Ads2 Holographic Dictionary.Pdf

GARYFEST, Gravitation, Solitons and Symmetries Tours, March 22-24, 2017

AdS2 Holographic Dictionary: An application to the subtracted geometry of non-extremal black holes

Mirjam Cvetič Honored to be able to participate in the celebration of Gary’s numerous contributions to gravitational physics, including classical and quantum aspects of black holes, topological defects and the role of (super)symmetry.

Cherish him as a friend and a colleague, since we met in the 90-ies. During my many visits at Cambridge U., and his visits at Penn, which also led to his association with Penn, we had the opportunity to not only collaborate but also to explore Philadelphia’s dynamic cultural as well as wining and dining scenes. His curiosity and interest in every aspect of human endeavor are boundless, which led to many animated discussions and arguments in different settings, and not only on topics pertinent to science… He is a generous and a patient collaborator, with encyclopedic knowledge of numerous aspects of physics and mathematics. Our association resulted in a productive collaboration with over 30 papers.

Our collaboration covered many aspects of gravitational physics, with applications to supergravity and string theory as a common thread: special holonomy spaces, non-linear Kaluza-Klein reduction in supergravity theories, and extensive work on black holes there; topics that fit well into celebrating Gary’s contributions to science.

The topic of my talk Outline: I. Motivate Subtracted Geometry of general asymptotically flat black holes – prototype STU black holes II. Stepping stone toward holography: Variational Principle for Subtracted Geometry conserved charges and thermodynamics

III. Dual Field Theory of Subtracted Geometry via Holography of 2D Einstein-Maxwell-Dilaton gravity holographic dictionary & new insights & beyond IV. Outlook Background:

Initial work on subtracted geometry M.C., Finn Larsen 1106.3341, 1112.4846, 1406.4536 M.C., Gary Gibbons 1201.0601 M.C., Monica Guica, Zain Saleem 1301.7032 … Recent: Toward holography of subtracted geometry: Variational principle; conserved charges & thermodynamics Ok Song An, M.C., Ioannis Papadimitriou, 1602.0150

Subtracted geometry and AdS2 holography M.C., Ioannis Papadimtiriou,1608.07018 I. General non-extremal, asymptotically flat black holes in effective string theory in D=4

specified by

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 [sector of toroidally compactified effective 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 truncationrelevant of thefor STUdescribing model thatthe subtracted is geometries. We will do so in the duality frame discussed in relevant for describing the subtracted geometries. We will do so[14], in the2TheSTUmodelanddualityframes where duality both frame charges discussed are in electric, as well as in the one used in [17], where there is one electric and [14], where both charges are electric, as well as in the one used in [17],one where magnetic there is charge. one electric We and will refer to these frames as ‘electric’ and ‘magnetic’ respectively. As it one magnetic charge. We will refer to these frames as ‘electric’ and ‘magnetic’ respectively. As it will becomeIn this section clear from we review the subsequent the bosonic analysis, sector of in the order 2-charge to compare truncation the thermodynamics of the STU model in that the is two will become clear from the subsequent analysis, in order to compare therelevant thermodynamics for describing in the twothe subtracted geometries. We will do so in the duality frame discussed in frames, it is necessary to keep track of boundary terms introduced by the duality transformations. frames, it is necessary to keep track of boundary terms introduced by[14], the duality where transformations. 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 2.1 Magnetic frame 2.1will Magnetic become clear frame from the subsequent analysis, in order to compare the thermodynamics in the two The bosonic Lagrangian of the STU model in the duality frame uThesed inframes, bosonic [17] is it given Lagrangian is necessary by of to the keep STU track model of boundary in the termsduality intro frameduced used by inthe [17] duality is given transformations. by 1 1 2 2ηa a a 2 1 1 2ηa a a 2κ4 4 = R ⋆ 1 ⋆ dηa dηa e ⋆ dχ dχ 2.1 Magnetic2κPrototype:= R frame⋆ 1 Black⋆ dη holesdη ofe STU⋆ dχ Modeldχ L − 2 ∧ − 2 ∧ 4 4 a a LagrangianL − 2[A sector∧ of toroidally− 2 compactified∧ effective string theory] 1 η0 0 0 1 2ηa η0 a a 0 a a 0 e− ⋆ F F e − ⋆ (F + χ F ) (FThe+ bosonicχ F ) Lagrangian1 ofη0 the STU0 model0 1 in2ηa theη0 dualitya framea u0sed ina [17] isa given0 by − 2 ∧ − 2 ∧ e− ⋆ F F e − ⋆ (F + χ F ) (F + χ F ) − 2 1 ∧ − 2 1 ∧ 1 1 1 2 2ηa a a + C χaF b F c + C χaχbF 0 F c + C χaχbχcF 0 F2κ0,4 4 = R1(2.1)⋆ 1 ⋆ dηa dηa1 e ⋆ dχ dχ 1 2 abc ∧ 2 abc ∧ 6 abc ∧ L + C −χa2F b F∧c + −C2 χaχbF 0 ∧F c + C χaχbχcF 0 F 0, (2.1) 1abc 1 abc abc 3 2 η0 ∧0 0 2 2ηa η0 ∧a a 06 a a 0 ∧ where ηa (a =1, 2, 3) are dilaton fields and η0 = ηa.ThesymbolCabc is pairwise symmetrice− ⋆ F F e − ⋆ (F + χ F ) (F + χ F ) a=1 where η (a =1, 2, 3) are− dilaton2 fields∧ and−η2 = 3 η .Thesymbol∧ C is pairwise symmetric with C123 =1andzerootherwise.TheKaluza-Kleinansatzforobtainina gthisactionfromthe 0 a=1 a abc 1 a b c 1 a b 0 c 1 a b c 0 0 ! with C123 =1andzerootherwise.TheKaluza-Kleinansatzforobtainin+ C χ F F + C χ χ F F + C χ χ χgthisactionfromtheF F , (2.1) 6-dimensional action (1.1) is given explicitly in [17]. Thisframepossessesanexplicittrialitysym-2 abc ∧ 2 abc! ∧ 6 abc ∧ a a6-dimensional actiona (1.1) is given explicitly in [17]. Thisframepossessesanexplicittrialitysym-(a=1,2,3; C -anti-symmetric tensor) metry, exchanging the three gauge fields A ,thethreedilatonsη and the three axions χ .Inthis 3 abc frame, the subtracted geometries source all three gauge fields Aa metry,magnetically,where exchangingηa while(aw/=1 A the,0 2 &is, 3) threeelectri- are dilaton gauge fields fieldsA anda,thethreedilatonsη0 = a=1 ηa.Thesymbolηa and theCabc threeis pairwise axions symmetricχa.Inthis cally sourced. Moreover, holographic renormalization turns out toframe, bewith much theC more123 subtracted=1andzerootherwise.TheKaluza-Kleinansatzforobtainin straightforward geometries source all three gauge fields Aa magnetically,gthisactionfromthe while A0 is electri- Black holes: explicit solutions of equations! of motion for the above in this frame compared with the electric frame. cally6-dimensional sourced. Moreover, action (1.1) holographic is given explicitly renormalization in [17]. Thi tursframepossessesanexplicittrialitysym-ns out to be much more straightforward Lagrangian w/ metric, four gaugea potentials anda three axio-dilatons a In order to describe the subtracted geometries it suffices toin consider thismetry, frame a exchanging truncation compared of the with (2.1), three the gauge electric fields frame.A ,thethreedilatonsη and the three axions χ .Inthis corresponding to setting η = η = η η, χ = χ = χ χ,andframe,A1 = theA2 = subtractedA3 A.The geometries source all three gauge fields Aa magnetically, while A0 is electri- 1 2 3 ≡ 1 2 3 ≡ In order toPrototype, describe≡ thefour subtracted-charge geometriesrotating black it su holeffices, tooriginally consider obtained a truncation via of (2.1), resulting action can be written in the σ-model form cally sourced. Moreover, holographic renormalization turns out to be much1 more2 straightforward3 corresponding tosolution setting generatingη1 = η2 = techniquesη3 η, χ1 = χ 2 = χ 3 χ ,and M.C.,A =YoumA 9603147= A A.The 1 1 in this frame compared with the electric≡ frame. Chong,≡ M.C., Lü, Pope 0411045≡ S = d4x√ g R[g] ∂ ϕI ∂µϕJ F Λ F Σµν resultingϵµνρσ actionF Λ F Σ can+ S be, written(2.2) in the σ-model form 4 2 IJ µ ΛΣ µν ΛΣ In orderµν ρσFour to describe- SO(1,1)GH thetransfs subtracted. cosh geometriesi sinh i it suffices to consider a truncation of (2.1), 2κ4 − − 2G − Z − R H = , 1 2 (34)3 "M # corresponding1 time$ to-reduced setting ηKerr= BH1η = η sinhη,i χ cosh= χi = χ χ,andA = A = A A.The S = d4x√ g R[g]1 2 ∂ 3✓ϕI≡∂µϕJ 1 2◆F Λ F3Σ≡µν ϵµνρσF Λ F Σ +≡S , (2.2) where 4 resulting2 action can be written inIJ theµσ-model formΛΣ µν ΛΣ µν ρσ GH 1 2correspondsκ4 to: − − 2G − Z − R 3 "MFull four-electric# and four-magnetic charge solution only recently obtained$ SGH = 2 d x√ γ 2K, (2.3) 2κ − 1 4 p1+↵2x0 1 1 p1+↵I2x µ J ChowΛ Σ, µCompèreν 1310.1295;1404.2602µνρσ Λ Σ 4 ∂ whereS = y0 =d y,x√ g e R[g] = ⇤ e ∂ ϕ ∂, ϕ F F ϵ F F + S , (2.2) " M 4 2 IJ µ 1 ΛΣ µν ΛΣ µν ρσ GH is the standard Gibbons-Hawking [36] term and we have defined the doublets2κ4 1 − − 22Gp1+↵2x −2 Z23 − R2 2 2p1+↵2x 0" = ⇤ [ +# (SeGH = (1 + ↵ ) d )]x ;√⇤ =(γ 2K, + 1) e . (35)$ (2.3) M p 2 2 0 where 1+↵ 2κ4 ∂ − I η Λ A " M ϕ = ,A= ,I=1, 2, Λ =1, Note,2, this transformation(2.4) can also be determined1 as an analytic3 continuation of transformations χ A is the standard Gibbons-Hawking [36]SGH term= and wed havex√ definedγ 2K, the doublets (2.3) # $ # $ given in Section 2 of [19]. A Harrison transformation2κ2 in the limit− of an infinite boost corresponds 4 "0∂ as well as the 2 2matrices to 1 5. In theI case ofη↵ = 1 , weΛ shall actA withM (33) on the Schwarzschild solution with × is the standard! Gibbons-Hawkingϕ = ,Ap3 [36] term= and we,I have defined=1, 2, theΛ doublets=1, 2, (2.4) 30 1 e 3η +3e ηχ2 3e ηχ e12U =1χ3 23mχ,2 = 0, =χ 0. The transformationA (34) with =1resultsin⇤ = 2m , and the − − − 2r # $ # $0 r IJ = 2η , ΛΣ = η η , ΛΣ =metric3 (6)2 with the. subtracted(2.5) η geometry warp factor:A G 03e Z 4 3e− χ 3e− R 4 χ 3χ ϕI = ,AΛ = ,I=1, 2, Λ =1, 2, (2.4) # $ # $ as well as# the2 2 2matrices$ χ A × # $ 4 # $ 3 As usual, ϵµνρσ = √ g εµνρσ denotes the totally antisymmetric Levi-Civita tensor,30 where εµνρσ = 1 es0 =3ηr+3esη=(2χ2 m3)er,ηχ 1 χ3(36)3 χ2 − as= well as the 2 ,2matrices= − ! − − , = 2 . (2.5) 1istheLevi-Civitasymbol.ThroughoutthispaperwechoosetheIJ orientation03e2 inη × so thatΛΣ 3e ηχ 3e η ΛΣ 3 χ2 3χ ± G and the scalarM field andZ the electric4 field strength3η − : η 2 −η R 4 23 3 2 εrtθφ =1.WenoteinpassingthattheLagrangian(2.2)isinvariantunder# the global30$ symmetry #1 e− +3e− χ 3e− χ$ 1 #χ 2 χ $ IJ = 2η , ΛΣ = η η , ΛΣ = 3 2 . (2.5) transformation As usual, ϵµνρσ03= √e g εµνρσ denotes2 the2 totallym3e χ antisymmetric2 3e1 Levi-Civita tensor,χ 3 wheχ re εµνρσ = G Z p43 − − R 4 2 # −$ e #= , Ftr = ,$ # (37) $ η 2 η 2 0 3 0 1istheLevi-Civitasymbol.Throughoutthispaperwechoose2 2 r r r3 2m the orientation in so that e µ e , χ µ− χ,A µ A ,A µA,± As ds usual,ds ,ϵµνρσ = √ g(2.6)εµνρσ denotes the totally antisymmetric Levi-Civita tensor, whereMεµνρσ = → → → → ε =1.WenoteinpassingthattheLagrangian(2.2)isinvariant→ − under the global symmetry rtθφ 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. ± The subtracted geometry for the Kerr spacetime can be obtained by reducing the spacetime M transformationε =1.WenoteinpassingthattheLagrangian(2.2)isinvariantunder the global symmetry rtθφon the time-like Killing vector and acting on the Kerr black hole with an infinite boost Harrison η 2 η 2 0 3 0 2 2 transformationtransformatione forµ Lagrangiane , χ densityµ− χ (1),,A where weµ setA,A= = µA,, ds' = ' ds= ', (2.6) → → → 1 2 → 3 ⌘ 1 → 2 3 ⌘ 5 2 η 2 η 2 2 0 3 0 2 2 , F1 = F2e= µ1 e , Fχ andµ−2 χ=,Ap2 , i.e.µ anA Einstein-Dilaton-Axion,A µA, ds gravityds , with (2.6) where µpis3 an⇤ arbitrary→⇤ non-zeroF ⌘ 3 constant→ F parameter.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 2.1. The Black Hole Metric

The setting for our discussion is the rotating black hole solution of four dimensional string theory with four independent U(1) charges [5]. The asymptotic charges of the black hole are parametrized as:

1 3 G M = m cosh 2δ , 4 4 I !I=0 1 (2.1) G Q = m sinh 2δ , (I =0, 1, 2, 3) , 4 I 4 I G J = ma(Π Π ) , 4 c − s where we employ the abbreviations 3 3 Π cosh δ , Π sinh δ . (2.2) c ≡ I s ≡ I I"=0 I"=0 The parametricCompact mass and form angular of the momentum metric form, rotating a both have four dimension-charge ofblack length. holes We write the 4D metric as a ﬁbration over a 3D base space M.C. & Youm 9603147 Chong, M.C., Lü & Pope 0411045 2 − dr X ds2 = ∆ 1/2G(dt + )2 + ∆1/2 + dθ2 + sin2 θdφ2 , (2.3) 4 − 0 A 0 X G # $ where for the black holes we consider X = r2 2mr + a2 , − G = r2 2mr + a2 cos2 θ , − 2ma sin2 θ = [(Πc Πs)r +2mΠs] dφ , A G − (2.4) 3 3 ∆ = (r +2m sinh2 δ )+2a2 cos2 θ[r2 + mr sinh2 δ +4m2(Π Π )Π 0 I I c − s s I"=0 !I=0 2m2 sinh2 δ sinh2 δ sinh2 δ ]+a4 cos4 θ . − I J K I

The rather complicated conformal factor ∆0 simplifies in some special cases. The benchmark is the non-rotating case a =0whereonlythefirsttermremains.However,the expression also simplifies with rotation when the four charges are equal in pairs

2 2 2 2 2 ∆0 =[(r +2m sinh δ1)(r +2m sinh δ2)+a cos θ] . (2.5)

The generic case with rotation and four independent charges does not simplify.

3 2.1. The Black Hole Metric

The2.1. setting The Black for our Hole discussion Metric is the rotating black hole solution of four dimensional string theoryThe with setting four for independent our discussionU(1) is the charges rotating [5]. black The hole asymptotic solution ofcharges four dimensional of the black hole arestring parametrized theory with as: four independent U(1) charges [5]. The asymptotic charges of the black hole are parametrized as: 3 1 G M = m cosh3 2δ , 4 1 I G M4 = m cosh 2δ , 4 I4=0 I ! I=0 (2.1) 1 ! G4QI = m sinh1 2δI , (I =0, 1, 2, 3) , (2.1) G Q4 = m sinh 2δ , (I =0, 1, 2, 3) , 4 I 4 I G4J G= Jma=(maΠc(Π Πs)Π,) , 4 −c − s where wewhere employ we employ the abbreviations the abbreviations 2.1. The Black Hole Metric

3 3 The setting for3 3 our discussion is the rotating black hole solution of four dimensional Π cosh δ , Π sinh δ . (2.2) Πc c ≡cosh stringδI , I theoryΠs withs ≡ foursinh independentδII . U(1) charges [5]. The asymptotic(2.2) charges of the black ≡ I=0 ≡ I=0 I=0 " hole are parametrizedI=0" as: The parametric mass and angular" momentum m, a"both have dimension of length. The parametric mass and angular momentum m, a both have dimension1 3 of length. Compact form of the metric for rotating fourG M -=chargem cosh black 2δ , holes We write the 4D metric as a fibration over a 3D base space4 4 M.C. & YoumI 9603147 We write the 4D metric as a fibration over a 3D base space I=0 2 Chong, M.C.,! Lü & Pope 0411045 (2.1) 2 −1/2 2 1/2 dr 2 X 1 2 2 ds4 = ∆ G(dt + ) + ∆ 2 + dθ G+4QI =sinmθsinhdφ 2δI ,, (I =0, 1,(22,.3) , 2 −1−/2 0 2 A 1/20 dr X 2 XG 42 2 ds4 = ∆0 G(dt + ) + ∆0 # + dθ + sin θdφ $ , (2.3) − A X GG4J = ma(Πc Πs) , where for the black holes we consider # − $ where we employ the abbreviations where for theX black= r2 holes2mr + wea2 consider,= 0 outer & inner horizon − 3 3 2 2 2 2 2 Π cosh δ , Π sinh δ . (2.2) X = r G =2mrr +2amr,+ a cos θ , c ≡ I s ≡ I − − I=0 I=0 2 " " 2 2ma sin2 θ 2 The parametric mass and angular momentum m, a both have dimension of length. G = r =2mr + a cos[(Πθ , Π )r +2mΠ ] dφ , − G c s s 2.1. The Black Hole MetricA 2 − We write the 4D metric as a fibration over a 3D base(2 space.4) 2ma sin3 θ 3 2 The setting for= our discussion is[(Π thec rotatingΠs)2r black+2m holeΠ2 s] sold2φution, 2 of four2 dimensional−1/22 22 1/2 dr 2 X 2 2 ∆ = (r +2m sinh δ )+2a cos θ[r + mrds4 = ∆sinh0 Gδ(dt+4+ m) (+Π∆0 Π )Π + dθ + sin θdφ , (2.3) A 0 G − I − I A c − sX s (2G.4) string theory with four independent3 I=0 U(1) charges [5]. The asymptotic charges3I=0 of the black # $ " where for the black! holes we consider hole are parametrized as: 2m2 2 sinh22δ sinh2 2 δ 2 sinh2 δ ]+a4 cos2 4 θ . 2 ∆0 = (r +2m sinh δI )+2a cosI θ[rJX += rmr2 K2mr +sinha2 , δI +4m (Πc Πs)Πs − 3 − − I=0 1 I

2 2 2 2 2 ∆0 =[(r +2m sinh δ1)(r +2m sinh δ2)+a cos θ] . (2.5)

The generic case with rotation and four independent charges does not simplify.

3 Thermodynamics of outer & inner horizons suggestive of weakly interacting 2-dim. CFT M.C., Youm ’96 w/ ``left-’’ & ``right-moving’’ excitations M.C., Larsen ’97

Area of outer horizon S = S + S 2 2 + L R SL =S⇡L m= ⇡(⇧mc +(⇧⇧cs+)(1)⇧s)(1) 2 2 [Area of inner horizon S- = SL – SR ] S =p⇡ m2pm 2 a (⇧ ⇧ )(2) SR = ⇡R m m 2 a2 (⇧c ⇧cs)(2)s 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 R =R = (⇧c +(⇧⇧cs+)(4)⇧s)(4) outer horizon βH = ½ (βL + βR ) Lp=2mL 2=2p⇡mm⇡a22(m⇧(ca⇧2 c ⇧s⇧)(3)s)(3) 3 3 2⇡2m2 3 3 ⇧c =⇧⇧c I==0⇧I =0coshcosh2I⇡,PimI ,Pis = ⇧sI==0⇧I =0sinh sinhI I (5) (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 =⇧c⇧=I Ω⇧=0+I, =0Ω-coshandcosh momentaI ,PiI ,Pi sJ+=,s J=-⇧. I⇧=0I =0sinhsinhII (5)(5) Ch.Ch. 29: 29: HWExercises:HW 7, Phys 7, 2,6,11,13,16,20,23,37 Phys 151 151 Depend onlyExercises: on four parameters: 2,6,11,13,16,20,23,37 m, a, Problems: 2,7 Problems: 2,7 Ch.Ch. 29: 29: Shown more recently, all independent of the warp factor Δo ! HWExercises: due on Wednesday, 2,6,11,13,16,20,23,37 October 22nd. Exercises:HW due 2,6,11,13,16,20,23,37 on Wednesday, October 22nd.M.C., Larsen ’11 Problems:Problems: 2,7 2,7 Reminder: Next 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: Next Midterm is on Thursday, October 23rd, 11 AM in A1. Reminder:(or equivalentlyhttp://www.physics.upenn.edu/phys151.htlmNext at Midterm http://www.physics.upenn.edu/ is on Thursday, October cvetic) 23rd, 11 AM in 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) M.C., Larsen 1112.4856 Motivation for Subtracted Geometry Focus on the black hole “by itself” à enclose the black hole in a box (à la Gibbons Hawking) à an equilibrium system w/ conformal symmetry manifest * The box chosen to lead to a ``mildly’’ modified geometry changing only the warp factor Δ0à Δ [maintains the same horizon thermodynamic quantities]

*Determination of new warp factor Δ0 à Δ Via scalar field wave eq.: separable & radial part solved by hypergeometric functions w/ SL(2,R)2 à unique Δ 3.2. The Subtracted Geometry

2.1. The BlackThe di Holefferential Metric equation (3.2) with the effective potential (3.3) has simplifying features Thebeyond setting the seperability. for our discussion The radial is the equation rotating has black two holeregular solutionsingularities, of four dimensional at r = r+ and 2.1. The Black Hole Metric iβ± stringr theory= r−.Theindicesattheseregularsingularitiesare with four independent U(1) charges [5]. The asymptotic2π ,ie.essentiallytheouterand charges of the black The setting for our discussion is the rotating black hole solution of four dimensional hole areinner parametrized horizon temperatures. as: The radial equation has a third singularity at infinity, but string theory with four independent U(1) charges [5]. The asymptotic charges of the black this singularity is irregular. If it had been regular the radial equation would have been the hole are parametrized as: 1 3 hypergeometric equation,G4M with= itsm SL(2cosh,R)symmetrypermutingthethreesingularities. 2δI , 4 1 3 G M =I=0m cosh 2δ , This situation is desirable because4 ! it4 would hintI at an underlying conformal symmetry. 1 I=0 (2.1) The irregular singularityG Q = atm infinitysinh! 2δ is, due(I =0 to, the1, 2 asymptoti, 3) , cbehavior∆(2.1) r4 at 4 I 4 1 I 0 ∼ G4QI = m sinh 2δI , (I =0, 1, 2, 3) , 2 large r which encodes the asymptotic4 flatness of spacetime. If instead ∆0 ∆ r the G4J =GmaJ =(Πmac (Π Πs)Π,) , → ∼ singularity at infinity in the radial4 equation− c − woulds be regular. For even more special warp where we employwhere we the employ abbreviations the abbreviations factors with ∆0 ∆ r at large r the radial equation maintains its hypergeometric char- → ∼ 3 3 3 3 acter but, in addition, the angularΠc equationcosh δI , simplifiesΠs sinh to thδIefamiliarsphericallysymmetric. (2.2) Πc ≡cosh δI , Πs ≡ sinh δI . (2.2) ≡ I=0 ≡ I=0 form I=0 " I=0" The parametric mass1 and" angular momentum1 2 m, a"both have dimension of length. The parametric mass and angular∂θ sin θ∂ momentumθ + ∂φm,χ a(bothθ, φ)= havel(l dimension+1)M.C.,χ (Larsenθ, φ of) . 1112.4856 length. (3.4) We write thesin 4Dθ metric as a fibrationsin2 θ over a 3D base− space Subtracted! geometry for rotating" four2 -charge black holes We write the 4D metric as− a fibration over a 3Ddr base spaceX When ∆ r theds geometry2 = ∆ 1/2 thusG(dt + indicates)2 + ∆1 an/2 unbroken+ dθ2 +SU(2)sin2 R-symmetry.θdφ2 , In(2.3) this case ∼ 4 − 0 A 0 2X G − dr# X $ the indicesds2 of= the∆ regular1/2G( singularitydt + )2 + at∆1 infinity/2 are+ d (l,θ2 +l 1).sin2 θdφ2 , (2.3) where4 for the− black0 holes we considerA 0 X − −G It wasX = shownr2 2 inmr section+ a2 , 2 that both the# causal structure and the$ thermodynamics of where for the black holes− we consider black holesG is= independentr2 2mr + a2 cos of the2 θ , conformal factor ∆0.Weinterpretthisasademonstration 2 − 2 Xthat= r an alternate2mr + a∆,2 ∆ corresponds to a black hole with the same internal structureas − 2ma sin0 →θ 2 = 2 2[(Πc Πs)r +2mΠs] dφ , Gthe= originalr A2mr one,+ a butG cos a blackθ ,− hole that finds itself in a different external environment.(2.4) − 3 3 2 2 2 2 2 2 2 We2ma∆ willsin0 = focusθ (r +2 onm thesinh warpδI )+2 factorsa cos θ∆[r that+ mr preservesinh δI separabilit+4m (Πc yofthescalarwaveΠs)Πs − = I=0[(Πc Πs)r +2mΠs] dφ , I=0 AequationG and" also analyticity− in the coordinates.! These technical assumptions identify a 2m2 sinh2 δ sinh2 δ sinh2 δ ]+a4 cos4 θ . (2.4) warp3 factor with− the asymptoticI behaviorJ ∆ Kr3uniquely as ∆ = (r +2m sinh2I

3 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 sector2.1 of the Magnetic 2-charge truncation frame 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 wellThe as in bosonic the one Lagrangian used in [17], where of the there STU is model one electric in the and duality 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, in order2 toκ compare4 = R ⋆ the1 thermodynamics⋆ dηa dηa in thee two⋆ dχ dχ 4L − 2 ∧ − 2 ∧ frames, it is necessary to keep track of boundary terms introduced by the duality transformations. 1 η0 0 0 1 2ηa η0 a a 0 a a 0 e− ⋆ F F e − ⋆ (F + χ F ) (F + χ F ) − 2 ∧ − 2 ∧ 2.1 Magnetic frame 1 a b c 1 a b 0 c 1 a b c 0 0 + Cabcχ F F + Cabcχ χ F F + Cabcχ χ χ F F , (2.1) The bosonic Lagrangian of the STU model in the duality frame used2 in [17] is given∧ by 2 ∧ 6 ∧ 3 2 1 where1 2ηηaa (a =1a , 2,a3) are dilaton fields and η0 = a=1 ηa.ThesymbolCabc is pairwise symmetric 2κ4 4 = R ⋆ 1 ⋆ dηa dηa e ⋆ dχ dχ L − 2 ∧ with− 2 C123 =1andzerootherwise.TheKaluza-Kleinansatzforobtainin∧ gthisactionfromthe 1 1 ! η0 0 0 6-dimensional2ηa η0 a actiona 0 (1.1) isa givena 0 explicitly in [17]. Thisframepossessesanexplicittrialitysym- e− ⋆ F F e − ⋆ (F3Asymptoticallyconicalbackgrounds+ χ F )3Asymptoticallyconicalbackgrounds(F + χ F ) − 2 ∧ −metry,2 exchanging the three∧ gauge fields Aa,thethreedilatonsηa and the three axions χa.Inthis 1 1 1 + C χaF b F c +frame,C χ theaχb subtractedF 0TheF generalc + C geometries rotatingχaχbχ subtractedcF source0 F 0, all geometry three(2.1) backgrounds gauge fields areAa solutionsmagnetically, of the equations while A of0 is motion electri- 2 abc ∧ 2 abc following∧ from6 Theabc the general action (2.2)∧ rotating or (2.10) subtracted and take the geometry form [14, 17] backgrounds5 are solutions of the equations of motion cally sourced. Moreover, holographic renormalization turns out to be much more straightforward5 3Matter fieldsfollowing (gauge from potentials the action and (2.2) scalars) or (2.10) and take the form [14, 17] where ηa (a =1, 2, 3) are dilaton fields andinη this0 = framea=1 η compareda.Thesymbol√∆ withCabc theG is electric pairwise frame. symmetric X ds2 = dr¯2 (dt¯+ )2 + √∆ dθ2 + sin2 θdφM.C.,¯2 , Gibbons 1201.0601 with C123 =1andzerootherwise.TheKaluza-KleinansatzforobtaininIn order3Asymptoticallyconicalbackgrounds to describeX the− √ subtractedgthisactionfromtheA geometries itG suffices 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 22 ¯2 3 correspondingScalars: toη setting(2m) dsη1 ==η2 a=(Πdηcr¯3 Πsη), χ1(d=t +χ2 =) χ+3 ∆χ,anddθ +A =sinA θ=dφA , A.The metry, exchanging the three gauge fields Aa,thethreedilatonsThe generale = rotatingηa and, the subtractedχ three=X axions geometry−≡−χ√a.Inthiscos∆ backgroundsθ, A are s≡olutions of theG equations of motion≡ resulting action can be√∆ written in the2σm-model form !5 " frame, the subtracted geometries source all three gaugefollowing fields fromAa themagnetically, action (2.2) while or (2.10)2 A0 andis electri- take the form [14, 17] 4 (2m) a2 (Π2 Π ) 2 4 0 (2m) a (Πηc Πs) 2 (2ma) cos cθ (Πc s Πs) +(2m) ΠcΠs cally sourced. Moreover, holographic renormalization1 Running turnsA4 out =dilaton to be: e much=− more1 sin straightforward,θdφ¯I +µχ J= −Λ −Σµcosν θ, µνρσ dΛt,¯ Σ S = d√x∆√ g RG∆[g] ∂ ϕ ∂ ϕ X F(Π2F Π2) ∆ ϵ F F + S , (2.2) in this frame compared with the electric frame.4 2 2 2 ¯ √IJ∆2 µ√ 2 ΛΣ2mµν¯c2 s ΛΣ µν ρσ GH 2κ4 ds = d−r¯ (d−t +2G) + ∆ dθ +− Zsin θdφ − , − R "M X 2m−cos#√θ∆ A 42 2 G2 ¯ 2 2 $ 2 4 In order to describe the subtracted geometries it suffiAces= to consider∆ a(2(2 truncationmma))a(Π(Πc !c ofΠs (2.1),Π) ssin) θ dφ 2"ma(2(2mmaΠs)+¯cosr(Πc θ (ΠΠs))c dt¯ Π, s) +(2m) ΠcΠs 2 ∆ 0 2 ¯ where (2m) A 1=a −(Πc 2 Πs) 3 −− sin θd−φ + − − dt,¯ corresponding to setting η1 = η2 = η3 η, χ1 = χGauge2eη==χ3 potentials:χ,,andχ A= = A −= A2 cos θ,A.The 2 2 2 2 ≡ ≡ 1 #$ (2ma) (∆Π≡c 1 Πs) 3% (2ma) (Πc Πs)[2mΠs +¯(rΠ(Πcc ΠΠ&ss)]) cos∆ θ A√=∆ r¯ m 2SmGH = − ddt¯+x√ γ 2K,− −− dt¯ (2.3) resulting action can be written in the σ-model form 2m (2m)3(Π +2Π ) 2m∆ −4 − −2m cos θ c2κ4 2s ∂ 2 2 − 2 2 4 2 (2m) a (Πc! Πs) (2ma)"cosM" θ (Πc Πs) +(2m) ΠcΠs ¯ ¯ 1 1 A0 = −A =sin2 θdφ¯ + ∆ (2ma) −(Π(2mac )2Π(Πs) sinΠ )2θcosdt,¯d2 φθ 2ma (2mΠs +¯r(Πc Πs)) dt , 4 isI µ theJ standardΛ Gibbons-Hawking'Σµν µνρσ Λ [36]∆Σ term and− we2 have2 defined2 − cthe doubletss − ¯ − S4 = 2 d x√ g R[g] IJ∂µϕ ∂ ϕ ΛΣFµν F ∆ ΛΣϵ Fµν F+ρσa(Π+c SGHΠs,)sin(2.2)θ(Π1+c Πs) ∆ − dφ, (3.1) 2κ4 − − 2G − Z − R − −2 ∆ 2 2 "M # 2m cos θ 21 $ #$2 20 (2ma¯! ) (Πc Πs) % (2ma" ) (Πc Πs)[2mΠs +¯r(Πc Π&s)] cos θ A = I ∆A (2η=ma) (Πc r¯ΛΠs) msinA θ dφ 2ma (2m−Πs +¯r(Πdct¯+Πs)) dt¯ , − − dt¯ where where ∆ ϕ =− ,A− = ,I− 3 =1, 2, Λ =1−, 2, (2.4) 1 χ −2m − −A (2m) (Πc + Πs)Magnetic frame 2m∆ 3 1 #$ (2ma)2(Π! Π ) % (2ma)2(Π Π )[2m"Π +¯r(Π Π& )] cos2 θ SGH = 2 d x√ γ 2K, # $ c s # (2.3)¯ $ c 2mas s c s 2 ¯ 2 2 2κ ANon−= -extremal2r¯ mblack hole2 immersed− din2t +constant2 magnetic− field −(2ma) (2Πcdt¯ Πs) cos θ as4 well∂ as the 2 −X22matricesm=¯r −2m'−r¯ +(2am,G)3(Π +=ΠX) a sin θ, = ((Π2mc2∆Πs)¯r +2mΠs)sin θdφ, ¯ " M × ! − c s −" + a(ΠAc ΠGs)sin −θ 1+ − dφ, (3.1) is the standard Gibbons-Hawking [36] term and we have defined the3 doublets2 2 3η 4 2 η 2 2 (2maη)2−(Π2 Π2 )2 cos2 θ 3 3 2 ∆ 30' w/ ∆ =(2m) (Πc Πs1)¯r +(2e m)+3Πs e2 (2χma) (3Πec χΠs) c cos sθ, ! 1 ¯ χ χ (3.2) " − + a(Πc − Πs)sin−−θ 1+ −− − dφ, 2 (3.1) η IJA0= 2η , ΛΣ = − η η , ∆ ΛΣ = 3 2 . (2.5) ϕI = ,AΛ = G ,I03e=1, 2, whereΛ =1Z , 2, 4 3e− χ(2.4)! 3e− R "4 2 χ 3χ χ A # and Πc$, Πs, a and m are parameters# of the solution. $ # $ # $ # $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 whemare εµνρσ = as well as the 2 2matrices parameters− are fixed by the2 boundary conditions2 in the variational2 problem.2 A full analysis of the 2 ¯ × 1istheLevi-Civitasymbol.Throughoutthispaperwechoose2 X2 =¯r 2mr¯2+ a2 ,G=2maX a sintheθ orientation, = 2 in ¯ ((Πsoc thatΠs)¯r +2mΠs)sin θdφ, 1 3η± η 2 Xvariationalη=¯r 2m problemr¯ + a ,G for1 the3=− actionsX3 2a (2.2)sin θ or, (2.10)= require−((sknowledgeofthegeneralasymptoticΠc Πs)¯r +2mΠAs)sin θGdφM, − 30 e− ε+3rtθφe−=1.WenoteinpassingthattheLagrangian(2.2)isinvariantχ 3e− χ − χ 32−χ 2 2 A G4 2 − under2 the global2 2 symmetry IJ = 2η , ΛΣ = η solutionsη , and3 2 isΛΣ beyond∆=2 =(2 the3 m2 scope)4 (2Π of the. Π(2.5) present2)¯r +(2 paper.m2) Π Howev2 (2er,ma we) can(Π considerΠ ) thecos variationalθ, (3.2) G 03e Z 4 3e− χ 3∆e−=(2m) (ΠR Π )¯r4+(2χm) Π3χc (2mas) (Πc Πs) cos sθ, c s (3.2) # $ # transformationproblem$ withinc − thes class# of2 stationarys −$− solutions (3.1).− To this− end, it is convenient− to reparameterize As usual, ϵµνρσ = √ g εµνρσ denotes the totally antisymmetricthese backgroundsη Levi-Civita2 η by means tensor, of2 whe a suitablere εµ0νρσ coordinate=3 0 transformation, accompanied2 2 by a relabeling − and Πc, Πse, a andµandme are,Πc parameters,χΠs, µa−andχ of,A them are solution. parametersµ A ,A ofµA, the solution. ds ds , (2.6) 1istheLevi-Civitasymbol.Throughoutthispaperwechooseof the free→ parameters.the orientation In→ particular, in weso introduce→ that the new→ coordinates → ± In order to studyIn the order thermodynamics to studyM of the these thermodynamics backgrounds it is of necessary these to backgrounds identify which it is necessary to identify which εrtθφ =1.WenoteinpassingthattheLagrangian(2.2)isinvariantwhere µparametersis an arbitrary are fixed non-zero byunder the4 boundary the constant global3 conditions parameter.symmetry2 2 in the variat4 2 ional problem.2 A2 full analysis of the ℓ r =(2m) (Π Π )¯r +(2m) Π (2ma) (Πc Πs) , transformation variational problemparameters for the actions are fixed (2.2)c − or by (2.10)s the boundary requiresknowledgeofthegeneralasymptotics − conditions− in the variational problem. A full analysis of the k 1 2ma(Πc Πs) η 2 η 2 0 solutions3 0 and is beyondvariational2 thet scope=2 problem of the present for thet,¯ paper. actionsφ = Howevφ¯ (2.2)er, we or can− (2.10) considert,¯ require the variationalsknowledgeofthegeneralasymptotic(3.3) e µ e , χ µ− χ,A µ A ,A µA, ds ℓ3 ds ,(2m)3(Π2 Π(2.6)2) − (2m)3(Π2 Π2) → → → problem within→ thesolutions class of→ stationary and is beyond solutionsc − s (3.1).the5 scope To this of end, the it is present convenientc − s paper. to reparameterize However, we can consider the variational where µ is an arbitrary non-zero constant parameter.thesewhere backgroundsℓ andproblemk byare means additional within of a suitable non-zero the class coordinate parameters, of stationary transfo whosermation, role solutions will accompanied become (3.1).clear To by shortly. t ahis relabeling end, Moreover, it is convenient to reparameterize of thewe free define parameters. thethese new parameters In backgrounds particular, we by introduce means the of new a suitable coordinates coordinate transformation, accompanied by a relabeling 4 3 2 2 4 2 2 2 of4 theℓ r =(2 free3m parameters.) (2Π 2Π )¯r +(2 Inm particular,2 ) Π (22ma we) (Π introducec2 Π2s) , 3 the2 new2 coordinates 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 Πs),Bt,¯4=2m,φ = φ¯ 3 2 ( 2 t,¯ 4 2 (3.3)2(3.4) 2 3 −3 2 2 ℓ r =(2m) (Π 3 Π2−)¯r +(22 m) Π (2ma) (Π Π ) , ℓ (2m) (Πc Πs) − (2cm) (Πcs Πs) s c s 5 − − − − − In order to compare this background withk the expressions given1 in eqs. (24) and (25) of [14], one should2ma take(Π into Π ) whereaccountℓ and thek are field additional redefinition non-zeroA →−A, χ parameters,→−χ,beforethedualizationof whose role will becomeA, as¯ mentionedclear shortly. in footnote¯ Moreover, 4, and addc a s ¯ 3 t = 3 2 2 t, 2φ =2 φ¯ 3 2− 2 t, (3.3) constant pure gauge term. Moreover, there isℓ a typo in(2 eq.m (25) )of[14]:theterm2(Π Π ) mΠs cos θdt should− (2m be) replaced(Π Π ) we define the new− parameters2 ¯ c s c s by 2mΠs(Πs Πc)cos θdt. − − 4 3 2 2 2 2 2 2 3 2 2 ℓ r =(2wherem) mℓ (andΠc +kΠsare) (2 additionalma) (Πc non-zeroΠs) m parameters,a (2m) (Πc whoseΠs), role will become clear shortly. Moreover, ± − − ± − − 3 ℓ ω =2wema define(Πc theΠs),B new parameters=2m, ( (3.4) − 5 In order to compare this background4 with the expressions3 giv27 en in eqs.2 (24) and (25)2 of [14], one should2 take into2 2 3 2 2 account the field redefinition A →−ℓA,rχ →−=(2χ,beforethedualizationofm) m(Πc + Πs) A,(2 asma mentioned) (Πc in footnoteΠs) 4, and addm a a (2m) (Πc Πs), ± − 2 2 −¯ ± − − constant pure gauge term. Moreover, there3 is a typo in eq. (25)of[14]:theterm2mΠs cos θdt should be replaced − 2 ¯ ℓ ω =2ma(Πc Πs),B=2m, ( (3.4) by 2mΠs(Πs Πc)cos θdt. − 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 3AsymptoticallyConicalMetricsBrief Remarks:

The scaling limit,Asymptotic or equivalently geometry the subtraction of subtracted process, alters geometry the environment is of that our black holes find themselvesLifshitz in [12,-type 13]. In w/ fact a deficit the subtracted angle: geometry m etric is asymptotically of the form R 2p ds2 = dt2 + B2dR2 + R2 dθ2 + sin2 θ2dφ2) (39) − R ! 0 " ! " p=3, B=4 with R0 aconstant,B =4andp = 3. In general, metrics with asymptotic form (39) may be referred as Asymptoticallyà black Conical hole(AC). in an The `àsymptotically spatial metric is conical conical because box’’ the radial distance BR is a non-trivial multiple of the area distance R. Restricted to the equatorialM.C., Gibbons plane 1201.0601 the spatial metric is that of a flat two-dimensional cone 2 à the box conformal to AdS2 x S 2 2 2 2 2 (confining, butdsequ softer= B thandR + AdSR d)φ (40) 2 with deficit angle à lift on a circle: locally AdS3 x S M.C., Larsen 1112.4856 1 2 Conformal symmetry2π(1 of AdS)=83 πηpromoted. to Virasoro algebra (41) − B of dual CFT2 , à la Brown-Hennaux à A characteristic featurereproduces of conical metrics entropy of of the 4D form black (39) holes is that à theyla Cardy admit a Lifshitz scaling. That is they admit a homothety,adiffeomorphism under which the pulled back metric goes into a constant multiple of itself. In our case the homethety is

1 p 2 2 2 R λR, t λ − t, ds λ ds (42) → → ⇒ → or if one introduces isotropic Cartesian coordinates x = RB(sin θ cos φ, sin θ sin φ, cos θ) which render the spatial metric conformally ﬂat, we have scaling under,

x λx ,tλzt, (43) → → where the difference of the Lifshitz scaling exponent z from unity is a measure of how space and time scale differently. We have 1 p z = − (44) B 1 and so z = 2 in our case. The nonstandard scaling of time is reminiscent of the Lifshitz symmetry that− has recently been developed for applications of holography to condensed matter systems. (See, e.g., [24, 25, 26] and references therein.). Since asymptotically Conical (AC) metrics of the type (39) may be unfamiliar to the string theory community, we briefly recall some of their properties and give some examples. AC metrics typically arise when the energy density T00 of a static four dimensional spacetime falls off as 2 η 5 T00 R2 . Because of this slow fall off the metric cannot have finite total energy and cannot be asymptotically→ flat (AF). At large distances such spacetime metrics typically take the form of (39) near infinity. Examples where (39) is exact are

2 2γ √1+6γ+γ p = 1+γ , B = 1+γ , gives Bisnovatyi-Kogan Zeldovich’s gas sphere [27, 28] Here, γ • is the constant ratio of pressure to density of. the gas for which

γ2 1 P (45) ∝ 1+6γ + γ2 2πR2

5These components in a pseudo-orthonormal frame. The coordinate R is the area distance. We use throughout signature +++andunitsinwhichNewton’sconstant G =1. −

11 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 dilationthree dimensional coupling ↵. base, The Lagrangian independent density of the is 4 charge.: parameters, and the warp factor denoted by ∆0. 1 1 2 1 2↵ 2 In the static casep g oneR sets a(@=) 0 ande the solutionF . simplifies significantly(24) [17]: three dimensional base, independent of the charge parameters, and the warp factor denoted by 4 2 4 2 ⇣2 1/2 1/2 dr 2 ∆0. − 2 2 2 Making the Ansatz ds4 = ∆0s Xdt + ∆0s X + dθ + sin θdφ , (7) 2 2U 2 − 2U i j In the static case one sets a = 0 and the solution simplifies significantly [17]: ds = e dt + e dx dx ,F!= @ " (25) ij i0 i 2 where 2 1/2 1/2 dr 2 we obtain an e↵ective action density in three dimensions of the form − 2 2 2 ds4 = ∆0s Xdt + ∆0s X + dθ + sin θdφ , (7) − 2 ! " ij X = r 2U 2mr2↵ , (8) p R(ij) 2 @iU@jU + @i@j e −e @i @j (26) where 4 ⇣ ⇣ 2⌘⌘ Defining ∆0s = (r +2m sinh δI ) , (9) X = r2 2mr , (8) U + ↵ ↵U + − x ,y I#=1 , (27) 4 ⌘ p1+↵2 ⌘ p1+↵2 2 and the scalar fields and the gauge potentials take the form: ∆0s = (r +2m sinh δI ) , the e↵ective action density(9) becomes I#=1 1 2 ij 2 2p1+↵2x 2 p R(ij)ϕ1 2 (@ri+2x@jxm+sinh@iy@jδy1)(re+2 m sinh@i δ@3j) . (28) and the scalar fields and the gauge potentials take the form: χi =0,e = 2 2 , ⇣ $⇣(r +2m sinh δ2)(r +2m sinh δ4)%⌘⌘ 1 Evidently we can consistently set y = 0 and we obtain a sigma1 model, whose fields x, map 1 2 2 2 2 2 2 2 2 2 1 (r +2m sinh δ1)(r +2m sinh δ3) into the target SL(2, )/SO(r (1+2, 1),m coupledsinh δ2)( tor three+2m dimensionalsinh δ3) Einstein gravity.(r +2 Them non-trivialsinh δ1)(r +2m sinh δ2) χ =0,eϕ = , ϕ2 R ϕ3 i 2 2 e = 2 2 ,e= 2 2 , $(r +2m sinh δ2)(r +2m sinh δ4)% action of an SO(1, 1) subgroup$(r +2 ofm SLsinh(2,δR1))( isr called+2m sinh a Harrisonδ4)% transformation.$(r +2m sinh δ3)(r +2m sinh δ4)% 1 More concretely, and following1 [19] but making2 2 some changes necessitated2 3 by˜ considering˜ a 2 2 2 2 2 m sinh2 δI cosh+2δm ˜I a˜ cos θ [e0 dt˜ a˜ sin θ sinh δ cosh δ0 dφ] . (18) ϕ2 (r +2m sinh δ2)(r +2m sinh δ3) ϕ3 (r +2mreductionsinh δ1)( onr +2 time-like,mA sinh= ratherδ2) than a space-likedt, Killing(I = 1, vector2, 3, 4) we. define a matrix (See also, e.g., (10) e = ,e= I , 2 − } $ 2 2 % $ [20] and2 references therein.):2 r +2% m sinh δI (r +2m sinh δ1)(r +2m sinh δ4) (r +2m sinh δ3)(r +2Here:m sinh δ4) 2m sinh δI cosh δI 2p1+↵2x 2 2 2 A = dt, (I = 1, 2, 3, 4) . 2.1 Staticp1+ Case↵(10)2(x+y) e (1 + ↵ ) p1+↵ I 2 P = e 2 2 2 , 2 (29) r +2m sinh δI e = sinh δ˜coshp1+δ˜cosh↵2 δ˜ sinh δ˜ (cosh1 δ˜ + sinh δ˜) ✓ It is0 straightforward0 ◆ to show that a Harrison transformation: We shall first demonstratesinh3 theδ˜cosh scalingδ˜(sinh limit,2 δ˜ leading+ 2 sinh to2 theδ˜ + subtracte 2 sinh2dgeometryofthegeneralδ˜sinh2 δ˜ ), so that static solution. We perform a scaling limit on the static0 solutions (7)-(10) wh0ere without loss 2.1 Static Case Origin of− subtractedT3 ˜ 3 ˜ geometry2p21+˜↵2y 2 ˜ ˜ ˜ 410˜ 6 ˜ e0 =P sinh= P ,δ coshdet P =δ(coshe δ0 +. sinh δ0) sinh δ0 cosh(30)δ0(3H sinh= δ + 2 sinh, δ) . (19) (34) of generality we take three equal charges and the fourth one different by defining F1 =1F2 = We shall first demonstrate the scaling limit, leading to the subtractedgeometryofthegeneral 3 − ∗ Taking H SO1(1, 1)Fi. which Subtractedand acts2 on P .geometryas We use – theas “tilde” a scaling notation limit forof near all the-horizon variables, with the✓ choice◆ of static solution. We perform a scaling limit on the static solutions (7)-(10) wh2∗ereFAgain, without≡ we loss takeF ≡˜ theF scaling˜ ˜ limit˜ (11),˜ and˜ furthermore we take for the rotational parameter: charge parametersblack holeδ1 = w/δ2 three= δ3 -largeδ andT chargescorrespondsδ4 δ0. WeQ, to: take(mapped the following on m, a, scaling Π , Π limit) with ϵ 0: of generality we take three equal charges and the fourth one different by defining F = F = P HPH≡ , ≡ (31)c s → ∗ 1 2 ! 3 ˜ 1 pM.C.,2 Gibbons 1201.0601p 2 1 F and 2 . We use the “tilde” notation for all the variables,it preserves with not theonly choice the properties of r˜ = (30)rϵ, butt = alsotϵ− the, Lagrangianm˜ = mϵ , densitya˜ = (28)aϵ . which1+↵ canx0 be cast1 1+↵ x (20) ∗F ≡ F ≡ F y0 = y, e = ⇤ 2e , charge parameters δ˜1 = δ˜2 = δ˜3 δ˜ and δ˜4 δ˜0. We take the following scaling limit with ϵ 0: Π in the form: 2 ˜ 1/3 2 2 1/31 2 ˜ 2p1+s ↵2x 2 2 2 2 2p1+↵2x ≡ ≡ → 2˜m sinh δ Q =2mϵ− ( Π0c =Πs)⇤ ,[ +sinh δ0 = (e2 2 , (1 + ↵ )(11) )] ; ⇤ =( + 1) e . (35) ≡1 ij 1− 2 Π Π 1 In terms ofp new R coordinates(ij)+ andTr( parameters@iP @jP ) . the metricp1+ takes↵ thec(32)forms (3), where only the warp r˜ = rϵ, t˜= tϵ− , m˜ = mϵ , ii. Subtracted geometry1+↵2 - as an infinite boost Harrison − Π2 factor changes:⇣ ⌘ 2 ˜ 1/3 2 2 1/3 2 ˜ It is straightforwards where the to “tilde”showtransformations that coordinates a Harrison on and transformation: the parameters originalNote, this of BH the transformation scaledM.C., solutio canGibbonsn also are 1201.0601 related be determined to those as of an the analytic continuation of transformations 2˜m sinh δ Q =2mϵ− (Πc Πs) , sinh δ0 = 2 2 , (11) ≡ − Πc Πssubtracted geometry for the four-chargegiven static in Sectionblack hole. 2 of [20]. In the AVirmani Harrison latter 1203.5088 case transformation the metric of in the the limit of an infinite boost corresponds − ∆ ∆ =(2m10)3r(Π2 Π2)+(2m)4Π2 (2m)2a2(Π Π )2 cos2 θ . (21) (unsbtracted) blackSO(1,1):0 hole solution H is of the to , c form s1. (7), One but may with verifySahay, thes su that Virmanibtracted (34) 1305.2800(33) with geometryc 1s inthe the metric Einstein-Maxwell gravity (↵ = 0) takes the where the “tilde” coordinates and parameters of the scaled solution are related to those of the → ⇠ 1 −! − −! (7) is the same, except for the✓ warp factor:◆ Schwarzschild metricM.C., Guica, to the Saleem Robinson-Bertotti 1302.7032.. one. This type of transformation was employed subtracted geometry for the four-charge static black hole. In the latter case the metric of the 4We choose theThis units geometry in which 4⇡G with=1.NotethatintheLagrangiandensity(1)16 the subtracted warp factor is sourced⇡G =1andthefield by the scalars: 2 iii. Subtracted geometry – as recentlyturning3 inoff2 [22]. certain2 For anotherintegration4 2 work, constants relating the Schwarzschild geometry to AdS2 S , see [23]. (unsbtracted) black hole solution is of the form (7), but with the subtractedstrengths geometry di↵er by a factor the metric of p2. ∆0s ∆s =(2m) r(Πc Πs)+(2m1 ) Πs . (12) ⇥ 2U 2m in harmonic functions→ of asymptoticallyIn the case− offlat↵ =blackp , weholes shall act with (34) on the2 Schwarzschild solution with e =1 r , (7) is the same, except for the warp factor: 2ma(Πc Πs)cosθ 3 Q = 0, = 0. The transformationϕ1 ϕ (35)2 withϕ3 =1resultsin⇤ = 2m , and the metric (6) with The sources supportingχ1 = thisχ2 geometry= χ3 = are obtainedBaggio, −de by2 Boer, taking Jottar, the,e Mayerson scaling= e limit1210.7695= (11)e in= (10)1 (with, r(22) 3 2 2 4 2 9 − Q 2 ∆0s ∆s =(2m) r(Π Π )+(2m) Π . “tilde” coordinates(12) and parameters): the subtractedAn, geometry M.C., Papardimitriou warp factor: 1602.0150 ∆ → c − s s and the gauge potentials: 2 4 3 The sources supporting this geometry are obtained by taking the scaling limit (11) in (10) (with ϕ1 ϕ2 ϕ3 Q s0 = r s =(2m) r, (36) à non-extremal blackχ1 =holeχ2 =microscopicχ3 =0,eproperties= e = e associated= 1 , with its! “tilde” coordinates and parameters): 2 2 2 2 2 ∆2s horizon are rcaptured(2 mby) aa anddual[2m the Πfields scalar theoryr(Π fieldc of andΠ subtracteds) the]cos electricθ geometry field strength : 2 3 A = dt + − − dt ϕ1 ϕ2 ϕ3 Q While one can in principle perform a scaling limit with three unequal large charges Qi (I =1, 2, 3), by χ1 = χ2 = χ3 =0,e = e = e = 1 , −Q 1 Q∆ 3 3 2 2 replacing in the scaling limit (11) Q (Π =1QI ) 2,appropriatepowersof2 2 QI in the scalar2 fields22mϕi (i =12, 2, 3) 1 ∆s 2ma(Πc→ ΠI s) sin θ (2m) a (Πc eΠsp)3 =cos θ , Ftr = , (37) and gauge field strengths F1,F2, 1 can be removed without loss of generality, resulting inr the same3 gauge 2m 3 ∗ ∗F − 1+ − r dφr, While one can in principle perform a scaling limit with three unequal large charges Qchoicei (I =1 for, sources2, 3), by (15). − Q ! ∆ " 3 1 replacing in the scaling limit (11) Q (Π QI ) 3 ,appropriatepowersofQI in the scalar fields ϕi (i =1, 2, 3) 3 2 i.e.,2 this is the static2 2 subtracted geometry3 of Subsection2 2.1, with ⇧c =1, ⇧s = 0. → I=1 Q [(2m) ΠcΠs + a (Πc Πs) cos θ] Q 2ma(Πc Πs) sin θ and gauge field strengths F1,F2, 1 can be removed without loss of generality, resulting in the same gauge= The subtracted− geometrydt + for the Kerr spacetime− can bed obtainedφ , (23) by reducing the spacetime choice for sources (15). ∗ ∗F A 2m(Π2 Π26)∆ ∆ onc − thes 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 resulting in field strengths with both electric and magnetic components. The (formally infinite)⌘ ⌘ 6 2 , F = F = 2 F and = p2 , i.e. an Einstein-Dilaton-Axion gravity with factors of Q can again be removedp from3 ⇤ gauge1 potentials2 ⇤F1 ⌘ by3 removinF2g correspondingF factors from two U(1) gauge fields and respectiveq dilaton couplings ↵ = 1 and ↵ = p3. The subtracted scalar fields, and thus the sources take the canonical form: 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 of2 (1). This has recently been confirmed a(Πc Πs)cosθ ϕ1 ϕ2 ϕ3 (2m) χ1 = χ2 = χ3 =[14]. − ,e = e = e = 1 , (24) −These results2m demonstrate that the subtracted∆ geometry2 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 r (2m)a2[2mΠ2 r(Π Π )2]cos2 θ A = dt + any physicals − propertyc − s of the originaldt black hole solution which is invariant under the duality −2m transformation∆ of M-theory remains the property of the subtracted geometry. For example the (2m)2a2(Π Π )2 cos2 θ a(Π Π ) sin2 areaθ(1 + of the horizon isc − unchanged.s ) dφ, − c − s ∆ 4 2 2 2 2 4 2 (2m) ΠcΠs +(2m) a (Πc Πs) cos θ (2m) a(Πc Πs) sin θ = 232 Asymptotically− dt + Conical Metrics− dφ , (25) A (Πc Πs)∆ ∆ The− scaling limit, or equivalently the subtraction process, alters the environment that our black Again, the sources supporting thisholes geometry find themselves are inthose [12, 13]. of the In fact minim the subtractedal supergravity geometry in metric five- is asymptotically of the dimensions, where F and are theform five-dimensional Maxwell and Kaluza-Klein field strengths, F R 2p respectively. ds2 = dt2 + B2dR2 + R2 d✓2 +sin2 ✓2d2) (38) R0 The scaling limits (11),(20) are reminiscent of the near-BPS dilute gas approximation [8], which were generalized to rotating four-dimensional black holes in [18]. As10 a natural consequence, 2 the subtracted geometry of general black holes is a Kaluza-Klein coset of AdS3 4S just as in the dilute-gas approximation [18]. Furthermore, there is an analogous microscopic× interpretation

8 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 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 No time iii) Vacuum polarization <φ2> 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 1612.06766 iv) Thermodynamics of subtracted geometry

via Komar integral: M.C., Gibbons, Saleem 1412.5996 à Systematic approach via variational principle highligts II. Thermodynamics via variational principle An, M.C., Papadimitriou 1602.0150 Following lessons from AdS geometries Heningson, Skenderis ’98; Balasubramanian ,Kraus ’99; deBoer,Verlinde2 ’99; Skenderis, Solodukhin ’99… achieved through an algorithmic procedure for subtracted geometry:

• 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 transf.)

• Covariant boundary term, Sct, to the bulk action - determined by solving asymptotically the radial Hamilton-Jacobi eqn. à Skenderis,Papadimitriou ’04, Papadimitriou ’05 • Total action S+Sct independent of the radial coordinate • First class constraints of Hamiltonian formalism lead to conserved charges associated with Killing vectors.

• Conserved charges satisfy the first law of thermodynamics 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 − √√∆ A G 2 ∆ 2 G ¯ ! 2 √ 2 X" 2 ¯2 ds2 = dr¯ (dt + ) + ∆ dθ + sin θdφ , (2m) X a (−Πc√∆Πs) A G eη = , χ = − cos θ, ! " √∆ (2m)2 2m a (Π Π ) eη = , χ = c − s cos θ, (2m)4a (Π √∆Π ) (2ma2m)2 cos2 θ (Π Π )2 +(2m)4Π Π A0 = c − s sin2 θdφ¯ + c − s c s dt,¯ ∆ 4 (Π22 Π22) ∆ 2 4 0 (2m) a (Πc Πs) 2 ¯ (2mac) cos s 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 − dt¯+ − − dt¯ −2m − 1− (2m#$)3(Π +(2maΠ ))2(Π Π ) % (2ma)2(Π2m∆Π )[2mΠ +¯r(Π Π& )] cos2 θ !A = r¯ m c s " c − s dt¯+ c − s s c − s dt¯ −2m − − (2m)3(Π + Π ) 2 2 2 2m∆ ' ! 2c s (2"ma) (Πc Πs) cos θ ¯ + a(Πc Πs)sin θ 1+ − 2 d2φ, 2 (3.1) − (2∆ma) (Πc Πs) cos θ ' + a(Π !Π )sin2 θ 1+ − " dφ¯, (3.1) c − s ∆ where ! " where 2 2 2 2 2ma 2 X =¯r 2mr¯ + a ,G= X a sin θ, = ((Πc Πs)¯r +2mΠs)sin θdφ¯, − 2 2 − 2 A 2 G −2ma 2 ¯ 3X =¯2 r 2 2mr¯ + a 4,G2 = X 2 a sin θ,2 2 = ((Πc Πs)¯r +2mΠs)sin θdφ, ∆ =(2m) (Πc Π−s)¯r +(2m) Πs (2ma) −(Πc Πs) cos Aθ, G − (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 solutions and is beyond the scope of the present paper. However, we can consider the variational problem• withinIdentify the class normalizable of stationary solutions and (3.1). non To-rormalizable this end, it is convenient modes to reparameterize these backgroundsproblem by within means the of class a suitable of stationary coordinate solutions transfo (3.1).rmation, To this accompanied end, it is convenient by a relabeling to reparameterize of the freethese parameters. backgrounds In particular, by means we of introduce a suitable the coordinate new coordinates transformation, accompanied by a relabeling of the free parameters. In particular, we introduce the new coordinates Introduce new coordinates: 4 3 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, Πc, Πs for: we define the new parameters 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) − 5In order to compare this background with the expressions given in eqs. (24) and (25) of [14], one should take into r , 5rIn, 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,beforethedualizationofA, 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 constant pure2 gauge term. Moreover, there is a typo in eq. (25)of[14]:theterm2mΠs cos θdt¯ should be replaced by 2mΠ (Π B− -Πnon)cos-renormalizableθdt¯. 2 mode s s by 2mc Π (Π − Π )cos θdt¯. (fixed ups tos bulk cdiffeomorphisms & global gauge 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 2TheSTUmodelanddualityframesa = ,m= B/2. (3.5) which can be inverted in order to2 express2 1 the old parameters2 intermsofthenewones,namely In this section we review the bosonic2 ℓ ω sector+ of√ ther+ 2-charge√r truncation of the STU model that is 4 − − relevant for describing the subtractedℓ2 $ 1 geometries. We will do so in the1 duality frame discussed in % 2& 2 2 6 Rewriting[14], where the both background chargesΠc,s are= electric,2 (3.1) as( in√ wellr terms+ as+ in√ the ofr the one) us newedℓ inω coordinat [17],+ where(√esr there+ and√ is oner parameters) electric, and we obtain one magnetic charge. We willB refer!2 to these frames− as± ‘electr" ic’ and4 ‘magnetic’− respectively.− # As it will become clear from2 the2 subsequent analysis,3 in order to compare the thermodynamics in the two η B /ℓ Bℓℓωω frames,e = it is necessary toa = keep track, ofχ boundary= 2 cos termsθ, introduced,m by the= dualityB/2. transformations. (3.5) r + ℓ2ω2 sin2 θ 2 2 1 B 2 2 ℓ ω + 4 √r+ √r 2.1 Magnetic3 frame3 − − 0 ' B /ℓ $ 2 2 A = √r+r kdt%+ ℓ ω sin θd&φ , 6 RewritingThe bosonic ther + Lagrangian backgroundℓ2ω2 sin2 ofθ the (3.1) STU in− model terms in the of the duality new frame coordinat used ines [17] and is given parameters by we obtain

2B 2cos2θ %1 3 1 2ηa a &a ηA = 2κ4B4 =/ℓR ⋆ 1 ( ⋆ωd√ηa r+drηaℓ kdtω e + rd⋆ dφχ) , dχ e = r + ℓL2ω2 sin2 θ− 2,− χ∧ =−− 2 cos θ, ∧ 2 2 1 2 B12 r + ℓ ω sin θη0 0 0 2ηa η30 a a 0 a a 0 3 ℓ 1 e− ⋆ F F e ωℓ− ⋆ (F + χωF√)r+(rF kdt+ χ F rd) φ ωℓ A = 3 r 3 − 2(r + r ∧) kdt− 2+ cos2 θ ∧ − − + dφ, 0 ' B /ℓ 1 + 1 2 2 1 2 2 2 A = −B −+2 C √χra+Fr−b kdtF c ++ Cℓ ωBχsinaχbFθ0dφF,c + rC+ ωχaℓχbχsincF 0 θ F 0, B(2.1) r + ℓ2)ω2 sin22θ abc −∧* 2 abc ∧ ) 6 abc ∧ * ℓ2dr2 (r r )(r r ) (2 B cos2θ 2 2 3 + 2 2 2 2 whereds η=a (a =1r +, 2ℓ, 3)ω aresin dilaton% θ fields and η0 = a=1 ηa.Thesymbol& − − −Cabc isk pairwisedt + symmetricℓ dθ A = 2 ( ω√(rr+r r kdt)(r+ rdr φ)) ,− r with C123 =1andzerootherwise.TheKaluza-Kleinansatzforobtainin2 2 − − + gthisactionfromthe r'+ ℓ ω sin θ ) − − − ! * 6-dimensional action (1.1) is given explicitly in [17].3 Thisframepossessesanexplicittrialitysym-2 2 ω 3 r r 2 ℓ 1 a ωℓ 2 ω√rℓa+rrsinkdtθ rdφ ωℓa √ + metry,A = exchangingr the three(r+ + gauger ) fieldskdtA+,thethreedilatonscos θ+ η and− the three− axionsdφ+ χ .Inthisdφ, − kdt . (3.6) −B − 2 − B r a+ ω22ℓ22 sin22 θ −0 B r frame, the subtracted) geometries source* all three gauge fiel)dsrA+magnetically,ℓ ω sin θ while) * A is electri- * cally sourced. Moreover, holographic renormalization2 2 turns out to be much more straightforward (2 2 ℓ dr (r r )(r r+) 2 2 2 2 indsSeveral this= framer comments compared+ ℓ2ω2 sin with areθ the in electric order frame. here. Firstly,−' the− two− parametk dters+ rℓ dθare the locations of the (r r )(r r+) − r ± outerIn and order inner to describe horizons the) subtracted respectively,− − geometries− and clearly it suffices corresp to considerond a to truncation normalizable of* (2.1), perturbations. A ' 2 2 1 2 3 2 corresponding to setting η1 = η2 = η3 η, χ1 = χ2 = χ3 χ,andA = A = Aω rA.Ther straightforward calculation shows≡ that ω is also a normalizableℓ≡ r sin θ mode. We√≡ + will explicitly confirm resulting action can be written in the σ-model form + dφ − kdt . (3.6) 2 2 2 − r this later`Vacuum’ on by showing solution that the long-distance divergencesr + ℓ ω sinofθ the) on-shell action are* independent 1 4 1 I µ J Λ Σµν µνρσ Λ Σ of ωS4.Settingthenormalizableparameterstozerowearriveatth= 2 d x√ g R[g] IJ∂µϕ ∂ ϕ ΛΣFµν F ΛΣϵ Fµebackgroundν Fρσ + SGH, (2.2) Several2κ4 commentsobtained− by are turning in− order2G off r here.+, r-, ω Firstly, −– Zthree' thenormalizable two− R paramet modes:ers r are the locations of the "M # $± outerwhere and inner horizons2 2 respectively, and clearly correspond to normalizable perturbations. A η B /ℓ 1 0 2 ℓ TheAsymptoticallye asymptotic= structure, conicalχ of=0 (stationary) box,A – conformal conical=03 ,A backgrou to AdS=nds2 BxS iscos parameterizedθdφ, A by= the threerkdt, non- straightforward calculation showsSGH that= 2ω is alsod x√ aγ normalizable2K, mode. We will(2.3) explicitly confirm zero constants B√,rℓ and k.Inthemostrestrictedversionofthevariationalproblem,2κ4 ∂ − −theseB three this later on by showing that the long-distance" MAsymptotically divergences conicalof the on-shellbox action are independent is theparameters standard Gibbons-Hawking should bedr kept2 fixed. [36] term However, and we there have is defined a 2-paramethe doubletster family of deformations of these of ω.Settingthenormalizableparameterstozerowearriveatthds2 = √r ℓ2 rk2dt2 + ℓ2dθ2 + ℓ2 sin2 θdφ2 , ebackground( (3.7) boundary data still leadingη2 to a well posedA0 variational problem, as we now explain. The first de- The asymptotic structure of (stationary)ϕI = r conical,A− backgrouΛ = nds is,I parameterized=1, 2, byΛ the=1 three, 2, non- (2.4) formation corresponds2 )2 toχ the transformationA of the boundarydatainducedbyreparameterizations* zero constants B, ℓ and k.Inthemostrestrictedversionofthevariationalproblem,B /ℓ # $ # $ these three ℓ which weof the shalle radialη = consider coordinate., as Namely, theχ =0 vacuum,A under the0 solution.=0 bulk,A diffeomorphi The= factBsmcos thatθdφ,the backgroundA = rkdt, (3.7) is singular parameters shouldas well be as kept the fixed. 2 2matrices However,√r there is a 2-parameter family of deformations of these −B × 4 boundary datadoes still not leading pose to any a well di posedfficulty variational since it3η prob shouldlem,r η as2λ only− wer, nowη beλ > explain.viewed0, The as an firstasymptotic de-3 3 2 solution(3.9) that helps us 30 2 1 e− +3e− χ 3e− χ 1 χ 2 χ formation correspondsIJ =Non to the2-normalizable transformation2η , 2 drΛΣ of(fourth)= the boundar2 mode2 ydatainducedbyreparameterizationsη 2 →B,2 along2 η with2, l and2 ΛΣ k=, fixed3 up2 to( . (2.5) 1/4 to properlyG ds03 formulate=e √r ℓ theZ variational4rk dt 3e+− problem.ℓχ dθ +3ℓe− Changingsin θdφR th, eradialcoordinateto4 χ 3χ ϱ = ℓr(3.7),the of the radial coordinate.theradial# boundary Namely, diffeomorphism$ under parameters ther2 bulk− transform: di#ffeomorphi as sm $ # 2 $ vacuumAs usual, metricϵµνρσ = becomes√ )g εµνρσ denotes the totally antisymmetric Levi-Civita* tensor, where εµνρσ = − 4 3 which1istheLevi-Civitasymbol.Throughoutthispaperwechoose we shall considerr asλ the− r, vacuumλ >k0, λ solution.k, 6ℓ λℓ The,B factB.the that orientationthe(3.9) background in so that (3.7)(3.10) is singular ± → 2 2 2→ ϱ →2 2 2→ 2 2 2 M doesεrtθφ not=1.WenoteinpassingthattheLagrangian(2.2)isinvariant pose any difficultyds since=4 itdϱ should onlyk bedt viewed+ ϱ d asθ under an+sinasymptotic theθ globaldφ , symmetry solution that helps us(3.8) the boundary parameterstransformationThisand transformtransformation global U(1) as issymmetry: a direct analogue− ofℓ the so called Penrose-Brown-Henneaux (PBH) diffeo- to properly formulate the variational problem.+ , Changing theradialcoordinatetoϱ = ℓr1/4,the morphisms inη asymptotically2 η AdS2 backgrounds0 3 0 [40], which% induce2 a Weyl2 transformation& on the which is a speciale k caseµλe3k,, ofχ theℓ λℓµ conical−,Bχ,A metricsB. µ A ,A discussedµA, in ds [14].(3.10)ds Di, fferent conical(2.6) geometries are vacuumboundary metric becomes sources.→→ The PBH→→ diffeomorphisms→ → imply that→ the bulk →fields do not induce boundary supportedwhere µ is by an arbitrarydifferent non-zero matter constant fields. Although parameter. we focus on the specific conical backgrounds obtained This transformationfields, is a direct but only analogue a conformal of the so structure called Pen,thatisboundaryfieldsuptoWeyltransformations[32].Thirose-Brown-Henneaux6 (PBH) diffeo- s 3 32 2 2 ϱ 2 2 2 2 2 2 morphisms inas asymptotically solutionsdictateswhich of that AdSkeep the the backgrounds STUk variationalBds/l model-=4fixed [40], problemdϱ here, which must in weduce be expectk formulated adt Weyl+ that transformationϱ ind ourθterms+sin analysi of conformal onθd thes,φ modified, classes rather accordingly than (3.8) for the representatives of the conformal class− [22].ℓ In the case of subtracted geometries, variations of the boundary sources.different The matter PBH diff sectors,eomorphisms applies imply to that general the+ bulk,5 asymptoticallyfields do not induceconical boundary backgrounds. fields, but only a conformalboundary structure parameters,thatisboundaryfieldsuptoWeyltransformations[32].Thi of the form % s & which6 is a special case of the conical metrics discussed in [14]. Different conical geometries are dictates that the variationalSince these problem solutions must carry be formulated non-zero in magneticterms of charge,conformal the classes gauge rather potential than A must be defined in the north supported by different matter fields.δ k Although=3ϵ k, δ weℓ = focusϵ ℓ, onδ theB =0 specific, conical backgrounds(3.11) obtained representatives(θ < ofπ the/2) conformal and south class hemispheres [22]. In the respectively case1 of subtracted as1 [38,1 39], geometries,A1 north 1= variationsA − Bdφ ofand the Asouth = A + Bdφ,whereA is the boundary parametersasexpression solutions of given the of form in the (3.6). STU model here, we expect that our analysis, modified accordingly for the differentcorrespond matter sectors, to motion applies within to the general equivalence asymptotically class (anisotrconicalopic conformal backgrounds. class) defined by the transformation (3.10), and therefore lead to a well posed variational problem. δ1k =3ϵ1k, δ1ℓ = ϵ1ℓ, δ1B =0, (3.11) 6 Aseconddeformationoftheboundarydatathatleadstoawellposed variational problem is Since these solutions carry non-zero magnetic charge,8 the gauge potential A must be defined in the north correspond(θ to< motionπ/2) and within south the hemispheres equivalence respectively class (anisotr asopic [38, conformal 39], Anorth class)= A defined− Bdφ and by theAsouth = A + Bdφ,whereA is the transformationexpression (3.10), given and therefore in (3.6). lead to a well posedδ2k =0 variational, δ2ℓ = problem.ϵ2ℓ, δ2B = ϵ2B. (3.12) Aseconddeformationoftheboundarydatathatleadstoawellposed variational problem is To understand this transformation, one must realize 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 ensures8 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. The significance of the2 parameterαη 2 B is twofold. It corresponds to the background magnetic field dsα = e ds , (3.13) in the magnetic frame and variations of B are equivalent to the global symmetry transformation for some α,whichwillbeimportantfortheupliftoftheconicalbackgr(2.6) or (2.12) of the bulk Lagrangian. Moreover,ounds as we willto five discuss dimensions. in the next section, it enters The significance of thein the parameter covariantB is asymptotic twofold. It second corresponds class constraints to the background imposing magnetic conical boundary field conditions. The in the magnetic frametransformation and variations (3.12) of B isare a variation equivalent of toB thecombined global symmetrywith a bulk transformation 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 modes k and B/ℓ fixed. The relevant bulk diffeomorphism is a rescaling of the radial coordinate of in the covariant asymptotic second class constraints imposing conical boundary conditions. The 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), accompanied4Boundarycountertermsandrenormalizedconservedcharge by a rescaling t λ3t of the time coordinate. 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 KangThe for canonical interesting momenta discussio conjugatens. M.C. to the would induced like fields to thank on Σ Garyu following from the Lagrangian Gibbons and Zain Saleem for discussions(A.2) are 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 ij δL 1 ij ij on Theoretical Frontiers in Black Holesπ = and Cosmology,= 2 √ γ JuneKγ 8–19,K 2015,, where part of this work (A.4a) δγ˙ij 2κ − − was carried out. This research is supported in part4 by the DOE Grant Award DE-SC0013528, δL 1 1! "J i J (M.C.), the Fay R. and Eugene L. LangbergπI = = EndowedN − Chair√ γ (M.C.IJ ϕ˙ )andtheSlovenianResearchN ∂iϕ , (A.4b) δϕ˙I −2κ2 − G − Agency (ARRS) (M.C.). 4 δL 2 ! " 2 πi = = N 1√ γ γij A˙Σ ∂ aΣ N F Σji √ γ ϵijkF Σ . (A.4c) Λ ˙Λ 2 − ΛΣ j j j 2 ΛΣ jk δAi −κ4 − Z − − − κ4 − R Appendices # # $ $ Λ Notice that the momenta conjugate to N, Ni,anda vanish identically, since the Lagrangian (A.2) Λ A Radial Hamiltoniandoes formalism 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. The canonical momenta (A.4) allow us to perform the Legendre transform of the Lagrangian (A.2) In this appendix we present in some detail the radial Hamiltonian formulation of the reduced STU to obtain the radial Hamiltonian σ-model (2.2). This analysis can be done abstractly, without reference to the explicit form of the IJ σ-model functions , ΛΣ and ΛΣ,anditthereforeappliestotheelectricLagrangian(2.10)H = d3x πijγ˙ + π ϕ˙ I + πi A˙ Λ L = d3x N +asN i + aΛ , (A.5) G Z R ij I Λ i − H iH FΛ well, provided AL, ΛΣ• andRadialΛΣ areHamiltonian replaced with% theirformalism electric frame analogues in(2.11).% Z R # $ ! " The first step towardsto a determine Hamiltonianwhere Sct formalism, to the bulk is pickingaction S a suitable radial coordinate u such that constant-u slices, which we will denote by Σu,arediffeomorphic to the boundary ∂ of . 2 M M Suitable radial coordinateκ u,4 such that constant1 -u slicesij kl Σu IJ Moreover, it is convenient to choose u to be= proportional2 toγik theγjl geodesicγijγkl distanceπ π + between(ϕ)πI anyπJ fixed H14 − √ γ − 2 G point in and a point in Σ ,suchthatΣu →Σ ∂M ∂as u →as ∞u. .Giventheradialcoordinateu, M u u →− M& & →∞ ' we then proceed with an ADM-like decomposition1 ΛΣ of the metric2and gauge fieldskl [47]M i 2 ipq N Decomposition of the metric+ and( ϕgauge) πΛ ifields:+ 2 √ γ ΛM (ϕ)ϵi Fkl πΣ + 2 √ γ ΣN (ϕ)ϵ Fpq 4Z κ4 − R κ4 − R 2 2 i 2 & i i j '& '' ds =(N + NiN )du√+2γ Nidudx +1γijdx dx ,I i J Λ Σij + − R[γ]+ IJ(ϕ)∂iϕ ∂ ϕ + ΛΣ(ϕ)F F , (A.6a) L Λ Λ i 2κ2 − 2G Z ij A = a du + Ai dx , 4 & (A.1)' The canonical momentai conjugateij to thei I inducedΛij fields on2Σ following fromkl theΣ Lagrangian = 2Djπ + πI∂ ϕ + F πΛj + √u γ ΛΣ(ϕ)ϵj F , (A.6b) where xi = t, θ, φ .Thisismerelyafieldredefinition,tradingthefullycovari(A.2) are H − κ2 ant− R fields g andkl { } { } Decomposition leads to the radial Lagrangian L w/& canonical4 momenta: µν ' AL for the induced fields N, N , γ , aΛ and AΛ on Σi .Insertingthisdecompositionintheσ-model µ i δLij 1 Λ = i DiπΛ.u (A.6c) πij = = F√ γ−Kγij Kij , (A.4a) action (2.2) and adding the Gibbons-Hawking2 term (2.3) leadstotheradialLagrangian δγ˙ij 2κ4 − − Since the canonical momenta conjugate to the fields N, N ,andaΛ vanish identically, the corre- δL 1 1! "J i J i π = = N − √ γ ϕ˙ N ∂ ϕ , Λ (A.4b) 1 I δϕ˙spondingI 2 Hamiltonw/ momenta1 equationsIJ conjugate leadi toto theN, N firsti, and class a vanish constrai ànts 3x √ 2 −2κ4 ij − G − I i I J j J L = 2 d N γ R[γ]+K KijK 2 IJ(ϕ) ϕ˙ N ∂iϕ ϕ˙ N ∂jϕ 2κ − i δL − 2 1 − 2N G! ij Σ −"Σ Σji − 2 ijk Σ 4 π = = N − √ γ γ A˙ ∂ a N F i √ γ ϵ F . (A.4c) ! " Λ ˙Λ 2 First classΛΣ constraints:j j =j = Λ 2=0, ΛΣ jk (A.7) δA −κ4 − Z #− − $# − κ4 − R$ 2 i ij ˙ Λ Λ # k# Λ ˙ Σ $ H ΣH $ Fl Σ ΛΣ(ϕ)γ Ai ∂ia N Fki Aj ∂ja N Flj (A.2) 2 Λ Notice−N thatZ which the momenta reflect conjugate respectively− − to N di, ffNeomorphismi,anda −vanish invariance identically,− under since radial the Lagrangian reparameterizations, (A.2) diffeomor- does not contain any% radial derivatives of1 these&% fields. It follows that the& fields N, N ,andaΛ are Λ 4 (phismsϕ)ϵijk alongA˙ Λ the∂ a radialΛ F Σ slices Σu (andϕ)∂ aϕUI ∂(1)iϕJ gauge invariance(ϕ)F ΛF forΣij every,i gauge field Ai . Lagrange− RΛΣ multipliers, implementingi − i jk three− 2 firstGIJ class consti raints,− whichZΛΣ we willij derive momentarily. The canonical momenta (A.4) allow us to perform the Legendre transform of the Lagrangian' (A.2) Hamilton-Jacobi% & formalism where to obtain the radial Hamiltonian 1 The first3 classij constraintsI (A.7)i ˙ Λ are particularly3 useful in thei Hamilton-JacobiΛ formulation of the KHij== d x (˙γπij γ˙ ij D+ iπNI ϕj˙ +DπjNA i) , L = d x N + Ni + a (A.3)Λ , (A.5) Λ i − H H F dynamics,% 2N where− the canonical− momenta are% expressed as gradients of Hamilton’s principal function Λ I # $ ! " is the extrinsic curvaturewhere of the radial[γ,A slices, ϕ ]asΣu, Di denotes a covariant derivative with respect to S δ δ δ the induced metric γij on Σu,whileadot˙standsforaderivativewithrespecttotheHami2 πij = , πi = , π = ltonian. (A.8) κ4 1 ij kl SIJ Λ SΛ I SI = 2 γ γ γijγ π π +δγij (ϕ)πI πJ δA δϕ ‘time’ u. H − √ γ ik jl − 2 kl G i − & & ' Λ Λ I 14 Since1 the momenta2 conjugate to N, N ,anda 2vanish identically, the functional [γ,A , ϕ ] We assume M to be a non-compact space+ ΛΣ with(ϕ) infiniteπ + volume√ suchγ that(ϕ) theϵ klF geModesici πi distance+ √ betweenγ ( anyϕ)ϵipq pointF N 4Z Λi κ2 − RΛM i kl Σ κ2 − RΣN pq S in the interior of M and a point in ∂Mdoesis infinite. not depend& on these4 Lagrange multipliers.'& Inserting4 theexpressions(A.8)forthecanonical'' √ γ 1 + − R[γ]+ (ϕ)∂ ϕI ∂iϕJ + (ϕ)F ΛF Σij , (A.6a) 2κ2 − 2GIJ i ZΛΣ ij 4 & ' i ij i I Λij 2 kl Σ = 2Djπ + πI∂ ϕ + F πΛj + √ γ ΛΣ(ϕ)30ϵj F , (A.6b) H − κ2 − R kl & 4 ' i Λ = DiπΛ. (A.6c) F − 29 Λ Since the canonical momenta conjugate to the fields N, Ni,anda vanish identically, the corresponding 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 (A.2) are ij δL 1 ij ij π = = 2 √ γ Kγ K , (A.4a) δγ˙ij 2κ4 − − ! " ij δL 1 ijδL ij 1 1 J i J π = = √ γ πKI γ= K= , N − √ γ IJ ϕ˙ N ∂iϕ , (A.4a) (A.4b) 2 δϕ˙I −2κ2 − G − δγ˙ij 2κ4 − − 4 ! " ! i δL " 2 1 ij Σ Σ Σji 2 ijk Σ δL 1 1π = = J N −i √ Jγ ΛΣ γ A˙ ∂ja NjF √ γ ΛΣϵ F . (A.4c) πI = = N − √Λ γ ˙ΛIJ ϕ˙ 2 N ∂iϕ , j 2 (A.4b) jk I 2 δAi −κ4 − Z − − − κ4 − R δϕ˙ −2κ4 − G − # # $ $ δL 2 ! " 2Λ πi = = NNotice1√ thatγ the momentaγij A˙Σ conjugate∂ aΣ toNNF, NΣiji,anda √vanishγ identically,ϵijkF Σ since. (A.4c) the Lagrangian (A.2) Λ ˙Λ 2 − ΛΣ j j j 2 ΛΣ jk Λ δAi −κ4 does not− Z contain any radial− derivatives− of these− fields.κ4 It− followsR that the fields N, Ni,anda are Lagrange multipliers,# # implementing$ three first$ class constraints, which we will derive momentarily. Λ Notice that the momentaThe conjugate canonical momenta to N, Ni,and (A.4) allowa vanish us to identically,perform the Legendresince thetransform Lagrangian of the (A.2) 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) The canonical momenta (A.4) allow us to performij theI LegendreΛ itransform− of the LagrangianH iH (A.2)FΛ % % to obtain the radial Hamiltonian # $ ! " where 3 ij Iκ2 i ˙ Λ 1 3 i Λ The canonicalH = d momentax π γ˙ ij conjugate+ πI ϕ˙ 4+ toπΛ theAi inducedL = fieldsd onx ijNΣuklfollowing+ NIJi + froma theΛ Lagrangian, (A.5) = 2 γikγ−jl γijγkl π πH+ H(ϕ)πI πJ F (A.2) are % H − √ γ − 2 % G # − & & $ ' ! " where 1 ΛΣ 2 kl M i 2 ipq N + (ϕ) πΛi + √ γ ΛM (ϕ)ϵi F π + √ γ ΣN (ϕ)ϵ F ij δL 1 ij ij 2 kl Σ 2 pq √ 4Z κ4 − R κ4 − R π = =2 2 γ Kγ K , & '& (A.4a) '' δγ˙ij κ 2κ − −1 4 4 √ γ ij kl 1 IJ I i J Λ Σij = 2 γik!γjl +γij−γ"kl πR[πγ]++ IJ((ϕϕ))∂πiIϕπ∂J ϕ + ΛΣ(ϕ)Fij F , (A.6a) δL γ 1 1 2 2J i J H − √ √ − 2κ4 − 2GG Z πI = I =− & 2 &N − γ IJ ϕ˙ '& N ∂iϕ , ' (A.4b) δϕ˙1 −2κ4 − 2G − 22 ΛΣ i = 2D πij + π ∂iϕklI + MF Λij πi + √ γ (ϕ)ϵ klipqF Σ N, (A.6b) +δL (ϕ2) πΛi + 2 √ ! γj ΛM (Iϕ)ϵi " Fkl πΣΛj+ 22√2 γ ΛΣΣN (ϕ)jϵ klFpq i 4Z 1 H κ− − ijR ˙Σ Σ Σji κκ4 − RR ijk Σ πΛ = = 2 N − √ γ 4ΛΣ γ Aj ∂ja Nj&F 4 2 √ γ ΛΣϵ Fjk'. (A.4c) ˙Λ κ & i '& κ '' δAi − 4 − =Z D π . − − − 4 − R (A.6c) √ γ FΛ 1 − #i Λ # I i J $ Λ $Σij + −2 R[γ]+ IJ(ϕ)∂iϕ ∂ ϕ + ΛΣ(ϕ)Fij F , (A.6a) 2κ − 2G ΛZ Λ Notice that the momenta4 &Since conjugatethe canonical to N momenta, Ni,and conjugatea vanish to identically, the fields' sinceN, N thei,and Lagrangiana vanish (A.2) identically, the corre- Λ does not contain anysponding radial derivatives Hamilton equations of these2 fields. lead to It the follows first class that constraithe fieldsntsN, Ni,anda are i = 2D πij + π ∂iϕI + F Λij π + √ γ (ϕ)ϵ klF Σ , (A.6b) Lagrange multipliers,j implementingI threeΛ firstj class2 constΛΣraints, whichj kl we will derive momentarily. H − κ4 − R i The canonical momenta (A.4) allow us& to perform the Legendre= transform= Λ'=0 of, the Lagrangian (A.2) (A.7) = D πi . H H F (A.6c) to obtainFΛ − theHamiltonian: radiali Λ Hamiltonian which reflect respectively diffeomorphism invariance under radial reparameterizations, diffeomor- Since the canonical momentaphisms along conjugate the radial to the slices fieldsΣ andN, aNU,and(1) gaugeaΛ invariancevanish identically, for every gauge the corre- field AΛ. H = d3x πijγ˙ + π ϕ˙ I + πi A˙ Λ uL = di3x N + N i + aΛ , (A.5) i sponding Hamilton equations leadij toI the firstΛ classi − constraints H iH FΛ % % Hamilton-Jacobi# formalism$ ! " where i First class constraints = = Λ =0 , - Hamilton Jacobi eqs.: (A.7) 2 The first class constraintsH H (A.7)F are particularly useful in the Hamilton-Jacobi formulation of the κ4 dynamics, where1 the canonicalij kl momentaIJ are expressed as gradients of Hamilton’s principal function which reflect& respectively= Momenta 2as di ffγgradientsikeomorphismγjl γ ijofγkl Hamilton’s invarianceπ π + underprincipal(ϕ)π rIadialπ Jfunction reparameterizations, (γ,AΛ,φI): diffeomor- H − √ γ Λ I − 2 G S [γ,A , ϕ ]as Λ phisms along the radial− S& slices& Σu and a U(1)' gauge invariance for every gauge field A . 1 2 ij δ i 2 δ δ i ΛΣ πkl =M S , i π = S , πI = ipqS . N (A.8) + (ϕ) πΛi + 2 √ γ ΛM (ϕ)ϵi Fkl πΣ +Λ 2 √ Λγ ΣN (ϕ)ϵ IFpq 4Z κ − R δγij κ δA−i R δϕ & 4 '& 4 '' Hamilton-Jacobi formalism Λ Λ I √ γSince the momenta1 conjugateI i J to N, Ni,andΛ Σija vanish identically, the functional [γ,A , ϕ ] + − R[γ]+ IJ(ϕ)∂iϕ ∂ ϕ + ΛΣ(ϕ)F F , (A.6a) S 2κ2does− not depend2G on these LagrangeZ multipliers.ij Inserting theexpressions(A.8)forthecanonical The first class constraints4 & (A.7) are particularly useful in the Hamilton-Jacobi' formulation of the dynamics, where the canonical momenta are expressed2 as gradients of Hamilton’s principal function i = 2D πij + π ∂iϕI + F Λij π +deBoer,Verlinde√ γ 2( ’ϕ99,…Skenderis,)ϵ klF Σ , Papadimitriou ’04,…(A.6b) [γ,AΛ, ϕIH]as − j I Λj κ2 − RΛΣ j kl S Solve asymptotically (for `vacuum’& 4 asymptotic solutions)' for i ij δ i δ δ 30 Λ = DiπΛ. π = S , π = S , π = S . (A.6c)(A.8) F − (γ,AΛ,φΛI) Λ I I δγS ij δAi= - Sct δϕ! Since the canonical momenta conjugate to the fields N, N ,andaΛ vanish identically, the corre- Λ I Λ i Λ I Sincesponding the momenta HamiltonS(γ,A , conjugateφ equations) coincides to lead Nwith to, N the thei,and first on- classshella vanish constrai action, identically,nts up to terms the that functional remain [γ,A , ϕ ] Λ I S does not dependfinite on as these Σu → Lagrange ∂M. In multipliers. particular, divergent Inserting thparteexpressions(A.8)forthecanonical of S[γ, A , φ ] coincides with that of the on-shell action.= i = =0, (A.7) H H FΛ which reflect respectively diffeomorphism invariance under radial reparameterizations, diffeomor- 30 Λ phisms along the radial slices Σu and a U(1) gauge invariance for every gauge field Ai .

30 space of equivalence classes of boundary data. We have therefore determined that a complete set of boundary counterterms for the variational problem in the magnetic frame is

1 3 B η/2 4 α η α 2η ij 1 4η 0 0ij S = d x√ γ e − +(α 1)e− R[γ] e− F F + e− F F . (4.11) ct −κ2 − 4 B2 − − 2 ij 4 ij 4 ! " # As wespace mentioned of equivalence at the beginning classes of of boundary this section, data. the We freedom have thereto choosefore thedetermined value of the that parameter a complete set α doesof boundary not correspond counterterms to a choice for theof scheme. variational Instead, problem it is a in dir theectmagnetic consequence frame of is the presence of the second class constraints (4.4). The scheme dependence corresponds to the freedom to include 1 3 B η/2 4 α η α 2η ij 1 4η 0 0ij space of equivalenceadditional classesS = of finite boundary locald x terms,√ data.γ whiche We have do− not there a+(ffectαfore the1) determinede divergen− R[γ] tpartofthesolution.Lateronwe thate− aF completeF + e− setF F . (4.11) ct −κ2 − 4 B2 − − 2 ij 4 ij of boundary countertermswill consider for the situations4 ! variational where problem additional" in conditions the magnetic on th frameevariationalproblemrequireaspecific is # valueAs for weα mentioned,orparticularfinitecounterterms. at the beginning of this section, the freedom to choose the value of the parameter 1 3 B η/2 4 α η α 2η ij 1 4η 0 0ij Sct = d x√ αγdoese not correspond− +(α to a1) choicee− R[ ofγ] scheme.e− Instead,FijF + it ise− a dirFectF consequence. (4.11) of the presence of −κ2 − 4 B2 − − 2 4 ij 4 ! 4.1.2the second The variational" class constraints problem (4.4). The scheme dependence corresponds# to the freedom to include

As we mentioned atGiven theadditional beginning the counterterms finite of this localS section,ct terms,,theregularizedactioninthemagneticframeisdefinedasth thewhich freedom do notto aff chooseect the the divergen valuetpartofthesolution.Lateronwe of the parameter esumof α does not correspondthewill on-shell to aconsider choice action situations of (2.2) scheme. and where the Instead, counterterms additional it is a dirconditions (4.11)ect on consequence the onregulating thevariationalproblemrequireaspecific of the surface presenceΣu,namely of the second class constraintsvalue for (4.4).α•,orparticularfinitecounterterms.Hamiltonian The scheme dependence Formalism corresponds with ``Renormalized’’ to the freedom to Action include additional finite local terms, which do not affect the divergenSreg = tpartofthesolution.LateronweS4 + Sct. (4.12) 4.1.2 The variational problem will consider situationsAs is shown where in additional appendix B, conditions the limit on thevariationalproblemrequireaspecific value for α,orparticularfinitecounterterms.Covariant Sct calculated for vacuum asymptotic solutions Given the counterterms Sct,theregularizedactioninthemagneticframeisdefinedasth2 esumof (conformal to AdS2 x S geometry) space of equivalence classes of boundarySren =lim data.Sreg We, have therefore determined that a complete(4.13) set the on-shellof boundary action (2.2)counterterms and the for counterterms the variationalr (4.11) problem on in the the regulatingmagnetic frame surface is Σu,namely 4.1.2 The variational problem →∞ exist and we will refer1 to its3 valueB asη/ the2 4 ‘renormalized’αSreg = S4 + on-η Sctshell. α action,2η followingij 1 4 standardη 0 0ij termi-(4.12) S = d x√ γ e − +(α 1)e− R[γ] e− F F + e− F F . (4.11) Given the countertermsnologyS in,theregularizedactioninthemagneticframeisdefinedasth the contextct −κ2 of the AdS/CFT− 4 duality.B2 − − 2 ij esumof4 ij ct 4 ! " # Agenericvariationoftherenormalizedon-shellactiontakAs is shown in appendix B, the limit es the form the on-shell action (2.2) and theAs we counterterms mentioned at the (4.11) beginning on the of thisregulating section, the surface freedomΣtou,namely choose the value of the parameter α does not correspond to a choice of scheme. Instead, it is a direct consequence of the presence of 3 Sijren =limiSregΛ, FiniteI – Independent of r (4.13) the second classδSSreg constraintsren==limS4 + (4.4).Sctd. x TheΠ schemeδγij r+ dependenceΠΛδAi + cΠorrespondsI δϕ , to the(4.12) freedom to(4.14) include r →∞ additional finite local terms,→∞ ! which do not affect the divergentpartofthesolution.Lateronwe exist and we will refer to its value as the$ ‘renormalized’ on-shell action,% following standard termi- As is shown in appendixwhere the B, renormalizedthewill limit considerRenormalized situations canonical canonical where momenta additional momenta: are given conditions by on thevariationalproblemrequireaspecific nology invalue the for contextα,orparticularfinitecounterterms. of the AdS/CFT duality. Agenericvariationoftherenormalizedon-shellactiontakijSren ij=limδSctSreg, i i δSct es theδS formct (4.13) Π = π +r , Π = π + , ΠI = πI + . (4.15) 4.1.2 The variational→∞ problemΛ Λ Λ I δγij δAi δϕ 3 ij i Λ I exist and we will refer to its valueGiven the as the counterterms ‘renormalized’δSrenS=limct,theregularizedactioninthemagneticframeisdefinedasth on-dshellx Π action,δγij + followingΠ δA + standardΠI δϕ , termi- esumof(4.14) Inserting the asymptotic form of ther backgrounds (3.6) into theΛ definitionsi (A.4) of the canonical nology in the context of the AdS/CFTthe on-shell action duality. (2.2) and the→∞ counterterms! (4.11) on the regulating surface Σu,namely momenta and in the functional derivatives (4.8) we$ obtain % Agenericvariationoftherenormalizedon-shellactiontakwhere the renormalized canonical momentaes are the given form by Sreg = S4 + Sct. (4.12) 2 t kℓ 1 α 2 2 2 φ k ℓω Π As is shown(r+ + inr 3 appendix)+ij ij−ij B,ℓ theωδS(1 limitcti + 3Λ cosi 2θ) i sinI δθS, ct Π √δrS+ctr sin θ, (4.16a) t δS 2=lim d Πx Π= πδγ++ Π ,δAΠ +=Ππ δϕ+ , , tΠI = πI2+ (4.14). (4.15) ∼−ren2κ4 4 − 8 ij Λ i Λ IΛ Λ ∼−2κ4 I − "r δγij # δAi δϕ 3 2 →∞ Sren =limS3 reg2, (4.13) kℓ ω ! φ r kℓ ω θ $ →∞% where the renormalizedΠInsertingθ canonical2 the((2 momenta asymptotic5α)cos2 areθ form+2 given of3α the by)sin backgroundsθ, Π φ (3.6)2 into((5αthe4) definitions cos 2θ +3α (A.4))sinθ of, the(4.16b) canonical ∼ 16κexist4 and− we will refer to− its value as the ‘renormalized’∼−16κ4 on-shell− action, following standard termi- momentanology and4 in the the context functional of the derivatives AdS/CFT duality. (4.8) we obtain 4 0t 1 ℓ δSct 2 2δSct2 0φ δ1Sctkωℓ ΠΠij = πij +Agenericvariationoftherenormalizedon-shellactiontaksin θ, √Πr+i r=+3πi ω+ℓ cos , θ ,Π Π= π + . es(r+ the+ formr )sinθ(4.15), (4.16c) ∼−2κ2 B3 Λ − Λ Λ I ∼−I 2κ2I 2B3 − 2 t 4 kℓδγij1 α δ2Ai2 2 δϕ4 φ k ℓω Π t 3 (r+ + r )+ − ℓ ω 2(13 + 3 cos 2θ) sin θ, Π t √r+r sin θ, (4.16a) 1 3ωℓ2 $ − 1 2αkω%ℓ 3 ij i Λ I 2 − t ∼−2κ4 4 φ δSren8 =lim d x Π δγij + ΠΛδAi + ΠI∼−δϕ ,2κ4 (4.14) Π "sin 2θ, Π r sin 2θ, # (4.16d) Inserting the asymptotic∼− form2κ2 of3B the2 backgrounds∼−2 (3.6)κ2 into→∞B !the definitions (A.4)3 2 of the canonical θ 4kℓ ω 4 $ φ kℓ ω % momenta and in the functionalΠ θ 1wherek derivativesℓ 2 the((2 renormalized5α)cos2 (4.8)θ we canonical+2 obtain3α momenta)sinθ, areΠ givenφ by 2 ((5α 4) cos 2θ +3α)sinθ, (4.16b) ∼ 16κ4 − 2 −2 ∼−16κ4 − Πη 2 sin θ 6(r+ + r )+ℓ ω ((13α 18) cos 2θ +7α 6) , (4.16e) ∼−2κ 8 − δSct − δSct − δSct kℓ 1 4α 1 2ℓ4 Πij = πij + , Πi = πi +k2ℓω , 1Πkωℓ= π4 + . (4.15) t 0t 2 2 2 δγ2 2 Λφ Λ0φ Λ I I δϕI Π t (r+ + Πr )+ −2 ℓ ω$sin(1θ +√ 3r cos+r 2+3θ) ωsinℓ θcosij, θΠ t, Π δAi √r+2 r% sin(rθ+, + r(4.16a))sinθ, (4.16c) ∼−2κ2 4 with all− other∼−2 components8κ B3 vanishing− identically. ∼−2∼−κ2 2κ 2−B3 − 4 " 4 # 4 4 3 2 Inserting the3 asymptotic$ form of the3 backgrounds2 2% 3 (3.6) into the definitions (A.4) of the canonical kℓ ω t 1 3ωℓ φ 1kℓ2αωkω ℓ Πθ ((2 5α)cos2Π θmomenta+2 3α and)sinsin in 2θ theθ,, functionalΠΠφ derivatives((5 (4.8)αsin we4) 2θ obtain cos, 2θ +3α)sinθ, (4.16b) (4.16d) θ 2 ∼−2κ2 B φ∼−2κ2 2B ∼ 16κ4 − 4 − ∼−164 κ4 13 − kℓ 1 α 2 k2ℓω 4 1t kℓ 2 2 4 φ 1 ℓ Π t 2 (r+ + r )+ −2 ℓ21ω (1kωℓ + 3 cos 2θ) sin θ, Π t 2 √r+r sin θ, (4.16a) 0t Πη 2∼−2 sin22κ4θ 246(r+ + −r 0)+φ ℓ8 ω ((13α 18) cos 2θ +7α ∼−6) ,2κ4 − (4.16e) Π 2 3 sin θ √r+∼−r 2+3κ ω8 ℓ cos" θ , Π− 2 −3 (r+ +#r )sinθ,− (4.16c) ∼−2κ B − 4 kℓ3ω2 ∼−2κ 2B −kℓ3ω2 4 Πθ ((2$ 5α)cos2θ +2 3α4)sinθ, Πφ ((5α %4) cos 2θ +3α)sinθ, (4.16b) 3 $ θ ∼ 16κ2 −2% 3 − φ ∼−16κ2 − t 1 3ωℓ with allφ other components1 24αkω ℓ vanishing identically. 4 Π sin 2θ, Π 4 sin 2θ, 4 (4.16d) 2 0t 21 ℓ 2 2 2 0φ 1 kωℓ ∼−2κ4 B ∼−Π 2κ4 Bsin θ √r+r +3ω ℓ cos θ , Π (r+ + r )sinθ, (4.16c) ∼−2κ2 B3 − ∼−2κ2 2B3 − 1 kℓ 4 4 2 1 23ωℓ3 $ 1 2αkω2%ℓ133 Πη 2 sin θ 6(r+ + r Π)+t ℓ ω ((13αsin 2θ18), Π cosφ 2θ +7α 6) sin, 2θ, (4.16e) (4.16d) ∼−2κ 8 − 2 − 2 − 4 ∼−2κ4 B ∼−2κ4 B $ 1 kℓ 2 2 % with all other components vanishingΠη identically.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 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 datain is terms the uniqueof boundary invariant data. To under this end both we need the to equivalence perform the integration classtransformation(3.11)andthetrans- over θ and remember that the magnetic potential A is not globally defined, as we pointed out in footnote 6. In particular, formationtaking A (3.12).north B(cos Weθ have1)dφ thereforeand Asouth demonstratedB(cos θ +1)dφ we that get by adding the counterterms (4.11) to the bulk action, the∼ variational− problem is∼ formulated in terms of equivalence classes of boundary data 1 3 3 δSren = dtdφ (r+ + r )kℓδ log kB /ℓ , (4.17) under the transformations (3.11)−2κ2 and (3.12). This− is an explicit demonstration of the general result 4 ! that formulating the variational problem in terms of" equiva# lence classes of boundary data under independently of the value of the parameter α.NotethatthecombinationkB3/ℓ3 of boundary 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- Angularshell velocityformation 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 the conservedbulk action, the charges, variational as problem well as is the formulated validity in terms of theof equivalence first law classesof thermodynamics. of boundary data 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 radial reparameterizations is achieved via the same canonical transformation that renders the on- 4.1.3 Conserved chargesg g shell action finite. As we will nowtφ demonstrate,tφ i the same bounr dary terms ensure the finiteness of Similarly,Ω4 the= Ω momentumH Ω = constraint =0in(A.6),whichcanbewritteninexplicitformas= ωk − . (5.3) the conserved charges,− as well∞ asgφφ the∂ validity− gφφ ofH the+ first lawrof+ thermodynamics. Let us now consider conserved! chargesM ! associatedH " with local conserved currents. This includes j ! ! 0 0j j In the coordinateelectric charges, system (3.6) as2 thereD welljπ is+ as noπη conserved∂ contributioniη +! πχ∂iχ quantities+ to! F thijeangularvelocityfrominfinity,butπ + relatedFijπ toasymptoticKillingvectors.Magnetic 4.1.3 Conserved− chargesi 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, the momentum√ constraintjkl 3 0 0 =0in(A.6),whichcanbewritteninexplicitformas2 0 2 0 in [21], whichLet is why us now our considerresult+ does2 conserved notγϵ fully charges agreeχ Fij associated withFHkl + theχ o withneFij obtainedF localkl +3conservedχ there.FijFkl currents.+ χ Fij ThisFkl includes=0, (4.21) Angular velocityas weelectric shall charges, see later as2κ well on.4 as− conserved quantities2 related toasymptoticKillingvectors.Magnetic2 j ! 0 0j j " Incharges the do radial not2D falljπ Hamiltonian in+ thisπη category,∂iη + π formulation butχ∂i theyχ + canF beπ of describe+ theFijdinthislanguageintheelectricframe, bulkπ dynamics, the presence of local conserved We defineElectric the physical chargesleads (diffeomorphism to− finite conservedi invariant) charges angular associated velocityij as with the di asymptotifference betweencKillingvectors.Notethattheterms the currentsas we is shall a see direct later on.consequence of the first class constraints Λ =0and i =0in(A.6).7 As angular velocity at thein outer the second horizon1 line and are at infinity, independentjkl namely3 0 of0 the3 canonical2 0 momenta and originate3 2 in the0 parity odd terms In the magnetic frameIn the+ there radial is√ Hamiltonian onlyγϵ one non-zero formulationχ F F electric+ of the chaχ bulkrgeF givenF dynamics,kl +3 byχ (4.20),FtheijF presencekl whose+F of valueχ localFij isF conservedH=0, (4.21) in the case of asymptotically2 AdS9 ij kl backgrounds,ij theseΛ constraintsi leadkl respectively7 to conserved incurrents the STU is a2 modelκ direct4 − consequence Lagrangian. of theHowever, first class2 for constrain asymptoticallyts =0and conical backgrounds2=0in(A.6). ofAs the form(3.6) gtφ ! gtφ 4 r F H 8 " electricinΩ the4 charges= caseΩH 0( ofe)Ω asymptotically and= charges2 AdS associated0 backgrounds,t =ℓωk with these− . asymptotic constr2aints2 lead Killing respectively(5.3) vectors. to conservedIn particular, the gauge these terms are∞ asymptoticallyg ∂ xg subleading,+ r the most dominant term being Q4− = φφ d− Πφφ = 3 √+r+r + ω ℓ . 8 (5.4) leads toelectric finite charges conservedΛ and− charges charges! M associated associated!H with4G4B asymptotic" with asymptoti− Killing vectors.cKillingvectors.NotethatthetermsIn particular, the gauge constraints in =0in(A.6)taketheform#∂ C constraints in AngularΛ =0in(A.6)taketheform velocityM!∩ ! jkl 1 In the coordinatein the system second (3.6) line thereF are is independent no contribution! of to! the th√eangularvelocityfrominfinity,but canonicalγϵ $ χF F momenta= (%r− ) a,nd originate in the parity odd(4.22) terms In the electric frame both electricF charges defined in (4.45) areij non-zero:kl there is in the original coordinate systemWe define (3.1). the The physical9 rotati (dionffeomorphism at− infinityi invariant) was not angular takenO 0i intovelocity account as the difference between the in the STU model Lagrangian. However,D πi =0D fori,Dπ asymptotically=0π0i,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 thei horizon one obtained and ati infinity, there. namely and so the momentum(e) constraint asymptotically3B 0(e) reduces0(e) to these terms arei asymptoticallyQ 0=i d2 subleading,x πt = , theQ most= dominaQ . nt term being(5.5) wherei π and4 π0i are respectively the canonical momentag4 conjugateg4 to ther gauge fields Ai and where π and π are− ∂ respectivelyC 4 theG4 canonicaltφ momentatφ conjugate to the gauge fields Ai and Electric charges• ConservedA0.Sincetheboundarycounterterms(4.11)aregaugeinvarian Charges# M∩ Ω4j= ΩH Ω = 0t, it0j= followsωk − fromj. (4.15) that these(5.3) 0 i 2Djπi + πη−jkl∂iη∞+ πgφφχ∂∂iχ −+gFφφijπ+ 1+ Fijr+π 0. (4.23) A .Sincetheboundarycounterterms(4.11)aregaugeinvarian& − √ γϵ χF& F! M= (!Hr− ), " t,≈ it follows from (4.15) that(4.22) these Magnetici chargeconservation laws hold for the corresponding& renormalizedij !kl momenta! as well, namely In the magneticConserved frame there currents, is only oneIn 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 presentℓ4 asymptotically in the magnetic frame reduces and to it is equal to one of the constant0(e) radialin [21], coordinate,2x which0t is why it our follows result does from not fully(4.15)2 2 agree that with thi thesconstraintholdsfortherenormalized one obtained there. electric chargesQConserved4 in the= electric currents frame:d forΠ gauge= potentials3 √:r +r + ωi ℓ . 0i (5.4) MassFΛ = 0This implies− ∂ thatC the quantities4G4B D−iΠ =0,DiΠ =0. (4.19) momenta as# well, j 0 0j j MElectric∩ charges2D π +j$π ∂ η + π ∂% χ + F 0π 0j+ F πj 0. (4.23) ((me)) j2Di3Π +ηΠi ∂ η +χΠ(0(e)ie∂) χ + Fij Π + FijΠ 0. (4.24) In the electric frameConserved both electriccharges: charges defined− in (4.45)j i are2x non-zero:ηt i χ i ij 211x 0ijt ≈ TheThis mass implies is the conserved thatIn the the magneticQQ quantities charge44 ==− frame associated there2 d is onlyΠ withF,Q one= non-zero theQ44 . Killing= electricvector charged givenΠζ by=, (4.20),≈∂t whose(5.6)Ω value∂φ(4.20).Since is −−2∂κ4 C∂ C i 12 − ∂ C − − ∞ Ω =0inthecoordinatesystem(3.6),(4.26)givesGiven an asymptotic Killing! M∩# vectorM∩ ζ satisfying the! M asymptotic∩ conditions Since∞ the counterterms(e) 2 (4.11)t 3 areB invariant0(e) with0(e) respectℓ4 to diffeomorphisms along the surfaces of whereQ =C denotes ad Cauchyx π = surfaceQ, that0(e) = extendsQ = toQd the2&x Π. boundary0t = ∂ r,arebothconservedandfiniter + (5.5)ω2ℓ2 . (5.4) 4 (e) 4 4 2 4 t 0(3 e√) + 2 0t Electricconstant potential radial− coordinate,∂ C it4 followsG4 from− ∂ (4.15)C that4G4B thiMsconstraintholdsfortherenormalized− and correspond# M∩ to theQ electric4 = charges2 associatedt #ΛMdt∩ x0j toΠ these,QΛt gaugeΛ4 ℓ fields.kj = I d xi Π I , (4.20) ζ γij M=4D=iζj + Djζi d −x0, 2Π ζtA+i Π=0Aζt ∂+jAΠi A+t A=j ∂i$ζ (r+−0,+ r %)ζϕ. = ζ ∂iϕ (5.9)0, (4.25) momenta as7 L well, ≈ ∂L C ≈ ∂−L C ≈ We define the electric& These constraints potentialIn the can− as electric be∂& derived frameC alternatively both! electricM&∩ by charges applying definedthe general in (4.45) variation8Gare4 non-zero: (4.14)! ofM the∩ renormalized action Magnetic charge ! M∩ j 0 0j j Hi = 0the to ConservedU conservation(1) gauge transformationscurrents: identity 2D and (4.24)jΠ transversei +" impliesΠη di∂ffeomorphisms,iη that+ Πχ the∂iassumingχ quantity+3#BFij theΠ invariance+ Fij ofΠ the renormalized0. action (4.24) Thewhere sameC resultdenotes is obtained a Cauchy in the surface electricQ(e) = that frame extends using3d2x πt = (4.50 to the)., As boundaryQ0( fore) = theQ0(e electric).∂ ,arebothconservedandfinite and magnetic(5.5) The only non-zero magneticunder such charge transformations. is present0(−e) This in the method magnetic will4 be usedframeB in orde andrtoderivetheconservedchargesintheelectricframe.r it is equal to one4 of4 the ≈ 8 0 i − i ∂ C 4G4 H M electric chargespotentials, inGivenandMass the electriccorrespond anIn the asymptotically asymptotic frame: gauge to potentials theΦ locally4 Killing electric= AdS atAi spaces, the vector charges boundary the= k Hamiltonianζ # satisfyingM associated do∩ not cons− conttraint, the toribute these asymptotic=0canbeusedinordertoconstruct to the gauge mass conditions fields. in the(5.7) gauge (3.6). conserved charges associated with conformalK &+ Killingℓ vectors2 ofr+ thet boundary0t data0& [22].t For asymptoticalj ly conical Conserved ``charges’’:Magnetic charge [ζ!]=H ' (d x" &2Π + Π A + Π A ζ , (4.26) The7 backgrounds, mass is the the Hamiltonian conserved constraint charge! leads associated to conserve withdchargesassociatedwithasymptotictransversedi thej Killingj vector11j ζ = ∂ ffeomor-Ω ∂ .Since These constraintsi (m) can3 be derivedQ alternativelyΛ∂(e) C j byΛ applyingΛ thej general0 i variationI (4.14)t i ofI theφ renormalized action whereAngular= ∂tphisms,+γ momentumΩH=∂ξφD,thatpreservetheboundarydatauptotheequivalenceclassisQζ theThe+ null=D onlyζ generator non-zero0, magnetic of!F the=A#Q charge outerM=.∩ ζ is horizon. present∂ A 12 in+ Note theA magnetictransformations∂ thatζ Aframe0,(5.6)is and (3.11). constant it isϕ equal−= over toζ− one∂ ϕ of∞ the 0, (4.25) KtoΩU(1)ζ=0inthecoordinatesystem(3.6),(4.26)givesij gauge transformationsi j4 −j 2iκ2 and transverseζ i 4 diffjeomorphisms,Asymptotic$i j Killingi assuming ivectorK the%ζ ζ invariancei of the renormalized action the horizon∞L [22] and so leadselectric to a charges well4≈#∂ defined in theC L electric electric frame: potential. However, as≈ we remarkedLi in the ≈ Theunder angularis such both 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, the electric potential is not gauge& invariant. Under gauge transformations it is the conservation8 identity (4.24) implies that(m) the3 quantity (e) ℓk Electric potentialζ = ∂φIn,whichgivesare asymptotically a few subtleties locally in evaluating AdS spaces,2 these theQ charges.t Hamiltonian14= t 0 Firstly,t F cons G=auss’Qtraint. theoremH =0canbeusedinordertoconstruct used to prove(5.6) conservation 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) −2κ4 ∂ C conservedfor the charges charges associated (4.26)− assumes with∂ Cconformal differentiabilityKilling of vectors# theM∩ integ of therand8 boundaryG across4 the data− equator [22]. For at the asymptotical boundary.ly conical We definein the the electric Smarr potential formula and as the first law.! M∩ 3 " 2 t 0t # 0& ωℓ t j Angularbackgrounds,If theMomentum: gauge the Hamiltonian potentialsElectricJ = potential are[ζ constraint]= magneticallyd2x (2Π leadst d+ sourced,toΠxt conserveA20Π+ Π as+tdchargesassociatedwithasymptotictransversediA isΠ)= theA case+ Π for.AAi ζin, the magnetic(5.10) frame, then(4.26)ffeomor- The samei result is obtained4 in the electric3 φ frame0 φ usingj φ (4.50j). As forj the electric and magnetic phisms,theξ gauge,thatpreservetheboundarydatauptotheequivalenceclass0( shoulde) be chosenQ∂ C suchB∂ thatCr Ai is continuous across−2G the4 transformations equator. In particular, (3.11). contrary Magnetic potential We define0 i the! M electric∩ potential as potentials, theΦ4 gauge= Ai potentials= 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 The same result is obtained inH the electric' ( " frame. 3 Similarly,is both the finite magnetic and potential conserved, is! defined i.e. it in is terms independent of0(e) the g0augei of field theAB choinice ther of electric Cauchy frame surface as C.However,there charges is the one given in (3.6), andΦ not= theA one= discussedk i in− ,footnote 6. (5.7) ! 4 i 0 i where = ∂ + ΩAngular∂ is the momentum null generator of! the outer horizon. NoteK that+ A ℓis constantr+ over K aret aH fewφ subtletiesSecondly, the in evaluating charges (4.26) these are not charges. generically Firstly,!H 14 invarianti K' G(auss’" under theorem the U(1) used gauge to prove transformations conservation Free energy (m) i ℓk ! 2 2 r the horizon [22] and soΛ leads toΛ aΦ well= definedAΛ electric= potent(r ial.r However,)+2ω! ℓ as we− remarked& . in the 0 i (5.8) for the chargesA A (4.26)+4 where∂ α assumes.Thesegaugetransformationsthoughmustpreserveboththei = ∂t + Ω diH ∂ffφ erentiabilityis the null generator+ of of the the outer integ horizon.rand Note across that A thei is equator constant over atradial the boundary. gauge Thei angulari momentumi KK is+ defined2B as− the− conserved charger+ correspondingK to the Killing vector previous section, the electric→ potentialthe horizonis not gauge [22]H and invar so leadsiant. to a Under well defined gauge electric transformations potential. However, it is as we remarked in the IfFinally, the gauge the full potentials Gibbs free are energy,! magnetically4,isrelatedtotherenormalizedEuclideanon-shellactioni' sourced, as" is the( case for A in the magneticn frame, then shifted by a constantζ =(4.3) (see∂φ,whichgives (4.27)) and the whichprevious asymptotic compensates section,! Killing theG the electric cor conditionsresponding potential is not (4.25). shift gauge of invar the Preseriant. chargesving Under (4.26) the gauge radiali transformations gauge implies it is that the the electric frame, where& all charges! are electric. Namely, Λ in the Smarrthe formula gaugegauge and should the parameter first belaw.shifted chosen must by a constant depend such (see that only (4.27))A oni whichis the continuous transversecompensates the coordina across corresponding thetes, equator. i.e. shiftα of the(x charges)(seee.g.[22]),while In particular, (4.26) contrary $ 3 respecting thein Killing the Smarr symmetry formula and the leads2E first to law.t the conditiont 0 t ωℓ to the variational problemJ4 we= discussedI4 = dS x (2= earlier,Π Sren+ Π= theAβ4 +4 gauge,Π Aφ)= that should. be used(5.11) to evaluate(5.10) these Magnetic potential ren −φ 0 φG −2G charges is the one givenMagnetic in potential (3.6),!∂ andC 21 not the one discussed in footnote4 6. M∩ ζi∂ αΛ =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$i A in the$ on-shell electric action frame 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. definedi 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 + ∂iα .ThesegaugetransformationsthoughmustpreserveboththeG radial gauge i Underi(m) suchi gauge$ transformationsℓk (m) the2 2i chargesr &ℓk (4.26) are shi2 2 ftedr & by the corresponding electric → Φ = Ai = (r r+Φ4)+2= ωAi ℓ =− . (r r+)+2ω ℓ (5.8)− . (5.8) Free energy4 E K + 2B − − r (4.3) andcharges the asymptotic (4.20).K + As2 willB Killing become− − conditions clear in sectionr (4.25).+ 5, this Preser compevingnsates 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 respectingwhere Sren thewas Killing given in symmetry(4.13). Evaluating leadsG this to the we obtain condition (see appendix B) the electric9In the frame, AdS/CFT where context all charges are interpreted are electric. as a gravitational Namely, anomaly in the dual QFT. β ℓk $ 21 r 21 4 iE 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 soAn that interesting their leading observation asymptotic is that contribution the value of to the the renormal Hamiltizedon-Jacobi action insolution the magnetic (4.7), as frame, well as to as well as the value of all other thermodynamic variables, is independent of the parameter α in the Mass the derivativesMass (4.8), vanishes by means of the asymptotic constraints (4.4). This is the reason boundarywhy any counterterms value of α leads (4.11). to This boundary property counterterms is necessary inorderforthethermodynamicvariables that remove the long-distance divergences. 11 11 The mass is thein conserved the electric charge andThe magnetic associated mass is frames the with conserved to agree, the Killing charge and in associatedvector order to mζ withatch= the with∂t Killing thoseΩ ∂vector ofφ.Since the 5Dζ uplifted= ∂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)givesblack holes that weΩ will=0inthecoordinatesystem(3.6),(4.26)gives discuss in section12 6. Recall that the terms12 multiplying α are designed ∞ in the unintegrated∞ value of the renormalized action. Nevertheless, α does not enter in any physical so that their leading asymptotic contribution to the Hamilton-Jacobi solution (4.7), as well as to observable. This observation results from the explicit2 comt putationt 0 oft the thermodynamicℓk variables, the derivatives (4.8),2 vanishest by meanstM40= of thet asymptoticd xℓk 2Π t c+onstraintsΠ0At + Π (4.4).At = This( isr+ the+ r reason) . (5.9) butM4 we= have not beend x able2Π t to+ findΠ0A at general+−Π ∂At argumentC= ( thatr+ + ensurer ) . sthissofar.8G4 (5.9) − why any− value∂ of Cα leads to boundary counterterms! M∩ 8G that4 remove− the long-distance divergences. ! M∩ " # However,An the interestingparameterThe same"α observationdoes result appear is obtained is in that the in therenormalized# the value electric of the momenta, frame renormal using as (4.50ized is clear). action As from for in (4. the the16), electric magnetic and and frame, magnetic The same resultin is5.2 the obtainedas unintegrated Thermodynamic well as in the thepotentials, value value electric of of the the all relations frame renormalized gauge other using potentials thermodynamic and (4.50 action. the at). the Never first As boundary variables, fortheless, law the do electricα isdoes notindependent cont not andribute enter magnetic toin of anythe the mass physical parameter in the gaugeα in the (3.6). potentials, the gaugeobservable.We potentials canboundary now This show counterterms observation at the that boundary the results thermodynamic (4.11). fromdo This not the cont property explicit variablesribute com is to necessaryputation we the just mass compof inorderforthethermodynamicvariables the inuted thermodynamic the gaugesatisfy (3.6). the variables, expected ther- butmodynamic wein have the not electric relations, beenAngular and able including magnetic momentumto 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 5.2so Thermodynamic that theirζ = leading∂ ,whichgives relations asymptotic and contribution the first law to the Hamilton-Jacobi solution (4.7), as well as to The angular momentumQuantum is statistical defined asφ relation the conserved charge corresponding to the Killing vector the derivatives (4.8), vanishes by means of the asymptotic constraints (4.4). This is the reason We can now show that the thermodynamic variables we just computed satisfy the expected3 ther- ζ = ∂φ,whichgivesIt is straightforward to verify that the total Gibbs2 free enet rgy t 40satisfiest the quantumωℓ statistical modynamicwhy any relations, value including of α leads the to first boundary lawJ4 = of black counterterms holed x (2 meΠchanics.φ + thatΠ0GA removeφ + Π Aφ the)= long-distan. ce divergences.(5.10) relation [36] ∂ C −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) QuantuminJ the4 statistical= unintegratedThe relationd 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 ∂ C −2G4 Similarly,observable. the! M thermodynamic This∩ observation potential results from4,whichwasobtainedfromtheon-shellactioninthe the explicit computation of the thermodynamic variables, It is straightforward to verify! that the total GibbsG free energy 4 satisfies the quantum statistical The same result ismagnetic obtainedbut we frame, in have theFree satisfies not electric energy been frame. able to find a general argumentG that ensuresthissofar. relation [36] 0(e) 0(e) Finally, the full Gibbs4 = M free4 energy,T4S4 0(Ω4e,isrelatedtotherenormalizedEuclideanon-shellactioni)4J40(e) Φ (mQ) (m.) (5.16) n 4 = M4 G T4S4 −Ω4J4 −ΦG Q − !Φ Q . (5.15) the electricG frame,− where− all charges− are electric.− 4 Namely,4 Free energy Note5.2 that Thermodynamic the shift of the mass relations and angular and momentum the first under law a gauge transformation (4.27) is Similarly, the thermodynamic potential 4,whichwasobtainedfromtheon-shellactioninthe$ compensated by that of! the electric potentialsG soI that= S theseE = relationsS = β are, gauge invariant. (5.11) Finally, the fullmagnetic GibbsWe free frame, can energy, now satisfies show4,isrelatedtotherenormalizedEuclideanon-shellactioni that the thermodynamic variables4 ren we− justren comp4G4uted satisfyn the expected ther- G 0(e) 0(e) the electric frame, wheremodynamic all charges relations, are electric. including4 = M Namely,4 theT4S first4 Ω law4J4 of blackΦ Q hole. mechanics. (5.16) with β4 =1/TG 4 and S−ren defined− in$ (4.33).− $ The Euclidean$ $ on-shell action in the magnetic frame First law $ Note that the shiftsimilarly of the defines mass and another angular thermodynamic momentum potential,under a gauge4,through transformation (4.27) is E G In orderQuantum to demonstrate statisticalI4 = Sren the relation= 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. I4 = S = Sren = β4 4, (5.12) andIt (3.12) is straightforward of the non-normalizable to verify that boundary the total data Gibbs thatren allow free− eneforrgy a wellG posedsatisfies variational the quantum problem. statistical with β4 =1/T4 and Sren defined in$ (4.33).$ The Euclidean$ $ on-shell action in the magneG4 tic frame FirstIn particular,• lawrelationThermodynamic [36] variationswhere S ofwasB, givenk relationsand inℓ (4.13).that are Evaluating and a combination the this first we obtain of law the (see two appendix transformationsB) (3.11) 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-normalizableI4 = 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 weimplies 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 I 2 =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 Smarr formula defined as the Lorentzian on-shell action in section0(e) 0( 5e of) that paper,(m) (m while) in section 6 it is defined as the Euclidean Moreover, (4.28) impliesSmarr’s that the Formula: on-shellM action4 =2 isS4 givenT4 +2Ω by4J4 + Q Φ + Q Φ , (5.18) First law on-shell action. We adopt the latter definition4 here.4 4 4 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-normalizable4M =2+ S∧T +2Ω boundaryJ− + Q Φ data+ thatQ allowΦ for, a well posed variational(5.18) problem. subtracted! metric4H in4 [21]4 has non-zero4 4 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 rotating beϵ 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 to the regulated Komar mass due to rotation, and in turn3 ensur3 ethevalidityoftheSmarrformulaandthefirstlaw transformationsVarying parameters: r+, r-, ω, and B, k, l subject to kB /l –fixed on-shell action. We adopttransformations, the latter definitionof thermodynamics. as here. well as arbitrary variations of the normalizable parameters r and ω,weobtain 12 ± In [21] the mass for static subtractedoriginal geometry parameters black holesm, a, w Πas, evaluatedΠ &0( ae )scaling from the0( eparameter) regulated(m) Komar(m integral) δM = ϵM , δS =2ϵS , δJ =2ϵJc , sδ23Q = ϵQ , δQ = ϵQ , (5.19) and the Hawking-Horowitz prescription4 4 and shown4 dM to4 be equivT 4dSalent.Ω4 BothdJ the4 Φ Smarr0(e)dQ4 formula0(e) Φ and4(m)d theQ(m first4) =0 law. of (5.17) 4 − 4 4 − 4 4 − 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 thermodynamics analysisSmarr of the rotating formula subtracted geometry. Furthermore,23 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 22 transformations

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

23 KK reduction ansatz III. Holography via 2D Einstein-Maxwell-Dilaton M.C., Papadimitriou 1608.07018

The 4D4D truncation STU fields of the can STU be model consistently can be consistentlyKaluza-Klein Kaluza-Klein reduced on reduced S2 by on 2 one-parameter family of Ansätze: S by means of the one-parameter family of KK ansatze¨ KK reduction ansatz 2⌘ 2 2 2 2 e = e + B sin ✓, = B cos ✓ 2⌘ 0 2 (2) 2 2 0 e A = e A + B sin ✓d,A+ A = B cos ✓d The 4D truncation of the STU model can be consistently Kaluza-Klein reduced on S2 by means of the one-parameter2 family of KK ansatze¨ ⌘ 2 22D Einstein-Maxwell-Dilaton2 2 sin ✓ (EMD) model(2) 2 e ds4 = ds2 + B d✓ + (d A ) 1+2⌘2B22e 2 sin2 22✓ 2 ✓ e = e + B sin ✓, = B◆cos ✓ 2⌘ 0 2 (2) 2 2 0 where is an arbitrary parameter. Fore anyA value= e ofA ,+ theB resultingsin ✓d,A 2D theory+ A = isB cos ✓d We would2 like(2 to) develop the holographic dictionary for the 2D sin2 ✓ the Einstein-Maxwell-Dilatonds2 , Ψ, A -fields theory of 2D we Einsteine⌘ consideredds2 = ds-2Maxwell+ B above2 d✓2-Dilaton –+ drops Gravity: out in 2D!(d A(2))2 Einstein-Maxwell-Dilaton (EMD) model 4 2 1+2B2e2 sin2 ✓ ✓ ◆

1 2 where is an arbitrary2 1 parameter.2 ab For any value of , the resulting 2D theory is However,S by2D comparing= d thexp KKge ansatz R[g]+ with the 4De blackFabF hole+ solutionsdtp wee see2K 3 232 ˆ the Einstein-Maxwell-DilatonL2 4 theory we consideredˆ above – drops out in 2D! that = !` /B ,2 i.e.✓ is the angular⇣ parameter of the 4D black⌘ hole. ◆

However, by comparing the KK ansatz with the 4D black hole solutions we see 3 3 The parameterB This= 2L; model λ-independent allows is rather any special solutionthat since = of!` it can the/B be 2Drotational, i.e. obtained EMD is theparameter theoryby angular circle to reductionof parameter besubtracted uplifted of of 3D geometry the to 4D a black hole. family ofEinstein-Hilbert 4D solutions, gravity i.e. it acts as a solution generating mechanism. The parameter allows any solution of the 2D EMD theory to be uplifted to a 1 3 2 S3D = dfamilyxp ofg3 4DR solutions,[g3] 2⇤ i.e.3 + it actsd asxp a solution2 2K[ generating2] mechanism. 22 ˆ ˆ 3 ✓ ⇣ ⌘ ◆

I. Papadimitriou AdS2 holography and non-extremal black holes 15 / 50

This ensures that a meaningfulI. Papadimitriou holographic dictionaryAdS2 holography exists and non-extremal for this theory, black holes but 15 / 50 certain aspects are in fact generic and apply to a wider class of 2D dilaton gravity models.

I. Papadimitriou AdS2 holography and non-extremal black holes 4 / 50 Web of theories Web of Theories

2 Subtracted geometry Locally: AdS3 x S S1 uplift 4D STU model 5D Einstein-Maxwell-Chern-Simons

2 2 5 = Rz4

B 3 Rz =2⇡Lk ` k!L Z 2

2 !-twisted 2 2 2 = 4 2 2 = 5 S reductionKK Ansatz2 ⇡L2 L =2B S reduction 3 ⇡L2 =0(! =0)

2 2  1  = 3 S reduction 2 Rz 2D Einstein-Maxwell-Dilaton 3D Einstein-Hilbert w/ speciﬁc BCs NCFT1 RG projected CFT2

truncation

RG Jackiw-Teitelboim Sachdev-Ye-Kitaev

I. Papadimitriou AdS2 holography and non-extremal black holes 16 / 50 Equations of motion in Fefferman-Graham gauge Equations of motion in Fefferman-Graham gauge Running dilaton solutions

In theIn the Fefferman-Graham Fefferman-Graham gaugeRunning gauge dilaton solutions TheGeneral general solution solution with running of 2D dilaton EMD takes Gravity the form – running dilaton 22 2 2 2 Feffeman-Graham gauge: 2 dsds 2== dudu ++tttt(u,2(u, t) tdt)dt2 ,A2 ,Au =0u =0 u/L m 0 (t)/↵ (t) 2 2u/L Q L 4u/L Thee general= (t)e solution with1+ running dilaton takesL e the form e 42(t) 44(t) Anayticthe generalthe equations equationss✓ solution: of of motion motion are are ◆ ↵(t) 2 m 2(t2)/↵2(t)2 2 3 Q2L2 p = @u/Lte 0 @ L 2e 2u/L+ Q e =0 4u/L e =(tt))e 1+ 2u 2L e 2 3 e 0 s 4@2u(t)L e + Qe44=0(t) ✓ 2 2u/L2 2 2 2 24◆ 2 ↵(t) 4L e @u L(t)+ m3Q0e(t)/↵p(t) =02Q/L At = µ(↵t)+(t) @t log 2 2 2 4 p = @ e 2 @2u/L 2L 3Q 2 e 2 p =0 2t0(t) 4L e⇣ u (t)+m 20 (t)/⌘ ↵ (t)+22 Q/L3 ! 0(t) ⇤t + K@u L e + Qe =0 ⇣ 2 ⌘ 2 3 where ↵(t), (t) and µ(t) are arbitrary functions2 2+u/LK@2 of time,L while me2 and+Q2Qare e =0 ↵(t) 4L⇤ te u(t)+m 0 (t)/↵ (t) 2Q/L arbitraryAt = constants.µ(t)+ @t log @u @te /p =0 2 (t) 4L 2e2u/L2(t)+ m 2(t)/↵2(t)+2Q/L 0 ⇣ p ⌘ 0 ! @@uuQ@t=e@tQ/=0 =0 Leading asymptotic behavior: Thewhere leading↵(t), asymptotic(t) and µ behavior(t) are arbitrary of this solution⇣ functions is of time,⌘ while m and Q are @u2Q = @tQ =0 arbitrary2 constants.where2u/L tt = (p ) , Ku/L= @u logu/Lp is the extrinsic curvature2u/L of tt, ⇤t is tt = ↵ (t)e + (1),e (t)e + (e ),At = µ(t)+1 (e 3) ut the covariantO Laplacian⇠ with respectO to tt, and Q = Op e F . running2 dilaton 2 and so thewhere arbitrarytt functions= (p↵(t), )(t,) Kand=µ(@t)ushouldlog p be identiﬁed is the with extrinsic the curvature of tt, ⇤t is The leading asymptotic behavior of this solution is 1 sources• Arbitrary of the corresponding functions α dual(t), operators.β(t) and µ(t) identified withQ the= p e 3 F ut theThe covariant general Laplacian solution of with this system respect of to equationstt, and can be obtained2 analytically.. 2 2u/L u/L u/L 2u/L tt =sources↵ (t)e of the+ corresponding(1),e ( tdual)e operators+ (e ),At = µ(t)+ (e ) O ⇠ O O AdS holography and non-extremal black holes 2 I. Papadimitriou• 4D uplift results in2 asymptotically conformally AdS2×S subtracted6 / 50 andgeometries, so theThe arbitrary general generalized functions solution↵ to(t of )include, this(t) and system arbitraryµ(t) should of time equations be-dependent identiﬁed can with sources be the obtained analytically. sourcesI. Papadimitriou of the corresponding dual operators.AdS2 holography and non-extremal black holes 5 / 50

I. Papadimitriou AdS2 holography and non-extremal black holes 6 / 50 I. Papadimitriou AdS2 holography and non-extremal black holes 5 / 50 Holographic dictionary for running dilaton solutions

Holographic dictionary for running dilaton solutions Since the asymptotic solutions with running dilaton are differentRadial from Hamiltonian those with formulation constant dilaton, the holographicHolographic dictionary is different dictionary for the for two running cases. dilaton solutions Since the asymptotic solutions with running dilaton are different from those with constant dilaton, the holographic dictionary is different for the two cases. For the running dilaton solutionsSince the the asymptotic boundary solutions counterterms with running are dilaton are different from those with Repeatconstant dilaton,Radial the Hamiltonian holographic dictionary Formalism is different in 2D for the two cases. Inserting the radial ADM decomposition 1 For the running1 dilaton solutions the boundary counterterms are Sct = dtp L (1 uoL t) e 2 ˆ 2 2⇤ t 2 2 Radial ADM decomposition: ds=(N + NtN )du +2Ntdudt + ttdt For2 the running dilaton solutions the1 boundary counterterms1 are Sct = dtp L (1 uoL t) e 2 ˆ ⇤ of the metric in the 2D action gives2 the radial Lagrangian The renormalized one-point functions are then defined1 as 1 Countertern Action: Sct = dtp L (1 uoL⇤t) e The renormalized one-point 2 ˆ functions are then defined as t 1 t 22 t˙ t 1 2 t 2 =2⇡t , = = dt⇡p ,N = K⇡ N @t e FutFu + 2⇤t e T L O22 ˆ J N t 2N 2 t t L2 RenormalizedThe renormalized one-point2 functions: one-point✓ functions=2⇣⇡t , are then =⌘ defined⇡ , as = ⇡ ◆ where T O J b where K = ttbK and the extrinsicb curvature K is given by where tt =2⇡t, = ⇡ , ttt = ⇡t 1 T t Ob bJ b t u/L 1 1 ⇡t = 2 lim e @ue te L1 u/L 1 2where u ⇡ =Ktt = lim (˙ett 2@DuteNt) e L 2 !1 t b 2 u2N b b ⇣ 22 !1⌘ eu/L 1 ⇣ ⌘ bt with the dott denotingt a derivative withu/Leu/L respect to the radial coordinate1 u, and Dt ⇡ =lim ⇡ ⇡t = bt2 lim e @tue e L u 2⇡ =limu ⇡ standingp for the covariant2 derivative!1u p with respect to the induced metric tt. !1 !1 ⇣ ⌘ eu/L 1 u/L bt 1 1 t u/L 1 ⇡b = lim e e ⇡ K=lim⇡b L= ⇡lim e e K L 2 u u p 2 u 2 !1 !1 2 !1 1 b b u/L 1 b ⇡ = 2 lim e e K L I. Papadimitriou AdS2uholography and non-extremal black holes 23 / 50 I. Papadimitriou AdS2 holographyI. Papadimitriou and non-extremal black holesAdS2 holography2 !1 and non-extremal black holes 23 / 50 20 / 50 b I. Papadimitriou AdS2 holography and non-extremal black holes 23 / 50 Holographic dictionary for running dilaton solutions Holographic dictionary for running dilaton solutions Holographic dictionary for running dilaton solutions Evaluating these expressions using the general solutions with running dilaton gives the one-pointEvaluating functions these expressions using the general solutions with running dilaton gives the one-point functions Evaluating these2 expressions using the general solutions with running2 dilaton L m 0 1 Q L m 0 0↵0 00 HolographicExplicitgives the dictionary one one-pointL-pointm functions functions: for 2t running dilaton1 Q solutionsL m 2 ↵ = 2 , 0 = 2 t , = 2 02 +20 0 00 T 2 = ↵2 J ,  ↵ = O , 2 = ↵2 ↵32 +2↵2 2 T✓ 22 2 ◆ ↵2 2J 2 ↵ O 2 ✓22 ↵2 ↵3 ◆↵2 L m 20✓ t ◆1 Q 2 L m 02 ✓ 0↵0 00 ◆ = , = , = 2 +2 T 22 ↵2 J 2 ↵ O 22 ↵2 ↵3 ↵2 All three2 ✓ operatorsAll three◆ operators are crucial are2 to crucial describe to describe the2 physics.✓ the physics. In particular, In particular, these◆ these one-pointAll three functions operatorsone-point are satisfy functions crucial the to satisfy Ward describe the identities Wardthe physics. identities In particular, these Holographic dictionary for running dilaton solutions one-point functions satisfy the Ward identities t @t @t log =0, t t =0 @t @Tt logO =0, t D=0J The renormalizedWard on-shell Identities: action can beT obtainedO (up to a constanttD J that depends @t @t log =0, t =0 on global properties) by integrating theT relations O D J L ↵ L + = 00 0 0 = @ 0 Conformal anomaly: L 00 2 0↵20 3 L 2 0 t Sren + =T LSOren00 20t↵0 ↵ 1 L=↵Sren @0 t2↵ ↵ ⌘ A = , +The= renormalized= 2 , on-shell2 ✓= action3= can@ bet2◆ obtained (up to✓ a constant◆ that depends TT OO 2 ↵2↵ ↵3 ↵ 2↵  ↵↵ ⌘↵A ⌘ A T ↵From theseO on global relations↵ properties)22✓✓ we deduceJ by integrating◆ that◆↵ the2 the scalarµ relations2✓ operator◆✓ ◆ is a marginally O FromFrom these theserelevant relations relations operator we we deduce deduce and that the that theory the the scalar has scalar operator a conformal operator is anomaly, a marginally is a solely marginally due to the using the above expressions for the one-point functions.Sren SOren t 1 Sren Exactrelevant generating operatorsource ofand thefunction the scalar theory operator. ( has = a conformal We , will anomaly,see = later solely that , O this due = anomaly to the matches ): relevant operator and the theoryT has↵ a conformalO ↵ anomaly, J solely ↵ dueµ to the sourcesource of of theprecisely the scalar scalar that operator. operator. of the We 2D We will CFT will see dual see later to later that 3D thisEinstein-Hilbert that anomaly this anomaly matches gravity, matches which provides the precisely thatUV completion of the 2D CFT of the dual 2D to dilaton3D Einstein-Hilbert theory. gravity, which provides the This gives precisely that of theusing 2D theCFT above dual expressions to 3D Einstein-Hilbert for the one-point functions. gravity, which provides the UV completion of the 2D dilaton theory. 2 UV completion of theL 2D dilatonm theory.↵ 0 2µQ Sren[↵, ,µ]= This givesdt + + + Sglobal 22 ˆ ↵ L 2 I. Papadimitriou 2 ✓AdS2 holography and non-extremalL ◆ blackm holes↵ 0 2µQ 24 / 50 Sren[↵, ,µ]= 2 dt + + I. Papadimitriou AdS2 holography and non-extremal black holes2 ˆ ↵ L 24 / 50 which is the exact generating function of the theory. 2 ✓ ◆ I. Papadimitriou whichAdS is2 theholography exact and generating non-extremal function black holes of the theory. 24 / 50

I. Papadimitriou AdS2 holography and non-extremal black holes 25 / 50 I. Papadimitriou AdS2 holography and non-extremal black holes 25 / 50 Running dilaton solutions

The PBHRunning transformations dilaton that solutions preserve the form of the general running dilaton solution are parameterized by three arbitrary functions "(t), (t) and '(t), and act on the sources as The PBH transformations that preserve the form of the general running dilaton solution are parameterized by three arbitrary functions "(t), (t) and '(t), and act PBH↵ = @t("↵)+↵/L, PBH = "0 + /L, PBHµ = @t("µ + ') on the sources as Asymptotic symmetriesRunning and conserved dilaton solutions charges These three functions correspond respectively to time reparameterizations, i.e. PBH↵ = @t("↵)+↵/L, AsymptoticPBHboundary = " diffeomorphisms,0 + symmetries:/L, PBH Weylsubsetµ transformations,= @t( "ofµ Penrose+ ') and-Brown gauge-Henneaux transformations.(PBH) Thetransformations, PBH transformations diffeomorphisms that preserveand the formgauge of thetransformations, general running dilaton These three functions correspondsolution respectively are parameterized to time reparameterizations, by three arbitrary i.e. functions "(t), (t) and '(t), and act boundary diffeomorphisms, Weylpreserving transformations,The asymptotic the Fefferman andsymmetries gauge- transformations.Graham obtained bygauge, imposing that preserve boundary onconditions the sources of asthe solution: PBH (sources)=0 The asymptotic symmetries obtainedPBH↵ by= imposing@t("↵)+↵/L, PBH = "0 + /L, PBHµ = @t("µ + ') are (sources)=0à constrain functions ε(t), σ(t) and φ(t) in term of two constants ξ1,2 ThesePBH three functions correspond respectively0 to time reparameterizations, i.e. boundary diffeomorphisms," = ⇠1 , Weyl transformations,/L = ⇠1 , and' = gauge⇠2 ⇠ transformations.1 µ are ↵ ↵ ↵ where ⇠1,02 are arbitrary constants. The symmetry algebra is therefore " = ⇠1 , Conserved/L = ⇠1 , Charges' = ⇠2 : boundary⇠1 µ terms obtained by varying the action ↵ Theu asymptotic(1) u↵(1), symmetries whose corresponding obtained↵ chargesby imposing are the mass and the electric charge: with respect to the asymptotic symmetries (and Ward identities) à where ⇠1,2 are arbitrary constants. The symmetry algebra is therefore2 u(1) u(1) PBHL (0sourcesmL)=0 t Q , whose correspondingU(1) chargesxU(1): are the1 = mass and the electric= charge:, 2 = ↵ = . Q T 22 ↵2 22 Q J 2 ✓ 2 ◆ 2 2 Lare 2 mL Q = 0 = , = ↵ t = . 1 2 2 2 2 2 Q T 2I. Papadimitriou↵ 2 Q AdS2 Jholography and non-extremal0 black holes 43 / 50 ✓ 2 ◆ 2 " = ⇠1 , /L =2 ⇠1 , ' = ⇠2 ⇠1 µ 3D perspective: two↵ copies of the Virasoro↵ algebra with↵ the Brown- ± I. Papadimitriou AdS holographyHenneaux and non-extremal central black holes charge. Only L 0 are realized non-trivially in 2D. 2 where ⇠1,2 are arbitrary constants. The symmetry43 / 50 algebra is therefore u(1) u(1), whose corresponding charges are the mass and the electric charge: 2 L 0 mL t Q 1 = = , 2 = ↵ = . Q T 22 ↵2 22 Q J 2 ✓ 2 ◆ 2 2

I. Papadimitriou AdS2 holography and non-extremal black holes 43 / 50 counterterterm that explicitly depends of the cuto↵, since from (2.4) we see that p e ⇤t ⇠ @ ( /↵), precisely matching the expression for the conformal anomaly . Notice that the conformal t 0 A anomaly is proportional to the source of the scalar operator dual to the dilaton , and hence, the dilaton plays a central role in AdS2 holography and the breaking of the symmetries of the vacuum.

Generating functional The renormalized one-point functions (3.14) can be expressed as

S S 1 S = ren , = ren , t = ren , (3.17) T ↵ O ↵ J ↵ µ in terms of the renormalized on-shell action6

L m↵ 2 2µQ S [↵, ,µ]= dt + 0 + , (3.18) ren 22 ↵ L 2 Z ✓ ◆ which is identiﬁed with the generating function of connected correlation functions in the dual theory. As the one-point functions (3.14), this expression for the generating functional is exact in the sources ↵(t), (t) and µ(t). Successively di↵erentiating the generating functional or the one-point functions with respect to these sources one can evaluate any n-point correlation function of the operators , and t in the dual theory. T O J Holographic dictionary for running dilaton solutions E↵ective action and the Schwarzian derivative The Legendre transform of the generating functional with respect to a particular source gives the 1PI e↵ective action for the corresponding operator. Setting the sources ↵(t) and (t) of the operators and equal probes the ‘pure gauge T O dynamics’ sector of the theory that is described bySchwarzian correlation functions derivative of the e↵ective effective operator action + , which through the traceThe renormalized Ward identity on-shell (3.16) is actionSchwarzian equal can to bethe obtained conformal derivative (up anomaly. effectiveto a constant action that depends T O Legendre transforming theon generating globalEffective properties) functional by integrating withaction respect theas to relations theSchwarzian combined sourcederivative↵(t)= (t) leads to the e↵ective action Sren Sren t 1 Sren = , = , = Under Penrose-Brown-HenneauxUnderUnder Penrose-Brown-HenneauxT PBH ↵transformationsO (PBH)↵ (PBH) transformationsthe transformations sources:J the↵ sources theµ sources = S + dt ↵ ( + ) 2 2 2 e↵ ren↵ = e (1+"0+T"0)+2O (" ), =e (1+"0)+ (" ),µ2 = '0+"0'0+"'00+ (" ), 2 ↵ =usinge (1+ the above"0+Z" expressions0)+ (" )O, for the= one-pointe (1+" functions.0)+O (" ),µ= '0+"0'O0+"'00+ (" ), L where the↵ primesO3 ↵ 2denoteµQ a derivative withprime respect O- toderivativet. with respect to t O = dt 00 00 m . (3.19) where the2 primes↵0 denote 2 ↵2 a derivativeL with respect to t. This gives2InsertingZ ✓ these expressions◆ in the renormalized action Inserting these expressions in the renormalized action2 and absorbing total 2 2 S 2 L m↵ 0 2µQ Expressing the boundary metric as ↵derivative(t)dtSren= terms[↵,( ind⌧,µ(globalt]=)) inwe terms obtain d oft the ‘dynamical+ + time’ ⌧(t) Inserting these expressions in the2 renormalized2 ˆ action↵ andL absorbing total [12], i.e. an arbitrary time reparameterization function,L the e↵ective2 action✓ (3.19) takes the form◆ derivativeand terms absorbing in SSren = totalwe obtaind derivativet ( ⌧,t m/2) terms + Sglobal in, S =logone⌧ 0 ,obtains: which is the exact generatingglobal2 ˆ function{ } of the theory. global L 2 e↵ = where2 d thet ( Schwarzian⌧,tL µQ/L derivativem) is, given by (3.20) 2S ={ } dt ( ⌧,t m/2) + S , =log⌧ , Zren 2 ˆ global 0 2 { } ⌧ 3 ⌧ 2 ⌧,t = 000 00 where 2 2 { } ⌧ 0 2 ⌧ 0 where the Schwarzian⌧ 000 derivative3 ⌧ 00 is givenSchwarzian by derivative ⌧,t = 2 , (3.21) The{ Schwarzian} ⌧ 0 derivative2 ⌧ 0 action is a manifestation of the conformal anomaly! c.f., Sadchev, Ye, Kitaev ’93,… Almeheiri, Polochinski2 ’14; denotes the Schwarzian derivative.I. Papadimitriou This form of theAdS e↵2ectiveholography action and non-extremal arises⌧ in black the holes3 infrared⌧ limit of the 25 / 50 Maldacena, Stanford, Yang⌧,t ’16,= Engelsoy000 , Merens00 , Verlinde ’16,… Sachdev-Ye-Kitaev model [15, 16] and is a key piece of evidence for the holographic2 identiﬁcation { } ⌧ 0 2 ⌧ 0 of this model with AdS2 dilaton gravityI. Papadimitriou [10, 11, 12]. AdS2 holography and non-extremal black holes 17 / 39 6The full expression for the renormalizedThe Schwarzian on-shell actionderivative includes action an integration is a manifestation constant that depends of the onconformal global anomaly! properties of the solution on which it is evaluated. This integration constant can be determined by explicitly evaluating the radial integral for any particular solution. This is necessary, for example, in order to compute the free energy. Note, however, that the value of this additive constant is renormalization scheme dependent.

I. Papadimitriou AdS2 holography and non-extremal black holes 17 / 39

12 Constant dilaton solutions and AdS2 holography c.f., Strominger ’98, …Castro, Grumiller, Larsen, McNees ’08,… Compère, Song, Strominger ’13,…Castro, Song’14,…

Holography depends on the structure of non-extremal constant dilaton solutions and choice of boundary conditions à

Provided systematic holographic dictionary for each choice M.C., Papadimitriou 1608.07018 no time

Q=mL/2 Note: non-extremal running dilaton solution à extremal running-dilaton solution

with RG flow to IR fixed point VEV of irrelevant scalar op. extremal constant dilaton solution à non-extremal constant dilaton branch (`Coulomb phase’) (does not lift into subtracted geometry) Summary/Outlook with focus on AdS2 Holography

• Provided consistent KK Ansätze that allow us to uplift any solution of 2D EMD gravity to 4D STU solutions, which are non-extremal 4D black holes, asymptotically 2 (conformally) AdS2×S – subtracted geometry. 3 [Works also for 5D solutions asymptotically (conformally) AdS2xS .]

• 2D EMD gravity has a well defined UV fixed point, described by a sector of 2D CFT.

• Constructed holographic dictionary of 2D EMD gravity theory obtained by an S2 reduction of 4D STU subtracted geometry – runing dilaton solution as well as constant dilaton solutions. • Many aspects of the holographic description are generic and should apply to generic 2D dilaton gravity theories. Thank you!

Congratulations Gary, and to many more productive, scientific contributions!