with(33) 1 in the Einstein-Maxwell gravity (↵ = 0) takes the ⇠ 1 ! Virmani 1203.5088 ! ✓ ◆ Schwarzschild Sahay metric & Virmani to the 1305.2800 Robinson-Bertotti.. one. This type of transformation was employed 4 2 We choose the units in which 4⇡G =1.NotethatintheLagrangiandensity(1)16recently in [22]. For another⇡G =1andthefield work, relating the Schwarzschild geometry to AdS2 S , see [23]. strengths di↵er by a factor of p2. In the case of ↵ = 1 , we shall act with (34) on the Schwarzschild solution with⇥e2U =1 2m , ii. Interpolating geometries between the original black holep3 r 2m and their subtracted geometries obtained:= 0, = 0. The transformation (35) with =1resultsin⇤ = r , and the metric (6) with 9 - via solution generating techniquesthe subtracted geometry warp factor:

= r4 =(2m)3r, (36) - dual CFT interpretation via deformation of irrelevant CFT operatorss0 ! s Baggio, de Boer, Jottar & Mayerson1210.7695 and the scalarM.C., Guica field & and Saleem the electric1302.7032.. field strength : 2 2m 2 1 e p3 = , Ftr = , (37) r r r3 2m

i.e., this is the static subtracted geometry of Subsection 2.1, with ⇧c =1, ⇧s = 0. The subtracted geometry for the Kerr spacetime can be obtained by reducing the spacetime on the time-like Killing vector and acting on the Kerr black hole with an infinite boost Harrison transformation for Lagrangian density (1), where we set = = , ' = ' = ' 1 2 3 ⌘ 1 2 3 ⌘ 2 , F = F = 2 F and = p2 , i.e. an Einstein-Dilaton-Axion gravity with p3 ⇤ 1 2 ⇤F1 ⌘ 3 F2 F two U(1) gauge fields and respectiveq dilaton couplings ↵ = 1 and ↵ = p3. The subtracted 1 p3 2 geometry of the multi-charged rotating black holes is expected to arise as a specific Harrison transformation on a rotating charged black solution of (1). This has recently been confirmed [14]. These results demonstrate that the subtracted geometry is a solution of the same theory as the original black hole. Furthermore the original black hole and the subtracted geometry clearly lie in the same duality orbit, specified above and passing through the original black hole. Thus any physical property of the original black hole solution which is invariant under the duality transformation of M-theory remains the property of the subtracted geometry. For example the area of the horizon is unchanged.

3 Asymptotically Conical Metrics

The scaling limit, or equivalently the subtraction process, alters the environment that our black holes find themselves in [12, 13]. In fact the subtracted geometry metric is asymptotically of the form R 2p ds2 = dt2 + B2dR2 + R2 d✓2 +sin2 ✓2d2) (38) R0 10 M.C., Larsen 1406.4536

2 Subtracted geometry [Δ0 à Δ= A r + B cos θ +C; A,B,C-horrendous] also works for most general black holes of the STU-Model (specified by mass, four electric and four magnetics charges, and

angular momentum) Chow & Compère 1310.1295;1404.2602

M.C. & Larsen 1106.3341 All also works in parallel for subtracted geometry of most general five-dimensional black holes

(specified by mass, three charges and two angular momenta) M.C., Youm 9603100 Further recent developments Quantum aspects of subtracted geometries: i) Quasi-normal modes - exact results for scalar fields two damped branches à no black hole bomb M.C., Gibbons 1312.2250; M.C., Gibbons, Saleem 1401.0544 ii) Entanglement entropy - minimally coupled scalar M.C., Satz, Saleem 1407.0310

2 iii) Vacuum polarization <φ > analytic expressions at the horizon: static M.C., Gibbons, Saleem, Satz 1411.4658 rotating M.C., Satz, Saleem 1506.07189 outside & inside horizon: rotating M.C., Satz 1605….

iv) Thermodynamics of subtracted geometry via Komar integral: M.C., Gibbons, Saleem, 1412.5996 (PRL) 3Asymptoticallyconicalbackgrounds 3Asymptoticallyconicalbackgrounds The general rotating subtracted geometry backgrounds are solutions of the equations of motion 5 following fromThe the general action rotating (2.2) or subtracted (2.10) and geometry take the backgroundsform [14, 17] are solutions of the equations of motion following from the action (2.2) or (2.10) and take the form [14, 17]5 √∆ G X ds2 = dr¯2 (dt¯+ )2 + √∆ dθ2 + sin2 θdφ¯2 , X − √√∆∆ AG G X ds2 = dr¯2 (dt¯+ !)2 + √∆ dθ2 + "sin2 θdφ¯2 , (2m)2 X a (−Π√ Π ) A G η c ∆ s ! " e = , χ =2 − cos θ, √∆ (2m) 2m a (Πc Πs) eη = , χ = − cos θ, 4 √∆ 2m 2 2 2 4 0 (2m) a (Πc Πs) 2 ¯ (2ma) cos θ (Πc Πs) +(2m) ΠcΠs ¯ A = − 4 sin θdφ + 22− 22 2 dt, 4 0 ∆ (2m) a (Πc Πs) 2 ¯ (2ma(Πc) cosΠs)θ∆(Πc Πs) +(2m) ΠcΠs ¯ A = − sin θdφ + − 2 − 2 dt, 2m cos θ ∆ 2 2 2 (Πc Πs) ∆ A = ∆ (2ma) (Πc Πs) sin θ dφ¯ 2ma (2mΠ−s +¯r(Πc Πs)) dt¯ , ∆ 2m−cos θ − 2 2− 2 ¯ − ¯ A = ∆2 (2ma) (Πc Πs) sin2 θ dφ 2ma (2mΠs +¯r(Πc Πs))2dt , 1 #$ ∆ (2ma) (−Πc Πs) ¯− % (2ma) (Πc −Πs)[2mΠs +¯r(Πc Π&−s)] cos θ ¯ A = r¯ m 3 − 2 dt + − 2 − dt 2 −2m − 1− (2m#$) (Πc +(2maΠs)) (Πc Πs) ¯ % (2ma) (Π2cm∆Πs)[2mΠs +¯r(Πc Π&s)] cos θ ¯ !A = r¯ m 3 " − dt + − − dt −2m − − (2m) (Πc + Πs) 2 2 2 2m∆ ' ! 2 (2"ma) (Πc Πs) cos θ + a(Πc Πs)sin θ 1+ − 2 d2φ¯, 2 (3.1) ' − 2 (2∆ma) (Πc Πs) cos θ ¯ + a(Πc !Πs)sin θ 1+ − " dφ, (3.1) − ∆ where ! " where 2 2 2 2 2ma 2 ¯ X =¯r 2mr¯ + a ,G= X a sin θ, = ((Πc 2maΠs)¯r +2mΠs)sin θdφ, − X =¯r2 2mr¯ + a2,G− = X a2 sinA 2 θ, G = − ((Π Π )¯r +2mΠ )sin2 θdφ¯, 3 2 2 4 2 2 2 2 G c s s ∆ =(2m) (Πc Π−s)¯r +(2m) Πs (2ma) −(Πc Πs) cos Aθ, − (3.2) ∆ =(2− m)3(Π2 Π2)¯r +(2− m)4Π2 (2−ma)2(Π Π )2 cos2 θ, (3.2) c − s s − c − s and Πc, Πs, a and m are parameters of the solution. In orderand toΠ studyc, Πs, thea and thermodynamicsm are parameters of these of the backgrounds solution. it is necessary to identify which parameters areIn fixed order by to the study boundary the thermodynamics conditions in the of variat theseional backgrounds problem. it Ais full necessary analysis to of identify the which variationalparameters problem for are the fixed actions by the (2.2) boundary or (2.10) conditions requiresknowledgeofthegeneralasymptotic in the variational problem. A full analysis of the solutions andvariational is beyond problem the scope for of the the actions present (2.2) paper. or (2.10) Howev requireer, wesknowledgeofthegeneralasymptotic can consider the variational problem withinThermodynamicssolutions the class and of is stationary beyond the solutions scope via of (3.1). thevariational present To this paper. end, it Howev isprinciple convenienter, we can to reparameterize consider the variational these backgroundsproblem by within means the of class a suitable of stationary coordinate solutions transfoAn, (3.1).M.C.,rmation, Papadimitriou To this accompanied end, it 1602.0150 is convenient by a relabeling to reparameterize of the free• Identifythese parameters. backgrounds normalizable In particular, by means and we non-rormalizable of introduce a suitable the coordinate new modes coordinates transfo: rmation, accompanied by a relabeling of the free parameters. In particular, we introduce the new coordinates Introduce new4 coordinates3 2 : 2 4 2 2 2 ℓ r =(2m) 4(Πc Πs)¯3r +(22 m)2 Πs (2ma4 ) 2(Πc Πs)2 , 2 Rescaled radial coord.: ℓ r =(2ß− m) (Πc Πs)¯r +(2− m) Πs (2−ma) (Πc Πs) , k 1 ¯ − ¯ 2ma(Π−c Πs) ¯ − t = k t,1 φ = φ −2ma(t,Πc Πs) (3.3) Rescaled time:ℓ3 (2m)3(tΠß=2 Π2) t,¯ − (2φm=)3φ(¯Π2 Π2) − t,¯ (3.3) ℓ3 c −(2ms)3(Π2 Π2) −c −(2ms)3(Π2 Π2) c − s c − s where ℓ and k are additional non-zero parameters, whose role will become clear shortly. Moreover, where ℓ and k are additional non-zero parameters, whose role will become clear shortly. Moreover, we define theTrade new four parameters parameters m, a, Π , Π for: we define the new parameters c s 4 3 2 2 2 2 2 2 3 2 2 ℓ r =(2m)4 m(Πc + Π3s) (22 ma)2 (Πc Πs)2 m 2 a (2m)2 (Πc2 Πs3), 2 2 ± ℓ r =(2m) m−(Πc + Πs) (2−ma) (Π±c Πs)− m a −(2m) (Πc Πs), 3 ± − − ± − − ℓ ω =2ma(3Πc Πs),B=2m, ( (3.4) ℓ ω =2− ma(Πc Πs),B=2m, ( (3.4) 5 − In order to compare5 this background with the expressions given in eqs. (24) and (25) of [14], one should take into r+, rIn-, ω order - normalizable to compare→− this backgroundmodes→− with the expressions given in eqs. (24) and (25) of [14], one should take into account the field redefinition A A, χ →−χ,beforethedualizationof→− A, as mentioned in footnote 4, and add a account the field redefinition A A, χ χ,beforethedualizationofA2, as2 mentioned¯ in footnote 4, and add a constant pure gauge term. Moreover, there is a typo in eq. (25)of[14]:theterm2mΠs cos θdt should2 2 be¯ replaced Bconstant− - non-renormalizable pure2 ¯ gauge term. Moreover, mode, there along is a with typo in eq.and (25 k)of[14]:theterm2 mΠs cos θdt should be replaced by 2mΠs(Πs Πc)cos θd−t. 2 ¯ l (fixedby 2m upΠs (toΠs radialΠc)cos diffeomorphismθdt. and gl. Symmetries)

7 7 which can be inverted in order to express the old parameters intermsofthenewones,namely

2 ℓ 1 2 2 1 2 Πc,s = 2 (√r+ + √r ) ℓ ω + (√r+ √r ) , B !2 − ± " 4 − − # Bℓω a = ,m= B/2. (3.5) which can be inverted in order to2 express2 1 the old parameters2 intermsofthenewones,namely 2TheSTUmodelanddualityframes2 ℓ ω + √r+ √r 4 − − In this section we review the bosonic2 $ sector of the 2-charge truncation of the STU model that is ℓ 1 % 2& 2 1 2 6 Rewritingrelevant for the describing backgroundΠc,s the= subtracted (3.1)( in√ geometries.r terms+ + √ ofr We the) will newdoℓ ω socoordinat in+ the(√ dualityres+ and frame√r parameters) discussed, in we obtain B2 2 − ± 4 − − [14], where both charges are electric,! as well as in the one us"ed in [17], where there is one electric# and 2 2 3 one magnetic charge.B /ℓ We will refer to theseBℓℓω framesω as ‘electric’ and ‘magnetic’ respectively. As it eη = a = , χ = cos θ, ,m= B/2. (3.5) will become clear from the subsequent2 analysis,2 in order to compare the thermodynamics in the two r + ℓ2ω2 sin θ 2 2 1 B 2 frames, it is necessary to keep2 trackℓ ω of+ boundary4 √r+ terms√ intror duced by the duality transformations. 3 3 − − 0 ' B /ℓ $ 2 2 A = √r+r kdt%+ ℓ ω sin θd&φ , 6 Rewriting2.1 Magnetic ther + backgroundℓ2ω frame2 sin2 θ (3.1) in− terms of the new coordinates and parameters we obtain The bosonic LagrangianB cos θ of the% STU model in the duality frame& used in [17] is given by A = B2/ℓ2 ( ω√r r ℓ3kdtω + rdφ) , eη = 2 2 2 1, χ+= 1 cos θ, r +2ℓ ω sin θ − − 2 2ηa a a r2κ+4 ℓ42=ω2Rsin⋆ 12 θ ⋆ dηa dηaB e ⋆ dχ dχ ℓ L 1 − 2 ∧ − 2 ωℓ3 ∧ ω r r kdt rdφ ωℓ3 3 3 1 1 2 √ + A = r (r η0+ r0 ) 0kdt +2ηa η0 cos aθ a 0 −a a−0 + dφ, 0 ' B /ℓ e−+ ⋆ F F e2 − ⋆2(F + χ F ) (F2 +2 χ F2 ) A = −B −−22 √r+r−∧ kdt−+2 ℓ ωBsin θdφ , r +∧ ω ℓ sin θ B 2)2 2 −* ) * r + ℓ ω sin 1θ a b c 2 1 2 a b 0 c 1 a b c 0 0 2 + Cabc2 χ F F ℓ+drCabcχ χ F (rF +r C)(abcrχ χr+χ )F 2 F2 , 2 2(2.1) ds( = rB+cosℓ2ωθ 2 sin2 % θ ∧ 2 ∧ & − 6− − k ∧dt + ℓ dθ A = ( ω√r+r kdt + rdφ) , 2 2 2 (r −r )(r r+3) − r where ηa r(a+=1ℓ ,ω2, 3)sin areθ dilaton− ) fields− and− η0 −= a=1 ηa.ThesymbolCabc is pairwise symmetric* ' 3 2 2 3 2 with C123 =1andzerootherwise.TheKaluza-Kleinansatzforobtaininℓ 1 ωℓ ω√rℓ+rrsinkdtθ gthisactionfromtherdφ ωℓω√r+r A = r (r + r ) kdt + !cos2 θ+ − − dφ+ dφ, − kdt . (3.6) 6-dimensional action (1.1)+ is given explicitly in [17]. Thisframepossessesanexplicittrialitysym-2 2 2 −B − 2 − a B r +a ω2 ℓ2 sin2 θ − Ba r metry, exchanging) the three gauge* fields A ,thethreedilatons) r η+ ℓandω thesin threeθ ) axions* χ .Inthis * frame, the subtracted geometries source2 all2 three gauge fields Aa magnetically, while A0 is electri- (2 2 2 2 ℓ dr (r r )(r r+) 2 2 2 2 callydsSeveral sourced.= r comments+ Moreover,ℓ ω sin are holographicθ in order renormalization here. Firstly, turns−' the out− two to be− paramet muchk moredters+ straightforwardℓr dθare the locations of the (r r )(r r+) − r ± outerin this and frame inner compared horizons with) the respectively, electric− − frame.− and clearly correspond to normalizable* perturbations. A ' 2 2 2 straightforwardIn order to describe calculation the subtracted shows that geometriesω is also it suffi aces normalizableℓ r tosin considerθ a mode. truncationω We√r of+ willr (2.1), explicitly confirm + 1 d2φ 3 − kdt . (3.6) corresponding to setting η1 = η2 = η3 η, χ1 = χ2 = χ3 χ,andA2 = A = A A.The ≡ r +≡ ℓ2ω2 sin θ − ≡ r thisresulting later action on by can showing be written that in the theσ-model long-distance form divergences of the) on-shell action are* independent of ω.Settingthenormalizableparameterstozerowearriveatth`Vacuum’ solution ebackground 1 1 ' Several comments4 are in order here.I µ J Firstly,Λ theΣµ twoν parametµνρσ ersΛ Σr are the locations of the S4 = 2 d x√ g R[g] IJ∂µϕ ∂ ϕ ΛΣFµν F ΛΣϵ Fµν Fρσ± + SGH, (2.2) outer and2κ4 inner horizons−2 2 respectively,− 2G and− clearlyZ corresp− Rond to normalizable perturbations. A "M B /#ℓ $ ℓ obtainedη by turning off r+, r-, ω – 0three normalizable modes: straightforwardwhere e calculation= , showsχ =0 that,Aω is also=0 a,A normalizable= B cos mode.θdφ, WeA will= explicitlyrkdt, confirm √r 1 −B The asymptotic structureS of (stationary)= conicald3x√ backgrouγ 2K, nds is parameterized by the(2.3) three non- this later on by showing that2 theGH long-distance2 divergences of the on-shell action are independent zero constants B, ℓ anddrk.Inthemostrestrictedversionofthevariationalproblem,2κ4 ∂ − these three of ω.Settingthenormalizableparameterstozerowearriveatthds2 = √r ℓ2 rk2dt2 +"ℓM2dθ2 + ℓ2 sin2 θdφ2 , ebackground( (3.7) is theparameters standard Gibbons-Hawking should be keptr2 − fixed. [36] term However, and we there have is defined a 2-paramethe doubletster family of deformations of these boundary data2 still)2 leadingη to a well posedA0 variational problem,* as we now explain. The first de- The asymptotic structureη of (stationary)B /ϕℓI = conical,A backgrouΛ = nds0 is,I parameterized=1, 2, byΛ the=1 three, 2, non- ℓ(2.4) which weformation shalle = corresponds consider, as toχ the theχ =0 transformation vacuum,AA solution.=0 of the,A boundar The= factydatainducedbyreparameterizationsB cos thatθdφ,the backgroundA = rkdt, (3.7) is singular zero constants B, ℓ andof thek.Inthemostrestrictedversionofthevariationalproblem, radial coordinate.√r # $ Namely, under# the$ bulk diffeomorphismthese three −B parameters shoulddoesas well not be as kept pose the fixed. 2 any2matrices However, difficulty there2 since is a 2-parame it shouldter only family be of viewed deformations as an of theseasymptotic solution that helps us 2 × 2 dr 2 2 2 2 4 2 2 2 ( 1/4 boundary datato still properly leadingds30 formulate to= a√ wellr posedℓ the variational variational1rkedt3η prob+3+ problem.lem,ℓer dηχθ as2λ−+ we3r,eℓ nowη Changingsinχλ > explain.θ0d, φ The th, eradialcoordinateto first1 χ de-3 3 χ2 (3.9)ϱ = ℓr(3.7),the 2 − −→ − 2 formation correspondsIJ =Non- to thenormalizable transformation2η , rΛΣ of(fourth)=− the boundar modeydatainducedbyreparameterizationsη B, alongη with, l andΛΣ k=, fixed3 up2 to . (2.5) vacuumG metric03e becomes) Z 4 3e− χ 3e− R* 4 2 χ 3χ of the radial coordinate.theradial# boundary Namely, diffeomorphism$ under parameters the bulk transform: di #ffeomorphi as sm $ # $ whichAs usual, we shallϵµνρσ consider= √ g εµνρσ asdenotes the vacuum the totally solution. antisymmetric6 The fact Levi-Civita that the tensor, background where εµνρσ (3.7)= is singular − 2 4 2 2 3 ϱ 2 2 2 2 2 2 does1istheLevi-Civitasymbol.Throughoutthispaperwechoose not pose any difficultyr dsλ since− =4r, λ itd>ϱ shouldk0, λ k, onlyℓk bedtλℓ viewed,B+ ϱ d asB.theθ an+sin orientationasymptotic(3.9)θdφ in , solutionso that (3.10) that helps us(3.8) ± → →− ℓ → → M toε properlyrtθφ =1.WenoteinpassingthattheLagrangian(2.2)isinvariant formulate the variational problem. Changing theradialcoordinatetounder the global symmetryϱ = ℓr1/4,the the boundary parameterstransformationThisand transformtransformation global U(1) as issymmetry: a direct analogue + of, the so called Pen% rose-Brown-Henneaux& (PBH) diffeo- vacuumwhich is metric a special becomes case of the conical metrics discussed in [14]. Different conical geometries are morphisms inη asymptotically2 3η AdS2 backgrounds0 3 0 [40], which induce2 a Weyl2 transformation on the e k µλek,, χℓ λℓµ−,Bχ,AB. µ A ,A µA, ds (3.10)ds , (2.6) supportedboundary by di sources.fferent→→ matter The PBH→→ fields. diffeomorphisms Although→ϱ →6 imply we focus that→ the on the bulk s→fieldspecific do conical not induce backgrounds boundary obtained wherefields,µ is an but arbitrary only a conformal non-zerods2 =4 constant structure2dϱ2 parameter.,thatisboundaryfieldsuptoWeyltransformations[32].Thik2dt2 + ϱ2 dθ2 +sin2 θdφ2 , s (3.8) This transformationas solutions is a direct of theanalogue STU of themodel so called here,− Pen weℓrose-Brown-Henneaux expect that our (PBH) analysi diffeo-s, modified accordingly for the dictateswhich that keep the k variationalB3/l3 - fixed problem must be formulated in terms of conformal classes rather than morphisms indiff asymptoticallyerent matter AdS sectors, backgrounds applies [40], to which general+ induce, asymptotically a Weyl transformation% conical on backgrounds.the & boundary sources.which is Therepresentatives a PBH special diffeomorphisms case of the of conformal the imply conical that class metricsthe [22]. bulk Infields the discussed case do not of s inubtractedinduce [14]. boundary Digeometries,fferent conical variations geometries of the are 6 5 fields, but onlysupported a conformalSinceboundary by these di structureff solutions parameterserent,thatisboundaryfieldsuptoWeyltransformations[32].Thi matter carry of the fields. non-zero form Although magnetic we charge, focus the on the gauge specific potential conicals A must backgrounds be defined obtained in the north − dictates thatas(θ the solutions< variationalπ/2) and of south problem the hemispheres STU must model be formulated respectively here, in weterms as expect [38, of conformal 39], thatAnorth our classes= A analysi ratherBdφs, thanand modifiedAsouth = accordinglyA + Bdφ,where forA theis the representativesexpression of the conformal given in (3.6).class [22]. In the caseδ1k of=3 subtractedϵ1k, δ1ℓ geometries,= ϵ1ℓ, δ1 variationsB =0, of the (3.11) boundary parametersdifferent matterof the form sectors, applies to general asymptotically conical backgrounds. correspond to motion within the equivalence class (anisotropic conformal class) defined by the 6 Sincetransformation these solutions (3.10), carry and non-zero therefore magnetic lead to a charge, well posed the gvaaugeriational potential problem.A must be defined in the north δ1k =3ϵ1k, δ1ℓ = ϵ1ℓ, δ1B =0, (3.11) (θ < π/2) andAseconddeformationoftheboundarydatathatleadstoawell south hemispheres respectively as [38, 39], A8north = A − Bdφposedand A variationalsouth = A + problemBdφ,where is A is the correspondexpression to motion given within in the (3.6). equivalence class (anisotropic conformal class) defined by the transformation (3.10), and therefore lead to a well posedδ2k =0 variational, δ2ℓ = problem.ϵ2ℓ, δ2B = ϵ2B. (3.12) Aseconddeformationoftheboundarydatathatleadstoawellposed variational problem is To understand this transformation, one must realize8 that theparametersB and ℓ do not correspond to independentδ2k =0 modes,, δ2ℓ but= ϵ rather2ℓ, δ2 onlyB = ϵ the2B. ratio B/ℓ,whichcanbeidentifiedwiththesourceof(3.12) the dilaton η.Inparticular,keepingB/ℓ fixed ensures that the variational problem is the same in To understand this transformation,all frames of the one form must realize that theparametersB and ℓ do not correspond to independent modes, but rather only the ratio B/ℓ,whichcanbeidentifiedwiththesourceof2 αη 2 dsα = e ds , (3.13) the dilaton η.Inparticular,keepingB/ℓ fixed ensures that the variational problem is the same in all frames of the formfor some α,whichwillbeimportantfortheupliftoftheconicalbackgrounds to five dimensions. 2 αη 2 The significance of thedsα parameter= e ds , B is twofold. It corresponds to the(3.13) background magnetic field in the magnetic frame and variations of B are equivalent to the global symmetry transformation for some α,whichwillbeimportantfortheupliftoftheconicalbackgrounds to five dimensions. (2.6) or (2.12) of the bulk Lagrangian. Moreover, as we will discuss in the next section, it enters The significance of the parameter B is twofold. It corresponds to the background magnetic field in the covariant asymptotic second class constraints imposing conical boundary conditions. The in the magnetic frame and variations of B are equivalent to the global symmetry transformation transformation (3.12) is a variation of B combined with a bulk diffeomorphism in order to keep the (2.6) or (2.12) of the bulk Lagrangian. Moreover, as we will discuss in the next section, it enters in the covariant asymptoticmodes k secondand B/ classℓ fixed. constraints The relevant impos bulking conical diffeomorphism boundary is conditions. a rescaling The of the radial coordinate of the form (3.9), accompanied by a rescaling t λ3t of the time coordinate. transformation (3.12) is a variation of B combined with a bulk diffeomorphism→ in order to keep the modes k and B/ℓ fixed. The relevant bulk diffeomorphism is a rescaling of the radial coordinate of the form (3.9), accompanied by a rescaling t λ3t of the time coordinate. 4Boundarycountertermsandrenormalizedconservedcharge→ s 4BoundarycountertermsandrenormalizedconservedchargeThe first law of black hole thermodynamics is directly relatedtothevariationalproblemandthes boundary conditions imposed on the solutions of the equations of motion. As we briefly reviewed The first law of blackin hole the thermodynamics previous section, is in directly non-compact relatedtothevariationalproblemandthe spaces, where the geodesic distance to the boundary is boundary conditionsinfinite, imposed the on the bulk solutions fields induce of the only equatio anns equivalence of motion. class As we of briefly boundary reviewed fields, which implies that in the previous section, in non-compact spaces, where the geodesic distance to the boundary is infinite, the bulk fields induce only an equivalence class of boundary fields, which implies that 9

9 Acknowledgments

O.A. would like to thank Jin U Kang for interesting discussions. M.C. would like to thank Gary Gibbons and Zain Saleem for discussions on related topics. M.C. and I.P. would like to thank the International Institute of Physics, Natal, for the hospitality and financial support during the School on Theoretical Frontiers in Black Holes and Cosmology, June 8–19, 2015, where part of this work was carried out. This research is supported in part by the DOE Grant Award DE-SC0013528, (M.C.), the Fay R. and Eugene L. Langberg Endowed Chair (M.C.)andtheSlovenianResearch Agency (ARRS) (M.C.).

Appendices

A Radial Hamiltonian formalism

In this appendix we present in some detail the radial Hamiltonian formulation of the reduced STU σ-model (2.2). This analysis can be done abstractly, without reference to the explicit form of the σ-model functions IJ, and ,anditthereforeappliestotheelectricLagrangian(2.10)as G ZΛΣ RΛΣ well, provided A , • andRadial areHamiltonian replaced with theirformalism electric frame analogues in(2.11). L ZΛΣ RΛΣ The first step towards to adetermine Hamiltonian Sct, to formalism the bulk action is picking S a suitable radial coordinate u such that constant-u slices, which we will denote by Σu,arediffeomorphic to the boundary ∂ of . Suitable radial coordinate u, such that constant-u slices Σ M M Moreover, it is convenient to choose u to be proportional to the geodesic distanceu between any fixed point in and a point in Σ ,suchthat Σ 14 →Σ ∂M ∂as u →as ∞u. .Giventheradialcoordinateu, M u u u → M →∞ we then proceed with an ADM-like decomposition of the metric and gauge fields [47] Decomposition of the metric and gauge fields: 2 2 i 2 i i j ds =(N + NiN )du +2Nidudx + γijdx dx , L Λ Λ i AThe= canonicala du + momentaAi dx , conjugate to the induced fields on Σu following from(A.1) the Lagrangian (A.2) are Decomposition leads to the radial Lagrangian L w/ canonical momenta: where xi = t, θ, φ .Thisismerelyafieldredefinition,tradingthefullycovariant fields g and ij δL 1 ij ij µν L { } { } π = = Λ 2 √ γ ΛKγ K , (A.4a) Aµ for the induced fields N, Niδ,γ˙γijij, a2κ4and−Ai on Σ−u.Insertingthisdecompositionintheσ-model action (2.2) and adding the Gibbons-HawkingδL 1 1! term (2.3)"J leadistotheradialLagrangianJ π = = N − √ γ ϕ˙ N ∂ ϕ , (A.4b) I δϕ˙I −2κ2 − GIJ − i 4 1 i δL 2 1 1 ! ij Σ "Σ Σji 2 ijk Σ 3 π = 2= N −ij√ γ ΛΣ γ A˙ I∂ja i NjIF J √j γ J ΛΣϵ F . (A.4c) L = 2 d xN√ γ RΛ[γ]+˙KΛ −Kκ2ijK − Z IJ(ϕ)j ϕ−˙ N −∂iϕ ϕ˙ − κN2 ∂−jϕR jk 2κ − δAi − 4 − 2N 2 G − − 4 4 ! " # # $ $ w/2 momenta conjugate to N, N , and a# vanish.Λ $# $ Notice that the momentaij ˙ Λ conjugateΛ toi Nk, NΛi,andΛ ˙ Σa vanish Σ identically,l Σ since the Lagrangian (A.2) ΛΣ(ϕ)γ Ai ∂ia N Fki Aj ∂ja N Flj (A.2) Λ does−N not2 Z contain any radial− derivatives− of these fields.− It follows− that the fields N, Ni,anda are Lagrange multipliers,% implementing three first&% class constraints, which& we will derive momentarily. ijk ˙ Λ Λ Σ 1 I i J Λ Σij The4 canonicalΛΣ(ϕ)ϵ momentaAi (A.4)∂ia allowFjk us to performIJ(ϕ) the∂iϕ Legendre∂ ϕ transformΛΣ(ϕ)F ofij theF Lagrangian, (A.2) to− obtainR the radial Hamiltonian− − 2G − Z % & ' where H = d3x πijγ˙ + π ϕ˙ I + πi A˙ Λ L = d3x N + N i + aΛ , (A.5) ij I Λ i − H iH FΛ % 1 % Kij = (˙#γij DiNj DjNi)$, ! (A.3)" where 2N − − is the extrinsic curvature of the radial2 slices Σ , D denotes a covariant derivative with respect to κ4 u i 1 ij kl IJ = 2 γikγjl γijγkl π π + (ϕ)πI πJ the induced metric γij on Σu,whileadot˙standsforaderivativewithrespecttotheHamiH − √ γ − 2 G ltonian − & & ' 1 2 2 ‘time’ u. + ΛΣ(ϕ) π + √ γ (ϕ)ϵ klF M πi + √ γ (ϕ)ϵipqF N 4Z Λi κ2 − RΛM i kl Σ κ2 − RΣN pq 14We assume M to be a non-compact space with infinite& volume4 such that the geodesic'& distance4 between any point '' √ γ 1 in the interior of M and a point in ∂M +is infinite.− R[γ]+ (ϕ)∂ ϕI ∂iϕJ + (ϕ)F ΛF Σij , (A.6a) 2κ2 − 2GIJ i ZΛΣ ij 4 & ' 2 i = 2D πij + π ∂iϕI + F Λij π + √ γ (ϕ)ϵ klF Σ , (A.6b) H − j I Λj κ2 − RΛΣ j kl & 4 ' = D πi . (A.6c) FΛ − i Λ Λ Since the canonical momenta29 conjugate to the fields N, Ni,anda vanish identically, the corre- sponding Hamilton equations lead to the first class constraints

= i = =0, (A.7) H H FΛ which reflect respectively diffeomorphism invariance under radial reparameterizations, diffeomor- Λ phisms along the radial slices Σu and a U(1) gauge invariance for every gauge field Ai .

Hamilton-Jacobi formalism The first class constraints (A.7) are particularly useful in the Hamilton-Jacobi formulation of the dynamics, where the canonical momenta are expressed as gradients of Hamilton’s principal function [γ,AΛ, ϕI ]as S ij δ i δ δ π = S , πΛ = SΛ , πI = SI . (A.8) δγij δAi δϕ Since the momenta conjugate to N, N ,andaΛ vanish identically, the functional [γ,AΛ, ϕI ] i S does not depend on these Lagrange multipliers. Inserting theexpressions(A.8)forthecanonical

30 The canonical momenta conjugate to the induced fields on Σu following from the Lagrangian (A.2) are

The canonical momenta conjugate to the induced fields on Σu following from the Lagrangian ij δL 1 ij ij (A.2) are π = = 2 √ γ Kγ K , (A.4a) δγ˙ij 2κ4 − − δL 1 ! " ij δL 1 π =ij =ij N 1√ γ ϕ˙ J N i∂ ϕJ , (A.4b) π = = √ γ KI γ I K , 2 − IJ i (A.4a) 2 δϕ˙ −2κ4 − G − δγ˙ij 2κ4 − − i δL 2 1 ! ij Σ "Σ Σji 2 ijk Σ δL 1 !π = = " N − √ γ γ A˙ ∂ a N F √ γ ϵ F . (A.4c) 1√Λ ˙Λ −Jκ2 i −J ZΛΣ j − j − j − κ2 − RΛΣ jk πI = I = 2 N − γδAiIJ ϕ˙ 4 N ∂iϕ , 4 (A.4b) δϕ˙ −2κ4 − G − # # $ $ ! " Λ i δL 2 Notice1 that the momentaij ˙Σ conjugateΣ to N, NΣiji,anda2 vanish identically,ijk Σ since the Lagrangian (A.2) πΛ = = N − √ γ ΛΣ γ Aj ∂ja NjF √ γ ΛΣϵ Fjk. (A.4c) Λ ˙Λ 2 does not contain any radial derivatives of these fields.2 It follows that the fields N, Ni,anda are δA −κ4 − Z − − − κ4 − R i Lagrange multipliers,# # implementing$ three first$ class constraints, which we will derive momentarily. The canonical momenta (A.4) allowΛ us to perform the Legendre transform of the Lagrangian (A.2) Notice that the momenta conjugate to N, Ni,anda vanish identically, since the Lagrangian (A.2) to obtain the radial Hamiltonian Λ does not contain any radial derivatives of these fields. It follows that the fields N, Ni,anda are Lagrange multipliers, implementing three first class constraints, which we will derive momentarily. H = d3x πijγ˙ + π ϕ˙ I + πi A˙ Λ L = d3x N + N i + aΛ , (A.5) ij I Λ i − H iH FΛ The canonical momenta (A.4) allow us% to perform the Legendre transform% of the Lagrangian (A.2) # $ ! " to obtain the radial Hamiltonianwhere 3 ij Iκ2 i ˙ Λ 1 3 i Λ The canonicalH = d momentax π γ˙ ij conjugate+=πI ϕ˙ 4+ toπΛ2 theAiγ inducedγ L =γ fieldsγd onxπijNΣπuklfollowing++ NIJi(ϕ)π+ fromπa theΛ Lagrangian, (A.5) H − √ γ ik −jl − 2 ij kl H G H I J F (A.2) are % # − & & $ % ' 1 2 ! 2 " where + ΛΣ(ϕ) π + √ γ (ϕ)ϵ klF M πi + √ γ (ϕ)ϵipqF N δL 1 Λi 2 ΛM i kl Σ 2 ΣN pq ij √ ij 4Zij κ4 − R κ4 − R π = =2 2 γ Kγ K , & '& (A.4a) '' δγ˙ij 2κ − − κ4 4 1 √ γ ij kl 1 IJ I i J Λ Σij = 2 γik!γjl +γij−γ2"kl πR[πγ]++ IJ((ϕϕ))∂πiϕI π∂J ϕ + ΛΣ(ϕ)Fij F , (A.6a) H −δL√ γ 1 1 − 2 2κ J − i J 2GG Z πI = = N − √ γ IJ ϕ˙4 & N ∂iϕ , ' (A.4b) δϕ˙I −−2&κ2 & − G −' 1 ΛΣ 4 i 2 ij i klI MΛij i 22 klipqΣ N = 2Djπ + πI∂ ϕ + F πΛj + √ γ ΛΣ(ϕ)ϵj F , (A.6b) +δL (ϕ2) πΛi + 2 √ ! γ ΛM (ϕ)ϵi " Fkl πΣ + 22√2 γ ΣN (ϕ)ϵ klFpq i 4 1 H κ− ij ˙Σ Σ Σji κκ4 − R ijk Σ πΛ = Z= N − √ γ 4ΛΣ −γ R Aj ∂ja Nj&F 4 −√ Rγ ΛΣϵ Fjk'. (A.4c) ˙Λ −κ2 & − Z i − −'& − κ2 − R '' δAi 4 Λ = Diπ . 4 (A.6c) √ γ F 1 − # Λ # I i J $ Λ $Σij + −2 R[γ]+ IJ(ϕ)∂iϕ ∂ ϕ + ΛΣ(ϕ)Fij F , (A.6a) 2κ − 2G ΛZ Λ Notice that the momenta4 &Since conjugate the canonical to N momenta, Ni,and conjugatea vanish to identically, the fields' sinceN, N thei,and Lagrangiana vanish (A.2) identically, the corre- does not contain anysponding radial derivatives Hamilton equations of these2 fields. lead to It the follows first class that constraithe fieldsntsN, N ,andaΛ are i = 2D πij + π ∂iϕI + F Λij π + √ γ (ϕ)ϵ klF Σ , i (A.6b) Lagrange multipliers,j implementingI threeΛ firstj class2 constΛΣraints, whichj kl we will derive momentarily. H − κ4 − R i The canonical momenta (A.4) allow us& to perform the Legendre= transform= Λ'=0 of, the Lagrangian (A.2) (A.7) = D πi . H H F (A.6c) to obtainΛ theHamiltonian: radiali Λ Hamiltonian F − which reflect respectively diffeomorphism invariance under radial reparameterizations, diffeomor- Since the canonical momentaphisms along conjugate the radial to the slices fieldsΣ andN, aNU,and(1) gaugeaΛ invariancevanish identically, for every gauge the corre- field AΛ. H = d3x πijγ˙ + π ϕ˙ I + πi A˙ Λ uL = di3x N + N i + aΛ , (A.5) i sponding Hamilton equations leadij toI the firstΛ classi − constraints H iH FΛ % % Hamilton-Jacobi# formalism$ ! " where i First class constraints = = Λ =0 , - Hamilton Jacobi eqs.: (A.7) 2 The firstThe class canonical constraintsH momentaH (A.7)F conjugate are particularly to the useful induced in the fields Hamilton-Jacobi on Σu following formulation from the of Lagrangian the κ4 dynamics,(A.2) are where1 the canonicalij kl momentaIJ are expressed as gradients of Hamilton’s principal function which reflect& respectively= Momenta 2as di ffγgradientsikeomorphismγjl γij ofγkl Hamilton’s invarianceπ π + underprincipal(ϕ)π rIadialπJ function reparameterizations, S(γ,AΛ,φI): diffeomor- H − √ γ [γ,AΛ, ϕI ]as− 2 G − & & δL 1' Λ phisms along the radialS slices ijΣu and a U(1)√ gauge invarianceij ij for every gauge field Ai . 1 ΛΣ π = 2 = 2 γ Kklijγ M δ K i , i 2 δ δipq N (A.4a) δγ˙ij 2κ − π = −S , π = S , πI = S . (A.8) + (ϕ) πΛi + 2 √ γ 4ΛM (ϕ)ϵi Fkl πΣ +Λ 2 √ Λγ ΣN (ϕ)ϵ IFpq 4Z κ − R δγij κ δA−i R δϕ δL4 1 1! "J 4i J & π = = N − √ γ '&ϕ˙ N ∂ ϕ , '' (A.4b) Hamilton-Jacobi w/ original. formalism I I 2 IJ Λ i Λ I √ γSince the momentaδ1ϕ˙ − conjugate2κ4I i J to− NG, Ni,andΛ−Σija vanish identically, the functional [γ,A , ϕ ] + −2 R[γ]+ δLIJ(ϕ)∂i2ϕ ∂ ϕ + ΛΣ(ϕ!)Fij F , " 2(A.6a) S The first class constraints2κ4 does− (A.7) notπi depend= are2G particularly on= theseN Lagrange1√ usefulγZ multipliers. in theγij Hamilton-JacobiA˙Σ Inserting∂ aΣ thNeexpressions(A.8)forthecanonical formulationF Σji √ ofγ the ϵijkF Σ . (A.4c) & Λ ˙Λ 2 − ΛΣ j ' j j 2 ΛΣ jk δA −κ4 − Z − − − κ4 − R dynamics, where the canonical momentai are expressed2 as grad# ients# of Hamilton’s$ principal$ function i = 2D πij + π ∂iϕI + F Λij π + deBoer,Verlinde√ γ (ϕ2’)99,ϵ kl…FSkenderis&Papadimitriou’04,Σ , (A.6b)… Λ I j I Λj 2 ΛΣ j kl Λ [γ,A , ϕ H]as − Notice that the momentaκ conjugate4 − R to N, Ni,anda vanish identically, since the Lagrangian (A.2) S Solve asymptotically& (for `vacuum’ asymptotic' sol.) for Λ i does notij containδ any radiali δ derivatives ofδ these30 fields. It follows that the fields N, Ni,anda are Λ = DiπΛ. π = S , π = S , π = S . (A.6c)(A.8) F − Lagrange multipliers,ΛΛ implementingI Λ threeI firstI class constraints, which we will derive momentarily. δγ ijS(γ,A ,φ ) δAi= - Sct δϕ ! The canonical momenta (A.4) allow us to performΛ the Legendre transform of the Lagrangian (A.2) Since the canonical momenta conjugate to the fields N, Ni,anda vanish identically, the corre- Since the momentaΛ conjugateI to obtain to theN, radialN ,and HamiltonianaΛ vanish identically, the functional [γ,AΛ, ϕI ] sponding Hamilton S(γ,A ,φ equations) coincides lead with to the thei first on-shell class constrai action,nts up to terms that remain S does not depend on these Lagrange multipliers.3 Insertingij thI eexpressions(A.8)forthecanonicali Λ Λ I 3 i Λ finite as Σu → ∂M. InH particular,= d x πdivergentγ˙ ij + πI ϕ ˙part+ π ofA ˙S[γ, AL, =φ ] coincidesd x N + Ni + a Λ , (A.5) i Λ i − H H F with that of the on-shell action.% = = Λ =0, % (A.7) H H# F $ ! " where which reflect respectively diffeomorphism2 invariance under radial reparameterizations, diffeomor- κ4 30 1 ij kl IJ Λ phisms along the radial slices Σ =and a U(1)2 gaugeγikγ invariancejl γijγkl for everyπ π gauge+ ( fieldϕ)πIAπJ. Hu − √ γ − 2 G i − & & ' 1 2 2 + ΛΣ(ϕ) π + √ γ (ϕ)ϵ klF M πi + √ γ (ϕ)ϵipqF N Hamilton-Jacobi formalism 4Z Λi κ2 − RΛM i kl Σ κ2 − RΣN pq & 4 '& 4 '' √ γ 1 The first class constraints (A.7) are+ particularly− R[ usefulγ]+ in the(ϕ) Hamilton-Jacobi∂ ϕI ∂iϕJ + ( formulationϕ)F ΛF Σij of, the (A.6a) 2κ2 − 2GIJ i ZΛΣ ij dynamics, where the canonical momenta are4 & expressed as gradients of Hamilton’s principal function' Λ I i ij i I Λij 2 kl Σ [γ,A , ϕ ]as = 2Djπ + πI∂ ϕ + F πΛj + √ γ ΛΣ(ϕ)ϵj F , (A.6b) H − κ2 − R kl S δ δ &δ 4 ' ij i i πΛ == DSiπ, Λ. πΛ = SΛ , πI = SI . (A.8) (A.6c) F −δγij δAi δϕ Since the canonical momenta conjugate to the fields N, N ,andaΛ vanish identically, the corre- Since the momenta conjugate to N, N ,andaΛ vanish identically, the functionali [γ,AΛ, ϕI ] sponding Hamiltoni equations lead to the first class constraints S does not depend on these Lagrange multipliers. Inserting theexpressions(A.8)forthecanonical = i = =0, (A.7) H H FΛ which reflect respectively diffeomorphism invariance under radial reparameterizations, diffeomor- Λ phisms along the radial slices30 Σu and a U(1) gauge invariance for every gauge field Ai .

Hamilton-Jacobi formalism The first class constraints (A.7) are particularly useful in the Hamilton-Jacobi formulation of the dynamics, where the canonical momenta are expressed as gradients of Hamilton’s principal function [γ,AΛ, ϕI ]as S ij δ i δ δ π = S , πΛ = SΛ , πI = SI . (A.8) δγij δAi δϕ Since the momenta conjugate to N, N ,andaΛ vanish identically, the functional [γ,AΛ, ϕI ] i S does not depend on these Lagrange multipliers. Inserting theexpressions(A.8)forthecanonical

30 space of equivalence classes of boundary data. We have therefore determined that a complete set spaceof boundary of equivalence counterterms classes for of the boundary variational data. problem We have in the theremagneticfore determined frame is that a complete set of boundary counterterms for the variational problem in the magnetic frame is 1 B 4 α α 1 3x√ η/2 η 2η ij 4η 0 0ij space of equivalence classesSct = of12 boundaryd data.γ Be We have4−2 α+( thereα fore1)e− determinedR[γ] αe− thatFij aF complete+ e1− F setijF . (4.11) S = −κ d3x√− γ 4 eη/2 B +(α− 1)e ηR[γ−] 2 e 2ηF F ij +4 e 4ηF 0 F 0ij . (4.11) ct 42 ! " −2 − − ij − ij # of boundary counterterms for−κ the4 variational− 4 problemB in the magnetic− frame− 2 is 4 As we mentioned! at the beginning" of this section, the freedom to choose the value of the parameter# 1 3 Asα does weB mentioned notη/2 correspond4 α at the to beginning a choiceη of of scheme. thisα section, Instead,2η theij itfreedom is1 a dir4toectη choose0 consequence0ij the value of the of the presence parameter of S = d x√ γ e − +(α 1)e− R[γ] e− F F + e− F F . (4.11) ct −κ2 αthe−does second4 not class correspondB2 constraints to− a choice(4.4). The of scheme.− scheme2 Instead, dependenceij it is4 c aorresponds directij consequence to the freedom of the to presence include of 4 " # ! theadditional second finite class local constraints terms, (4.4). which The do not scheme affect dependence the divergen ctpartofthesolution.Lateronweorresponds to the freedom to include As we mentioned at theadditionalwill beginning consider finite situations of local this section, terms, where which additional the freedom do not conditions atoffect choose the on divergen the thevariationalproblemrequireaspecific valuetpartofthesolution.Lateronwe of the parameter α does not correspondwillvalue to consider a for choiceα,orparticularfinitecounterterms. situations of scheme. where Instead, additional it is a conditions direct consequence on thevariationalproblemrequireaspecific of the presence of the second class constraintsvalue for (4.4).α,orparticularfinitecounterterms. The scheme dependence corresponds to the freedom to include additional finite local4.1.2 terms, The which variational do not aff problemect the divergentpartofthesolution.Lateronwe will consider situations4.1.2 where The additional variational conditions problem on thevariationalproblemrequireaspecific Given the counterterms S ,theregularizedactioninthemagneticframeisdefinedasthesumof value for α,orparticularfinitecounterterms.ct Giventhe on-shell the counterterms action (2.2) andSct,theregularizedactioninthemagneticframeisdefinedasth the counterterms (4.11) on the regulating surface Σu,namely esumof the on-shell action (2.2) and the counterterms (4.11) on the regulating surface Σu,namely 4.1.2 The variational problem Sreg = S4 + Sct. (4.12) Sreg = S4 + Sct. (4.12) Given the countertermsAs isSct shown,theregularizedactioninthemagneticframeisdefinedasth in appendix B, the limit esumof the on-shell action (2.2)As is and shown the`Hamiltonian in counterterms appendix B, Formalism (4.11) the limit on the regulatingwith `Renonormalized‘ surface Σu,namely Action Sren =limSreg, (4.13) r →∞ Sreg = S4 + Sct. Sren =limSreg, Finite-independent(4.12) of r (4.13) r exist and we will refer to its value as the ‘renormalized’→∞ on-shell action, following standard termi- As is shown in appendixexistnology B, and in thespace the we limit will of context equivalence refer of to the its classes AdS/CFT value of boundary as the duality. ‘renormalized’ data. We have on- thereshellfore action, determined following that a complete standard set termi- of boundaryCovariant counterterms S calculated for the variational for vacuum problem asymptotic in the magnetic sol.: frame is nologyAgenericvariationoftherenormalizedon-shellactiontak in the context of thect AdS/CFT duality. es the form Agenericvariationoftherenormalizedon-shellactiontak1 Sren =limB Sreg4 , α α es the form1 (4.13) 3x√ r η/2 η 2η ij 4η 0 0ij Sct = 2 d γ e −2 3 +(αij 1)e− R[γi ] Λ e− FijFI + e− FijF . (4.11) −κ δS−ren =lim4→∞ Bd x Π −δγij + Π δ−A2 + ΠI δϕ , 4 (4.14) 4 ! r " Λ i # exist and we will refer to its value as the ‘renormalized’→∞ on-3shellij action, followingi Λ standardI termi- As we mentioned atδ theSren beginning=lim! of thisd x section,Π δγij the+ freedomΠΛδAito+ chooseΠI δϕ the, value of the parameter(4.14) Renormalized canonicalr momenta:$ % nology in the contextwhere of the theα AdS/CFTdoes renormalized not correspond duality. canonical to a choice momenta→∞ ! of scheme. are given Instead, by it is a direct consequence of the presence of $ % Agenericvariationoftherenormalizedon-shellactiontakwhere thethe renormalized second class constraints canonical (4.4). momenta The schemees are the given dependence form by corresponds to the freedom to include ij ij δSct i i δSct δSct additional finiteΠ local= π terms,+ which, doΠ not= aπffect+ the divergen, ΠI tpartofthesolution.Lateronwe= πI + . (4.15) δγ Λ Λ δAΛ δϕI will consider situations3ij ijij whereδS additionalijct i Λi conditionsi Iδ onSi ct thevariationalproblemrequireaspecificδSct δSren =lim dΠx =Ππ δγ+ij + Π,ΛδAΠi += Ππ I δϕ+ , , Π = π + (4.14). (4.15) valuer for α,orparticularfinitecounterterms.δγ Λ Λ Λ I I δϕI Inserting the asymptotic→∞ ! form of theij backgrounds (3.6)δA intoi the definitions (A.4) of the canonical $ % where the renormalizedInsertingmomenta canonical4.1.2 the and asymptotic in momenta The the variational functional are form given ofproblem derivatives the by backgrounds (4.8) we obtain(3.6) into the definitions (A.4) of the canonical momentaGiven and the in the counterterms functionalSct derivatives,theregularizedactioninthemagneticframeisdefinedasth (4.8) we obtain 2 esumof ijt ijkℓ δS1ct i αi 2δS2 ct2 δSct φ k ℓω ΠΠ t= πthe+ on-shell(r, action+ +Πr (2.2))+= π and−+ theℓ ω counterterms,(1 +Π 3I cos= π (4.11)2θI)+sin onθ the,. regulatingΠ t surface√r(4.15)Σ+r,namelysin θ, (4.16a) ∼−2κ2 δγ4 Λ− Λ 8 Λ δϕI ∼−2κ2 u − t kℓ4 " 1ij α 2δA2i 2 # φ k4ℓω Π t 3 2 (r+ + r )+ − ℓ ω (1 + 3 cos 2θ) sin3θ,2 Π √r+r sin θ, (4.16a) kℓ ω2 Sreg =φ S4 + Skctℓ. ω t 2 (4.12) θ ∼−2κ4 4 − 8 ∼−2κ4 − Inserting the asymptoticΠ θ form of2 the"((2 backgrounds5α)cos2θ +2 (3.6)3α into)sinθthe, Π definitionsφ # (A.4)2 ((5α of4) the cos canonical 2θ +3α)sinθ, (4.16b) ∼ 163κ42 − − ∼−16κ34 2 − momenta and in the functionalθ Askℓ isω derivatives shown in appendix (4.8) we B, the obtain limit φ kℓ ω Π θ ((24 5α)cos2θ +2 3α)sinθ, Π ((54α 4) cos 2θ +3α)sinθ, (4.16b) 0t ∼ 161κ2 ℓ − −2 2 2 0φφ ∼−161κ2kωℓ − Π 4 sin θ √r+r +3ω ℓ cos θS , =limΠ S , 4 (r+ + r )sinθ, (4.13)(4.16c) 2 3 ren 2reg 2 3 kℓ 1 ∼−α2κ4 B24 − r ∼−k ℓω2κ4 2B 4 − t 0t 1 ℓ 2 2 2 2 2 φ →∞0φ 1 kωℓ Π (r+ + r )+ − ℓ3 ω (1 + 3 cos 2θ) sin θ, 2 Π3 √r+r sin θ, (4.16a) t 2 Π exist1 2 and3ωℓ3 wesin willθ$ √ referr+r to its+3 valueω1ℓ 2 ascosαk theωθ% ‘renormalized’ℓ , t Π 2 on-shell2 action,3 (r+ + followingr )sin standardθ, termi-(4.16c) ∼−2κ4 4 t−∼−2κ8 B φ− ∼−∼−2κ4 2κ 2B− − " Π nology24 in thesin context 2θ, Π of the AdS/CFT# 2 duality.sin 2θ, 4 (4.16d) 3 2 ∼−2κ4 B 3 $ ∼−2κ4 3B2 2% 3 θ kℓ ω t 1Agenericvariationoftherenormalizedon-shellactiontak3ωℓ φ φ 1 k2ℓαωkω ℓ es the form Π θ 2 ((2 5αΠ)cos2θ +212 kℓ 3αsin)sin 2θ,θ, ΠΠ φ 2 22 2 ((5αsin 24)θ, cos 2θ +3α)sinθ, (4.16b) (4.16d) ∼ 16κ4 − Πη ∼−2κ4 −Bsin θ 6(r+ + r ∼−)+∼−ℓ2κω416((13κB4α 18)− cos 2θ +7α 6) , (4.16e) ∼−2κ2 8 − 3 − ij i Λ − I 4 4 δSren =lim d x Π4 δγij + Π δA + ΠI δϕ , (4.14) 1 ℓ 1 kℓ r2 2 1 kωℓ Λ i Π0t sin θ Πη r r +3ω2sinℓ2 cosθ$ 6(2 rθ+ ,+ rΠ)+0φ ℓ →∞ω ((13! α 18)(r cos+ 2rθ +7)sinαθ, 6)% , (4.16c) (4.16e) 2 3 with√∼− all+ other2κ2 8 components vanishing− identically.2 −$ 3 + − % ∼−2κ4 B where− 4 the renormalized canonical∼− momenta2κ4 2B are given by− $ % 1 3ωℓ3 with$ all other components1 2αkω2%ℓ vanishing3 identically. Πt sin 2θ, Πφ ijsin 2ijθ, δSct i i δSct δSct (4.16d) 2 2 Π = π + , ΠΛ = πΛ + Λ , ΠI = πI + I . (4.15) ∼−2κ4 B ∼−2κ4 B δγij 13 δAi δϕ 1 kℓ Inserting the2 2 asymptotic form of the backgrounds13 (3.6) into the definitions (A.4) of the canonical Πη 2 sin θ 6(r+ + r )+ℓ ω ((13α 18) cos 2θ +7α 6) , (4.16e) ∼−2κ4 8 momenta− and in the functional− derivatives (4.8)− we obtain $ % 2 with all other components vanishingt k identically.ℓ 1 α 2 2 2 φ k ℓω Π t (r+ + r )+ − ℓ ω (1 + 3 cos 2θ) sin θ, Π t √r+r sin θ, (4.16a) ∼−2κ2 4 − 8 ∼−2κ2 − 4 " # 4 3 2 3 2 θ kℓ ω φ kℓ ω Π θ 2 ((2 5α)cos2θ +2 3α)sinθ, Π φ 2 ((5α 4) cos 2θ +3α)sinθ, (4.16b) ∼ 16κ4 − 13 − ∼−16κ4 − 4 4 0t 1 ℓ 2 2 2 0φ 1 kωℓ Π 2 3 sin θ √r+r +3ω ℓ cos θ , Π 2 3 (r+ + r )sinθ, (4.16c) ∼−2κ4 B − ∼−2κ4 2B − 3 $ 2% 3 t 1 3ωℓ φ 1 2αkω ℓ Π 2 sin 2θ, Π 2 sin 2θ, (4.16d) ∼−2κ4 B ∼−2κ4 B 1 kℓ 2 2 Πη 2 sin θ 6(r+ + r )+ℓ ω ((13α 18) cos 2θ +7α 6) , (4.16e) ∼−2κ4 8 − − − $ % with all other components vanishing identically.

13 Finally, we can use these expressions to evaluate the variation (4.14) of the renormalized action in terms of boundary data. To this end we need to perform the integration over θ and remember that the magnetic potential A is not globally defined, as we pointed out in footnote 6. In particular, taking A B(cos θ 1)dφ and A B(cos θ +1)dφ we get north ∼ − south ∼ 1 3 3 δSren = dtdφ (r+ + r )kℓδ log kB /ℓ , (4.17) −2κ2 − 4 ! " # independentlyFinally, we of can the use value these expressions of the parameter to evaluate theα variat.Notethatthecombinationion (4.14) of the renormalized actionkB3/ℓ3 of boundary in terms of boundary data. To this end we need to perform the integration over θ and remember datathat is the the magnetic unique potential invariantA is under not globally both defined, the equivalence as we pointed out clas instransformation(3.11)andthetrans- footnote 6. In particular, taking A B(cos θ 1)dφ and A B(cos θ +1)dφ we get formation (3.12).north ∼ We have− thereforesouth ∼ demonstrated that by adding the counterterms (4.11) to the bulk action, the variational problem1 is formulated in terms3 3 of equivalence classes of boundary data δSren = dtdφ (r+ + r )kℓδ log kB /ℓ , (4.17) −2κ2 − under the transformations (3.11)4 and! (3.12). This is an explicit demonstration of the general result " # thatindependently formulating of the the value variational of the parameter problemα.Notethatthecombination in terms of equivalencekB3 classes/ℓ3 of boundary of boundary data under radialdata reparameterizations is the unique invariant under is achieved both the equivalence via the same classtransformation(3.11)andthetrans- canonical transformation that renders the on- shellformation action finite. (3.12). We As have we therefore will now demonstrated demonstrate, that by the adding same the boun countertermsdary terms (4.11) to ensure the the finiteness of Angular velocitybulk action, the variational problem is formulated in terms of equivalence classes of boundary data the conserved charges, as well as the validity of the first law of thermodynamics. We define theunder physical the transformations (diffeomorphism (3.11) invariant) and (3.12). angular This isvel anocity expl asicit the demonstration difference between of the general the result angular velocitythat at formulating the outer horizon the variational and at problem infinity, in namely terms of equivalence classes of boundary data under 4.1.3radial Conserved reparameterizations charges is achieved via the same canonical transformation that renders the on- shell action finite. As we will nowg demonstrate,g the same boundary terms ensure the finiteness of Similarly, the momentum constrainttφ tφ i =0in(A.6),whichcanbewritteninexplicitformasr the conservedΩ4 charges,= ΩH asΩ well= as the validity of the= firstωk law of− thermodynamics.. (5.3) − ∞ gφφ ∂ − gφφH + r+ Let us now consider conserved! chargesM ! associatedH " with local conserved currents. This includes j ! ! 0 0j j electric charges, as2D welljπ + asπη conserved∂iη + πχ∂iχ quantities+ Fijπ + relatedFijπ toasymptoticKillingvectors.Magnetic In the coordinate4.1.3 system Conserved (3.6)− there chargesi is no contribution! to! theangularvelocityfrominfinity,but there is incharges the original do not coordinate fall in1 system this category, (3.1). The but rotatii theyon3 at can infinity be describe was not takendinthislanguageintheelectricframe,3 into account Similarly,Let us the now momentumconsider conserved√ constraintjkl charges3 0 associated0 =0in(A.6),whichcanbewritteninexplicitformas with2 0 local conserved currents.2 This0 includes in [21], which is why our result+ does2 notγϵ fully agreeχ Fij withFHkl + theχ oneFij obtainedFkl +3χ there.FijFkl + χ FijFkl =0, (4.21) as weelectric shall charges, see later as2κ well on.4 as− conserved quantities2 related toasymptoticKillingvectors.Magnetic2 Angular velocity j ! 0 0j j " Incharges the do radial not2D falljπ Hamiltoniani in+ thisπη category,∂iη + π formulation butχ∂i theyχ + canFij beπ of describe+ theFijdinthislanguageintheelectricframe, bulkπ dynamics, the presence of local conserved We defineElectric the physical chargesleads (diff toeomorphism− finite conserved invariant) charges angular associated velocity as with the di asymptotifference betweencKillingvectors.Notethattheterms the currentsas we is shall a see direct1 later on. consequence of the3 first class constraints Λ3 =0and i =0in(A.6).7 As angular velocity at thein outer theIn secondthe horizon radial line and Hamiltonian are at infinity, independentjkl formulation namely3 0 of0 the of the canonical2 bulk0 dynamics, momentathe a presencend originate of2 local in conserved the0 parity odd terms In the magnetic frame+ there2 is√ onlyγϵ one non-zeroχ FijF electrickl + chaχ rgeFij givenFkl +3 byχ (4.20),FijFkl whose+F valueχ Fij isFkl H=0, (4.21) in theincurrents the case STU is of a2 modelκ directasymptotically− consequence Lagrangian. of AdS9 theHowever, first backgrounds, class2 for constrain asymptotically thesets Λ =0and constr conicalaintsi backgrounds2=0in(A.6). lead respectively7 ofAs the form(3.6) to conserved 4 g g r F H in the case of asymptoticallytφ ! AdStφ backgrounds,4 these constraints lead respectively to conserved8 " electrictheseΩ4 charges= termsΩH0( aree)Ω and asymptotically= charges2 associated0 subleading,t =ℓωk with the− . asymptotic most domina2 2 Kintlling term(5.3) vectors.being In particular, the gauge Q − =∞ gφφ ∂ d−xgΠφφ =+ r√+r r + ω ℓ . 8 (5.4) leads toelectric finite charges conserved4 Λ and charges chargesM associated associatedH with asymptotic3 with+ asymptoti Killing vectors.cKillingvectors.NotethatthetermsIn particular, the gauge constraints in =0in(A.6)taketheform− ∂ ! C ! 4G4B" − constraints in AngularΛ =0in(A.6)taketheform# velocityM!∩ ! jkl 1 In the coordinatein the system second (3.6) line thereF areF is independent no contribution! of to! the th√eangularvelocityfrominfinity,but canonicalγϵ $ χFijF momentakl = (%r− ) a,nd originate in the parity odd(4.22) terms In the electric frame both electric charges defined in (4.45) are non-zero: there is in the original coordinate systemWe define (3.1). the The physical9 rotati (dionffeomorphism ati − infinityi invariant) was0i not angular takenO 0 intoi velocity account as the difference between the in the STU model Lagrangian. However,Diπ =0D fori,Dπ asymptotically=0iπ ,D=0, iπ =0 conical, backgrounds(4.18) of the form(3.6)(4.18) in [21], which is why our result doesangular not fully velocity agree at with the outer the horizon one obtained and at infinity, there. namely and so the momentum(e) constraint asymptotically3B 0(e) reduces0(e) to these termswhere areπi and asymptoticallyQ π0=i are respectivelyd2 subleading,x πt the= canonical, the momentaQ most= dominaQ conjugate. nt to term the gau beingge fields(5.5)A and i 4 0i g4tφ g4tφ r i where0 π and π are− ∂ respectivelyC 4 theG4 canonical momenta conjugate− to the gauge fields Ai and Electric chargesConservedA .Sincetheboundarycounterterms(4.11)aregaugeinvarian Charges# : Ω4j= ΩH Ω = 0t, it0j= followsωk fromj. (4.15) that these(5.3) 0 i M∩ 2Djπ + πη−jkl∂iη∞+ πgφφχ∂∂iχ −+gFφφ π+ 1+ Fijr+π 0. (4.23) √i ! M ij!H " Ai .Sincetheboundarycounterterms(4.11)aregaugeinvarianconservation& laws hold for the− corresponding& γϵ renormalizedχFij& F!kl =momenta(!r− as), well, namelyt,≈ it follows from (4.15) that(4.22) these In the magneticMagneticConserved frame charge there currents, is onlyIn one the a non-zero coordinateconsequence electric system cha (3.6) of−rge therethe given first is no by contributionclass (4.20),! constraints whoseO to! theangularvelocityfrominfinity,but value is conservationSince the counterterms laws hold for (4.11) the correspondingare invarianti with renormalized0i respect to diffmomentaeomorphisms as well, along namely the surfaces of there is in the original coordinateDiΠ =0 system,D (3.1).iΠ The=0 rotati. on at infinity was not taken into(4.19) account Theand only so non-zero the momentum magnetic charge constraint is present4 asymptotically in the magnetic frame reduces and to it is equal to one of the constant0(e) radialin [21], coordinate,2 which0t is why itℓ our follows result does from not fully (4.15)2 2 agree that with thi thesconstraintholdsfortherenormalized one obtained there. Q = d x Π = √r r + ωi ℓ . 0i (5.4) electricF chargesΛ = 0This Conserved4 in implies the electric currents that the frame: for quantities gauge potentials3 : + Mass momenta− as∂ well,C 4G4B D− iΠ =0,DiΠ =0. (4.19) # MElectric∩ charges j 0 0j j 2Djπi3+j$πη∂iη + πχ∂%iχ + Fij0π 0j+ Fijπ j 0. (4.23) In the electric frameConserved both electric charges charges: defined−(me)) in2D (4.45)jΠi +are2xΠ non-zero:ηt∂iη + Π(0(eχ)e∂) iχ + FijΠ 2+11x F0ijt Π ≈ 0. (4.24) TheThis mass implies is the conserved thatIn the the magneticQQ quantities charge4 ==− frame associated there2 d is onlyΠ withF,Q one= non-zero theQ44 . Killing= electricvector charged givenΠζ by=, (4.20),≈∂t whose(5.6)Ω value∂φ(4.20).Since is −−2∂κ4 C∂ C 12 − ∂ C − − ∞ Ω =0inthecoordinatesystem(3.6),(4.26)givesGiven an asymptotic Killing! M∩# vectorM∩ ζi satisfying the! M asymptotic∩ conditions Since∞ the counterterms(e) 2 (4.11)t 3 areB invariant0(e) with0(e) respectℓ4 to diffeomorphisms along the surfaces of whereQ =C denotes a Cauchyd x π = surface, that0(e) extendsQ = toQ the2&x . boundary0t ∂ ,arebothconservedandfinite(5.5)2 2 4 (e) Q4 = 4 2 d4 tΠ = 0(3 e√) r+r + ω ℓ . 2 0t (5.4) Electricconstant potential radial− coordinate,∂ C Q it=4 followsG4 from− ∂ d (4.15)Cx Π ,Q that4G4B thiℓMksconstraintholdsfortherenormalized=− d x Π , (4.20) and correspondγ # =MD∩ toζ the+ electricD4 ζ charges2 0, associatedt A#ΛM=t∩ ζ0j to∂ theseAΛt + gaugeAΛ4∂ fields.ζj 0, ϕI = ζi∂ ϕI 0, (4.25) ζ ij M4 =i j j i d −x 2∂Π ζt +Ci Π0At +j Πi At =j $i (r+−+ r∂ %)ζ . C i (5.9) Wemomenta define the electric& as7 L well, potentialIn the− as electric∂& frameC ≈ both! electricML &∩ charges defined in (4.45)8Gare4 ≈ non-zero:! −LM∩ ≈ Magnetic charge These constraints can! beM derived∩ alternativelyj by applying the general variation0 0j (4.14) of thej renormalized action to ConservedU(1) gauge transformations currents: 2D andΠ transverse+" Π di∂ffeomorphisms,η + Π ∂ assumingχ +# F theΠ invariance+ F ofΠ the renormalized0. action (4.24) Hwherei = 0the C conservationdenotes a identityCauchy (4.24)j surfacei implies(e) thatη i that extendsχ thei quantity to3B theij boundary0(e) ij0(e) ∂ ,arebothconservedandfinite The sameunder result such transformations. is obtained− inThis the method electricQ will= be frame used in using3 orded2xrtoderivetheconservedchargesintheelectricframe.πt = (4.50)., AsQ for= theQ electric. ≈ and magnetic(5.5) The only non-zero magnetic charge is present0(e) in the0 magnetici 4 frameB andr it is equal to one4 of4 the M 8 Φ = A = k− ∂ C − , 4G4 H (5.7) potentials,andMass correspondIn the asymptotically gauge to potentials the locally4 electric AdS ati spaces, the charges boundary the Hamiltoniani# M associated do∩ not cons conttraint toribute these=0canbeusedinordertoconstruct to the gauge mass fields. in the gauge (3.6). electric charges inGiven the electric an asymptotic frame: KillingK vector+ ζ satisfyingℓ r+ the asymptotic conditions conserved charges associated with conformalH& Killing vectors2 of thet boundary0t data0& [22].t For asymptoticalj ly conical Conserved ``charges’’:Magnetic charge [ζ!]= ' (d x" &2Π + Π A + Π A ζ , (4.26) 7 backgrounds, the Hamiltonian constraint! leads to conservedchargesassociatedwithasymptotictransversedij j 11j ffeomor- TheThese mass constraints isi the(m conserved) can3 be derived chargeQ alternatively associated∂(e) C with by applying the Killingthe generalvector0 i variationζ = ∂ (4.14)t Ω of∂ theφ.Since renormalized action whereAngular= ∂tphisms,+ momentumΩH ∂ξφ,thatpreservetheboundarydatauptotheequivalenceclassisQ the null= generator of!F the= QΛ outer. horizon.j Λ NoteΛtransformations thatj A (5.6)is (3.11). constantI overi ∞I ζ γij = Diζj4The+ D onlyjζ non-zeroi 2 0, magneticζ A#i charge4M=∩ ζ is present∂jAi 12 in+ theA magneticj ∂iζ framei 0, and itζ isϕ equal−= toζ− one∂iϕ of the 0, (4.25) KtoΩU(1)=0inthecoordinatesystem(3.6),(4.26)gives gauge transformations−2κ ∂ andC transverse diffeomorphisms,Asymptotic$ Killingassuming vectorK the%ζ invariance of the renormalized action the horizon∞L [22] and so leadselectric to a charges well4≈# defined in theL electric electric frame: potential. However, as≈ we remarkedL i in the ≈ Theunder angularis suchboth momentum transformations. finite and conserved, is defined ThisM∩ i.e. method as the it is conserved independentwill be used charge in of orde the corrertoderivetheconservedchargesintheelectricframe. chospondingice of Cauchy to the surface KillingC.However,there vector previous section,8 the electric potential is not gauge& invariant. Under gauge transformations it is Electric potentialthe conservationare a few subtleties identity in (4.24) evaluating implies these that charges.(m) the3 quantity Firstly, Gauss’(e) ℓk theoremH used to prove conservation ζ = ∂φIn,whichgives asymptotically locally AdS spaces,2 theQt Hamiltonian14= t 0 t F cons= Qtraint. =0canbeusedinordertoconstruct(5.6) shifted by a constantMass: (see (4.27)) M4 = which compensatesd x 2Π the4t + corΠresponding0A2t + Π At shift=4 of the(r charges+ + r ) (4.26). (5.9) for the charges (4.26)− assumesconformal differentiability−2 ofκ4 the∂ integC rand8G across the− equator at the boundary. We definein the the electric Smarrconserved formulapotential charges and as the associated first law. with∂ C Killing vectors# M∩ of the boundary4 data [22]. For asymptotically conical ! M∩ & 3 backgrounds,If the gauge the HamiltonianpotentialsElectric potential are constraint magnetically2 leads"t 2 sourced,tot conserve0 t astdchargesassociatedwithasymptotictransversedi is0 thet # 0 caseωℓ fort Ai inj the magnetic frame, then ffeomor- Angular Momentum:i J4 = [ζ]=d x (23 Π d+ Πx A2Π+jΠ+AΠφ)=Aj + Π .Aj ζ , (5.10) (4.26) phisms,The sameξ ,thatpreservetheboundarydatauptotheequivalenceclass result is obtained in theB electricrφ frame0 φ using (4.50−).2G As fortransformations the electric and (3.11). magnetic Magnetic potentialthe gauge should0(e)We define0 bei chosentheQ∂ electricC such potential∂ thatC asAi is continuous across the4 equator. In particular, contrary potentials, theΦ4 gauge= Ai potentials! M=∩k at# theM∩ boundary− , do not contribute to(5.7) the mass in the gauge (3.6). to the variationalK problem+ weℓ discussedr+ earlier,$ the gauge that should% be used to evaluate these Similarly,The the same magnetic result is potential obtained is! inH defined the electric' in terms( " frame. of the gauge field A 3in the electric frame as is both finite and conserved, i.e. it is independent0(e) 0 i of theB choi icer of Cauchy surface C.However,there charges is the one given! in (3.6), and not the one discussed in− footnote 6. ! Φ4 = Ai =0k i , (5.7) where = ∂t + ΩAngularH ∂φ is the momentum null generator of the outer horizon. NoteK that+ Ai ℓis constantr+ over K are a few subtletiesSecondly, the in evaluating charges (4.26)ℓ thesek are not charges. generically Firstly,!H 14 invariantKr' G(auss’" under theorem the U(1) used gauge to prove transformations conservation the horizon [22]Free and energy so leads to a( wellm) definedi electric potential. However,! 2 as2 we remarked& in the Λ ΛΦ4 = AΛ i = (r r+)+2ω ℓ − . 0 i (5.8) for the charges (4.26)where assumes= ∂t + Ω diH ∂ffφ erentiabilityis the null generator of of! the the outer integ horizon.rand Note across that A theis equator constant over at the boundary. previous section, theTheA electrici angular potentialAi momentum+ ∂iα isK not.ThesegaugetransformationsthoughmustpreserveboththeK gauge is+ defined invar2B iant. as− the− Under conserved gauge transformations charger+ corresponding it is i K to the Killingradial vector gauge → the horizon! [22]H and so leads' to a well defined electric" potent( ial. However, as we remarked in the Finally,(4.3) the full and Gibbs the asymptotic free energy, Killing4,isrelatedtotherenormalizedEuclideanon-shellactioni conditions (4.25). Preserving the radial gauge impliesn that the shifted by aIf constant theζ = gauge (see∂φ,whichgives (4.27)) potentials whichprevious compensates& are section,! magnetically theG the electric corresponding potential sourced, is not shift gauge as of is invarthe the chargesiant. case Under (4.26) for gaugeAi transformationsin the magnetic it is frame, then the electric frame, where all charges! are electric. Namely, Λ in the Smarrthe formula gaugegauge and should the parameter first be law.shifted chosen must by a constant depend such (see that only (4.27))A oni whichis the continuous compensatestransverse the coordina across corresponding thetes, equator. i.e. shift ofα the(x charges)(seee.g.[22]),while In particular, (4.26) contrary $ 3 to the variationalrespecting the problemin Killing the Smarr symmetry we formula discussed and the leads2E first earlier, to law.t the conditiont the0 gauget thatωℓ should be used to evaluate these Magnetic potential J4 = I4 = dSrenx (2=Π φSren+ Π=0Aβ4φ +4,Π Aφ)= . (5.11)(5.10) ∂ C 21 − G −2G4 charges is the one givenMagnetic in potential (3.6),! M and∩ not thei oneΛ discussed in footnote 6. ζ ∂iα =constant. (4.27) Similarly, the magneticwith β4 =1 potential/T4 and is definedSren defined in terms in of$ (4.33). the$ gauge The field Euclidean$ Ai in the$ on-shell electric actionframe as in the magnetic frame Secondly,The same the result chargesSimilarly, is obtained (4.26) the magnetic in the are potential electric not generically is frame. defined in terms invariant of the gaugeunder field Ai thein theU electric(1) gauge frame as transformations similarlyΛ Λ defines anotherΛ thermodynamic potential, 4,through A A (m+) ∂iα .Thesegaugetransformationsthoughmustpreserveboththeℓk r radial gauge i Underi suchi gauge transformations(m) the2 2i chargesG&ℓk (4.26) are shi2 2 ftedr by the corresponding electric Φ4 = Ai $ = (r r+Φ)+2= ωA ℓ =− . (r r )+2ω ℓ (5.8)−& . (5.8) → − 4 i + (4.3)Free andcharges energy the asymptotic (4.20).K + As2 willB Killing become− conditions clearE inK section+r (4.25).+ 2B 5,− this Preser− compevingnsatesr the+ a radial related gauge shift in implies the electric that the !H ' I4 = Sren = !"HSren(= 'β4 4, " ( (5.12) ! −! G Λ gauge parameterpotential& such must that depend the Smarr only formula on the& and! transverse the first law coordina are gautes,ge invariant. i.e. α (x)(seee.g.[22]),while Nevertheless, gauge Finally, the full Gibbs! free energy, 4,isrelatedtotherenormalizedEuclideanon-shellactionin where Sren was given in (4.13). EvaluatingG this we obtain (see appendix B) respectingthe electric9In the the Killingframe, AdS/CFT where symmetry context all charges are interpreted leads are to electric. as the a gravitational condition Namely, anomaly in the dual QFT. β ℓk $ 21 r 21 4 i E Λ 2 2 I4 = I (=r ζS∂irα+=)+2=constantSω ℓ= β − ., . (5.13)(5.11) (4.27) 8G 4 − −ren ren r4 4 4 % − & +G' 15 UnderMoreover,with suchβ4 (4.28)=1 gauge/T implies4 and transformations thatSren thedefined on-shell in$ the (4.33). action$ charges Theis given Euclidean$ (4.26) by are$ on-shell shifted action by in the the corresponding magnetic frame electric similarly defines another thermodynamic potential, ,through charges (4.20). As will become clear in section 5,G4 this compensates a related shift in the electric $ 3 (m) (m) potential such that the SmarrI4 = I4 + formula2 andAE F the= I first4 β law4Φ4 areQ4 gau. ge invariant. Nevertheless,(5.14) gauge 2κ4 I4 =+ Sren∧ = Sren− = β4 4, (5.12) !H − G 9In the AdS/CFT context are interpreted as a gravitational anomaly in the dual QFT. 11The overall minus sign relative$ to the Killing vector$ used in [22] can be traced to the fact that the free energy is where Sren was given in (4.13). Evaluating this we obtain (see appendix B) defined as the Lorentzian on-shell action in section 5 of that paper, while in section 6 it is defined as the Euclidean on-shell action. We adopt the latter definition here. 12In [21] the mass for static subtracted geometryβ4ℓk black holes was evaluated2 2 r from the regulated Komar integral I4 = (r r+)+2ω ℓ − . (5.13) and the Hawking-Horowitz prescription and shown8G4 to be− equiv− alent.15 Both ther Smarr+ formula and the first law of thermodynamics were shown to hold in the static case.% In the rotating case,& the' chosen coordinate system of the subtractedMoreover, metric (4.28) in [21] implies has non-zero that angular the on-shell velocity action at spatial is infinity given bwhichy was erroneously not included in the thermodynamics analysis of the rotating subtracted geometry. Furthermore, the evaluation of the regulated Komar integral in the rotating subtracted geometry would have to beperformed;thiswouldleadtoanadditionalcontribution 3 (m) (m) to the regulated Komar mass due to rotation, and in turn ensurethevalidityoftheSmarrformulaandthefirstlaw I4 = I4 + 2 A F = I4 β4Φ4 Q4 . (5.14) of thermodynamics. 2κ4 + ∧ − !H 11The overall minus sign relative$ to the Killing vector$ used in [22] can be traced to the fact that the free energy is defined as the Lorentzian on-shell action in section 5 of that paper, while in section 6 it is defined as the Euclidean on-shell action. We adopt the latter definition here.22 12In [21] the mass for static subtracted geometry black holes was evaluated from the regulated Komar integral and the Hawking-Horowitz prescription and shown to be equivalent. Both the Smarr formula and the first law of thermodynamics were shown to hold in the static case. In the rotating case, the chosen coordinate system of the subtracted metric in [21] has non-zero angular velocity at spatial infinity which was erroneously not included in the thermodynamics analysis of the rotating subtracted geometry. Furthermore, the evaluation of the regulated Komar integral in the rotating subtracted geometry would have to beperformed;thiswouldleadtoanadditionalcontribution to the regulated Komar mass due to rotation, and in turn ensurethevalidityoftheSmarrformulaandthefirstlaw of thermodynamics.

22 An interesting observation is that the value of the renormalized action in the magnetic frame, as well as the value of all other thermodynamic variables, is independent of the parameter α in the boundary counterterms (4.11). This property is necessary inorderforthethermodynamicvariables in the electric and magnetic frames to agree, and in order to match with those of the 5D uplifted black holes that we will discuss in section 6. Recall that the terms multiplying α are designed so that their leading asymptotic contribution to the Hamilton-Jacobi solution (4.7), as well as to An interesting observation is that the value of the renormalized action in the magnetic frame, Mass the derivatives (4.8), vanishes by means of the asymptotic constraints (4.4). This is the reason as well as the valueMass of all other thermodynamic variables, is independent of the parameter α in the why any value of α leads to boundary counterterms that remove the long-distance divergences. boundary counterterms (4.11). This property is necessary inorderforthethermodynamicvariables11 11 The mass is the conserved chargeThe associated mass is the with conserved the Killing chargevector associatedζ with= the∂t KillingΩ ∂vectorφ.Sinceζ = ∂t Ω ∂φ.Since However, the parameter α does appear in the renormalized momenta,− − as∞ is clear from (4.−16),− and∞ Ω =0inthecoordinatesystem(3.6),(4.26)givesin the electric andΩ magnetic=0inthecoordinatesystem(3.6),(4.26)gives frames to12 agree, and in order to match12 with those of the 5D uplifted ∞ blackin the holes unintegrated that we∞ will value discuss of the in renormalized section 6. Recall action. that Never thetheless,terms multiplyingα does not enterα are in designed any physical soobservable. that their leading This observation asymptotic results contribution from the to explicit the2 Hamilt comton-Jacobiputationt 0 solution oft the thermodynamic (4.7),ℓk as well as variables, to 2 t tM40 = t d xℓk 2Π t + Π0At + Π At = (r+ + r ) . (5.9) thebutM derivatives4 we= have not (4.8), beend vanishesx able2Π t to+ by findΠ means0A at general+ of−Π the∂At argument asymptoticC= ( thatr+ constraints+ ensurer ) . sthissofar. (4.4). This8G4 (5.9) is the− reason − ∂ C ! M∩ 8G4 − why any value! M of∩ α leads to boundary counterterms that" remove the long-distan# ce divergences. An interestingThe same" observation result is obtained is that in the# the value electric of the frame renormal using (4.50ized). action As for in the the electric magnetic and frame, magnetic However,5.2 Thermodynamic the parameter α does relations appear in and the renormalized the first law momenta, as is clear from (4.16), and The same result is obtainedas well as in the thepotentials, value electric of the all frame gauge other using potentials thermodynamic (4.50 at). the As boundary variables, for the do electric is notindependent cont andribute magnetic to of the the mass parameter in the gaugeα in the (3.6). potentials, the gaugein the potentials unintegrated at value the boundary of the renormalized do not cont action.ribute Never totheless, the massα does in the not gauge enter in (3.6). any physical observable.We canboundary now This show counterterms observation that the results thermodynamic (4.11). from This the property explicit variables com is necessaryputation we just compof inorderforthethermodynamicvariables theuted thermodynamic satisfy the variables, expected ther- Angular momentum butmodynamic wein have the not electric relations, been and able including magnetic to find a the general frames first argumentlaw to agree, of black that and hole ensurein order mechanics.sthissofar. to match with those of the 5D uplifted Angular momentum black holesThe that angular we will momentum discuss in is section defined 6. as the Recall conserved that the chargeterms corre multiplyingsponding toα theare Killing designed vector ζ = ∂ ,whichgives The angular momentum5.2Quantumso Thermodynamic that is statistical defined their leading asφ relation the relations asymptotic conserved and contribution charge the first corre law tosponding the Hamilt toon-Jacobi the Killing solution vector (4.7), as well as to the derivatives (4.8), vanishes by means of the asymptotic constraints (4.4).3 This is the reason ζ = ∂φ,whichgivesWeIt can is straightforward now show that to the verify thermodynamic that the total variables Gibbs we2 free just enet comprgy utedt 40satisfies satisfyt the expected quantumωℓ ther- statistical why any value of α leads to boundaryJ4 = countertermsd x (2Π φ + thatΠ0GA removeφ + Π Aφ the)= long-distan. ce divergences.(5.10) modynamicrelation [36] relations, including the first law of black∂ C hole mechanics. −2G4 However, the parameter α does appear! inM∩ the renormalized3 momenta, as is clear from (4.16), and 2 =t M t T 0S Ωt J Φ0(ωℓe)Q0(e) !Φ(m)Q(m). (5.15) inJ the4 = unintegratedThed samex (2 valueG result4Π φ of+ is4 theΠ obtained−0A renormalized4φ +4 −Π inA the4 φ4)= electric− action. frame. Never. − theless,4 4 α does(5.10) not enter in any physical Quantum statistical∂ C relation −2G4 Similarly,observable. the! M thermodynamic This∩ observation potential results from4,whichwasobtainedfromtheon-shellactioninthe the explicit computation of the thermodynamic variables, It is straightforwardFree to energy verify! that the total GibbsG free energy satisfies the quantum statistical The same result ismagnetic obtainedbut we frame, in have the satisfies not electric been frame. able to find a general argumentG that4 ensuresthissofar. relation [36] 0(e) 0(e) Finally, the full Gibbs4 = M free4 energy,T4S4 Ω4,isrelatedtotherenormalizedEuclideanon-shellactioni4J4 Φ Q . (5.16) n = M G T S −Ω J −ΦG0(e)Q0(−e) !Φ(m)Q(m). (5.15) Free energy 5.2 Thermodynamicthe electricG4 frame,4 − relations where4 4 − all4 charges4 and− the are electric. first− law4 Namely,4 Note that the shift of the mass and angular momentum$ under a gauge transformation (4.27) is Similarly, the thermodynamic potential 4,whichwasobtainedfromtheon-shellactionintheE Finally, the full GibbscompensatedWe free can energy, now by that show,isrelatedtotherenormalizedEuclideanon-shellactioni of! that the electricthe thermodynamic potentialsG soI variables that4 = S theseren = werelationsS justren = compβ4 are4uted, gauge satisfy invariant.n the expected ther-(5.11) magnetic frame, satisfiesG4 − G the electric frame, wheremodynamic all charges relations, are electric. including= M Namely, theT S firstΩ lawJ of blackΦ0(e)Q hole0(e). mechanics. (5.16) First law with β4 =1/TG44 and 4S−ren 4defined4 − in4 $4 (4.33).− $ The Euclidean$ $ on-shell action in the magnetic frame similarly$ defines another thermodynamic potential, ,through Note that the shift of the massE and angular momentum under a gaGuge4 transformation (4.27) is In orderQuantum to demonstrate statisticalI4 = Sren therelation= validitySren$ = ofβ the4 4 first, law we must recall the transformations(5.11) (3.11) compensated by that of the electric− potentials soG that theseErelations are gauge invariant. and (3.12) of the non-normalizable boundary dataI4 that= Sren allow= forSren a= wellβ4 posed4, variational problem. (5.12) It is straightforward to verify that the total Gibbs free− energy 4Gsatisfies the quantum statistical with β4 =1/T4 andIn particular,SrenThermodynamicdefined variations in$ (4.33).$ of B The, krelations Euclideanand$ ℓ that$ on-shell areand a combination the action first in the oflaw the magneG twotic transformations frame (3.11) First lawrelation [36]where S was given in (4.13). Evaluating this we obtain (see appendix B) similarly defines another thermodynamicren potential, 4,through 3 3 and (3.12) are equivalent to genericG transformations keepi0(eng) 0(kBe) /ℓ! (mfixed.) (m) Considering such a In order$ to demonstrate the validity4 = M of4 theT4 firstS4 lawΩ4J we4 mustΦ reQcall theΦ4 transformationsQ4 . (3.11) (5.15) transformations, as well as arbitrary variations ofβ the4ℓk normalizable parameters2 2 r r and ω,weobtain E G − I −= −(r r )+2−ω ℓ − . (5.13) and (3.12)Free of Energy: the non-normalizable I4 = Sren = boundarySren = β data4 44, that allow for a+ well posed variational(5.12)± problem. Similarly, the thermodynamic− potentialG 4,whichwasobtainedfromtheon-shellactioninthe8G4 − − r+ In particular, variations of B, k!and ℓ that are aG combination0(e) %0(e) of( them) two(m) transformations& ' (3.11) magnetic frame, satisfiesdM4 T4dS4 Ω4dJ4 Φ4 dQ4 Φ34 3dQ4 =0. (5.17) where Sren was givenand (3.12) in (4.13). are equivalent EvaluatingMoreover, to (4.28)− thisgeneric we implies transformations obtain− that (see the− on-shell appendix keepi actionngB)−kB is/ givenℓ fixed. by Considering such a Quantum statistical relation: = M T S Ω J Φ0(e)Q0(e). (5.16) transformations, as well as arbitrary variationsG4 4 of− the4 nor4 −malizable4 4 − parameters r and ω,weobtain Smarr formula 3 (±m) (m) β4ℓk I2 =2 I r+ A F = I β Φ Q . (5.14) Note thatI the= shift of(r the massr )+2 andω4 ℓ angular0(4e) − 0( momentum2e.) (m) under(m) 4 a4 ga4uge(5.13)4 transformation (4.27) is 4 + 2κ4 + ∧ − Finally,First one law: can explicitly8dGM4 T check4−dS−4 thatΩ4d theJ4 SmarrΦ4 rd formulaQ4 !H Φ4 dQ4 =0. (5.17) compensated by that4 %− of the− electric− potentials& + ' so− that these relations are gauge invariant. 11The overall minus sign relative$ to the Killing vector$ used in [22] can be traced to the fact that the free energy is Moreover, (4.28)Smarr implies formula that thedefined on-shell as the action Lorentzian is given on-shell b actiony in section0(e) 0( 5e of) that paper,(m) (m while) in section 6 it is defined as the Euclidean FirstSmarr’s law Formula: M4 =2S4T4 +2Ω4J4 + Q4 Φ4 + Q4 Φ4 , (5.18) on-shell action. We adopt the latter definition here. 12 Finally, one can explicitlyIn3 [21] check the mass that for the static Smarr subtracted formula geometry black holes was evaluated from the regulated Komar integral alsoIn holds. order This toand demonstrate identity the Hawking-Horowitz can be the derived validity prescription by of applying the and(m) first shown(m the) law to first be we equivlaw mustalent. to the re Bothcall one-parameter the the Smarr transformations formula family and the of first (3.11) law of I4 = I4 + A F = I4 β4Φ4 Q4 . (5.14) thermodynamics2 were shown to hold in0( thee) static0(e) case.(m In) the(m) rotating case, the chosen coordinate system of the transformationsand (3.12) of the2κ non-normalizable4M4 =2+ S∧4T4 +2Ω4 boundaryJ−4 + Q Φ data+ thatQ allowΦ for, a well posed variational(5.18) problem. subtracted! metricH in [21] has non-zero angular4 4 velocity at4 spatial4 infinity which was erroneously not included in the In particular, variations of B, k and ℓ that are a combination of the two transformations (3.11) 11The overall minus sign relative$ to thethermodynamics Killing vector analysis$ used of in the[22] rotating can be subtracted traced0( toe) geomet the factry.0(e) that Furthermore, the(m free) the energy evaluation(m) is of the regulated Komar also holds.δM This4 = identityϵintegralM4, inδ canS the4 =2 rotatingbeϵ derivedS4, subtractedδJ by4 =2 applying geometryϵJ4, wouldtheδQ4 first have=law toϵQ beperformed;thiswouldleadtoanadditionalcontribution to4 the, δ one-parameterQ34 3= ϵQ4 family, of(5.19) defined as the Lorentzianand on-shell (3.12) action are in equivalent section 5 of tothat genericpaper, while transformations in section 6 it keepi is definedng kB as the/ℓ Euclideanfixed. Considering such a transformationstransformations,Varying parametersto the regulated as well: r , as Komarr , arbitraryω, massand dueB, variations tok, rotation, subject of and theto in turnk norB3 ensurmalizable3 fixedethevalidityoftheSmarrformulaandthefirstlaw parameters r and ω,weobtain on-shell action. We adopt the latter definitionof thermodynamics. here.+ - l /l – 12 ± In [21] the mass for static subtracted original geometry parameters black holesm, a, w Πas, evaluatedΠ & 0(a escaling) from the0( parametere) regulated(m) Komar(m) integral δM4 = ϵM4, δS4 =2ϵS4, δJ4 =2ϵJc4, sδ23Q4 =0(eϵ)Q4 0(e, ) δQ4(m) = ϵ(Qm4) , (5.19) and the Hawking-Horowitz prescription and showndM to4 be equivT4dSalent.4 Ω4 BothdJ4 theΦ SmarrdQ formulaΦ andd theQ first=0 law. of (5.17) − − − 4 4 − 4 4 thermodynamics were shown to hold in the static case. In the rotating case, the chosen22 coordinate system of the subtracted metric in [21] has non-zero angular velocity at spatial infinity which was erroneously not included in the Smarr formula 23 thermodynamics analysis of the rotating subtracted geometry. Furthermore, the evaluation of the regulated Komar integral in the rotating subtractedFinally, geometry one can would explicitly have to check beperformed;thiswouldleadtoanadditionalcontribution that the Smarr formula to the regulated Komar mass due to rotation, and in turn ensurethevalidityoftheSmarrformulaandthefirstlaw of thermodynamics. 0(e) 0(e) (m) (m) M4 =2S4T4 +2Ω4J4 + Q4 Φ4 + Q4 Φ4 , (5.18)

also holds. This identity can be derived by applying the first law to the one-parameter family of transformations 22

0(e) 0(e) (m) (m) δM4 = ϵM4, δS4 =2ϵS4, δJ4 =2ϵJ4, δQ4 = ϵQ4 , δQ4 = ϵQ4 , (5.19)

23 Concluding Remarks

• Finite conserved charges and thermodynamic identities are a consequence of a well posed variational problem, formulated in terms of equivalence classes of boundary data under the asymptotic local symmetries. Demonstrated this for asymptotically conical backgrounds of the STU model.

• Uplifting these solutions to five dimensions, give a precise map between all thermodynamic variables of subtracted geometries and those of the BTZ black hole. Some free parameters (B, ω) of the four- dimensional black holes are fixed (B) or quantized (ω) in order for the solutions to be uplifted to five dimensions.

• Analysis does not assume or imply holographic duality for asymptotically conical backgrounds, but is a first step in this direction.

Work in progress