The causal structure of the 5 d black holes Three-charge black hole: For 0 ≤ µ < µ c we have a time-like naked singularity, the singularity is, however, behind r = 0 (one can extend the geometry past r = 0 ). At some critical µ , µ = µ c , we have an extremal solution with a finite size horizon (function f has double zeros at some r h � = 0 ). For µ > µ c the geometry has two inner and outer horizons. – p. 21/119
The causal structure of the 5 d black holes, Cont’d Three-charge black hole, Cont’d : From the 10 d viewpoint the three-charge case corresponds to a set of three smeared giant gravitons intersecting only on the time direction and the giants in each set moving on either of the three S 1 directions in the S 5 . If one of the charges is much smaller than the other two a better description of the system is in terms of two giants intersecting on an S 1 , but the third charge appears as a rotation on the S 1 . – p. 22/119
The near-horizon limit of the two-charge extremal solutions For the two charge case, with vanishing q 1 : f = r 2 L 2 + f 0 − µ − µ c , r 2 f 0 = 1 + q 2 + q 3 µ c = q 2 q 3 , L 2 . L 2 The horizon of the 5 d black hole is where g rr vanishes, or at the roots of r 4 / 3 f . For µ = µ c we have a double zero at r = 0 and hence the solution is extremal. For µ < µ c f is positive definite and for µ > µ c f has a single positive root. Radius of the horizon S 3 in the 5 d metric is ( H 2 H 3 ) 1 3 r 2 , hence the extremal case has vanishing horizon area. – p. 23/119
The near-horizon limit of the two-charge extremal solutions, Cont’d One can distinguish two extremal black holes (which have double horizons at r = 0 ) The BPS case, with µ = 0 and The extremal but non-BPS case with µ = µ c . Here we study the near-horizon near-BPS as well as near-horizon near-extremal but non-BPS limits of the two-charge 10 d solutions separately and argue that these lead to decoupled geometries involving AdS 3 × S 3 factors. – p. 24/119
The near-horizon near-BPS limit µ 1 ∼ 1 case µ − µ c = ǫ 2 M, q i = ǫ ˆ q i τ = t L µ i = ǫ 1 / 2 x i , i = 2 , 3 , L, r = q 3 ) 1 / 2 ǫρ, (ˆ q 2 ˆ while ǫ → 0 and keeping ˆ q i , M ; τ, ρ, x i , φ i , L fixed. In this limit µ 1 = 1 + O ( ǫ 2 ) or θ 1 ∼ ǫ 1 / 2 , θ 2 = fixed. µ 1 ∼ µ 0 1 � = 1 case 1 µ − µ c = ǫ 2 M, q i = ǫ ˆ q i , ψ i = ǫ 1 / 2 ( φ i − τ ) , L i − ǫ 1 / 2 ˆ θ i = θ 0 θ i , 0 ≤ θ 0 r = q 3 ) 1 / 2 ǫρ, i ≤ π/ 2 , i = 2 , 3 (ˆ q 2 ˆ q i , M, θ 0 while ǫ → 0 and keeping ˜ ρ, ˆ i , x i , L fixed. – p. 25/119
The near-horizon near-BPS limit, Cont’d Taking the above limits we arrive at + L 2 � � ds 2 = ǫ R 2 ds 2 BTZ + d Ω 2 ds 2 � � S 3 C 4 R 2 S where dρ 2 BTZ = − ( ρ 2 − γ 2 ) dτ 2 + ds 2 ρ 2 − γ 2 + ρ 2 dφ 2 1 with γ 2 = µ − µ c = M q 3 /L 2 , µ c = ˆ ˆ q 2 ˆ µ c µ c ˆ and the radius of the S 3 being � R 2 S = q 2 ˆ ˆ q 3 µ ≃ 1 for R 2 � q 3 µ 0 µ ≃ µ 0 S = q 2 ˆ ˆ for 1 1 – p. 26/119
The near-horizon near-BPS limit, Cont’d In either case C 4 is (locally) describing a T 4 and hence the solutions are AdS 3 × S 3 × T 4 . ds 2 C 4 have different forms for the two cases: µ 1 ∼ 1 case � ds 2 q i ( dx 2 i + x 2 i dψ 2 C 4 = ˆ i ) i =2 , 3 where ψ i = φ i − τ . µ 1 ∼ µ 0 1 � = 1 case � ds 2 q i ( dx 2 i + ( µ 0 i ) 2 dψ 2 C 4 = ˆ i ) i =2 , 3 where µ 0 2 = sin θ 0 1 cos θ 0 2 , µ 0 3 = sin θ 0 1 sin θ 0 2 , 2 d ˆ 2 d ˆ 1 d ˆ dx 2 = cos θ 0 1 cos θ 0 θ 1 , dx 3 = cos θ 0 1 sin θ 0 θ 1 + cos θ 0 2 sin θ 0 θ 2 . – p. 27/119
The near-horizon near-BPS limit, Cont’d For the metric dρ 2 BTZ = − ( ρ 2 − γ 2 ) dτ 2 + ds 2 ρ 2 − γ 2 + ρ 2 dφ 2 1 γ 2 = − 1 we have a global AdS 3 space, for − 1 < γ 2 < 0 it is a conical space, for γ 2 = 0 we have a massless BTZ and for γ 2 > 0 we are dealing with a static BTZ black hole of mass γ 2 . These geometries are, upon two T-dualities, related to standard the D1-D5 system and the corresponding arguments are applicable to this case. – p. 28/119
The Near-horizon limit, the near-extremal, but non-BPS case Here we keep µ c fixed, with the scalings � µ c L τ µ − µ c = ǫ 2 M √ f 0 r = ǫρ, t = ǫ , f 0 φ 1 = ϕ φ i = ψ i + ˜ q i τ ˜ ǫ , ǫ , i = 2 , 3 q i L and ǫ → 0 while ρ, τ, ϕ, ψ i , M, q i , L are kept fixed. In this limit q i /L 2 and hence f 0 , µ c /L 2 are fixed. In this limit L 4 f 2 H i = L 2 f 0 M ρ 4 · 1 1 ρ 2 · 1 q i ∆ = µ 2 0 f = f 0 (1 − µ c ρ 2 ) , ǫ 4 , ǫ 2 . 1 q 2 q 3 q 2 q 3 – p. 29/119
The Near-horizon limit, the near-extremal, but non-BPS case, Cont’d Taking the limit we obtain 3 ) + 1 ds 2 10 = µ 1 ( R 2 AdS 3 ds 2 3 + R 2 S d Ω 2 ds 2 M 4 µ 1 where dρ 2 3 = − ( ρ 2 − ρ 2 0 ) dτ 2 + ds 2 + ρ 2 dϕ 2 , ρ 2 − ρ 2 0 Note that ϕ ∈ [0 , 2 πǫ ] . d Ω 2 3 is the metric for a three-sphere of unit radius and M 4 = L 2 ds 2 q 2 ( dµ 2 2 + µ 2 2 dψ 2 2 ) + q 3 ( dµ 2 3 + µ 2 3 dψ 2 � � 3 ) . R 2 S In the above AdS 3 = R 2 S ≡ √ q 2 q 3 = 0 = M � R 2 R 2 S ρ 2 L 2 µ c , , . f 0 µ c – p. 30/119
The Near-horizon limit, the near-extremal, but non-BPS case, Cont’d The ϕ angle in the BTZ is coming from the part which was in the S 5 part of the original AdS 5 × S 5 , the rest of the six-dimensional part of metric comes from the original AdS 5 geometry; the M 4 is coming from the S 5 piece. Although ϕ ∈ [0 , 2 πǫ ] , the causal boundary of the near-horizon decoupled geometry is still R × S 1 , because at large, but fixed ρ the AdS 3 part of the metric takes the form AdS 3 ǫ 2 ρ 2 ( − dt 2 + dφ 2 ds 2 3 ∼ R 2 1 ) , t is the (global) time direction in the original AdS 5 . – p. 31/119
The Near-horizon limit, the near-extremal, but non-BPS case, Cont’d As the 10 d IIB solution, we have a constant dilaton field with the four-form B 4 = − L 2 � q 2 µ 2 q 3 µ 2 ∧ d 3 Ω 3 , � ˜ 2 dψ 2 + ˜ 3 dψ 3 where in the near-horizon, near-extremal limit 2 (1 + q 3 3 (1 + q 2 q 2 2 = q 2 q 2 3 = q 2 ˜ L 2 ) , ˜ L 2 ) . Note that even when M = 0 , that is for µ = µ c the near-horizon geometry is not preserving any SUSY. – p. 32/119
Addition of the third charge We discussed the near-horizon limits of the two-charge black holes, which lead to BTZ × S 3 geometries. Here we are going to turn on the third charge q 1 . Consider generic values for q 1 . That is, take all three charges to be of the same order, for some critical value for µ , µ c , we have an extremal (but non-BPS) black hole. In the near-horizon limit this extremal but non-BPS black hole goes over to AdS 2 × S 3 geometry. What we are going to consider here is the non-generic case, when q 1 ≪ q 2 , q 3 . That is perturbative addition of the third charge . – p. 33/119
Perturbative Addition of the third charge, the near-BPS case Let us turn on the third charge q 1 and scale it as q 1 = ǫ 2 ˆ q 1 while keeping ˆ q 1 fixed, and scale the rest of parameters the same as before. After shifting the ρ coordinate as ρ 2 → ρ 2 − ˆ q 1 ˆ q 2 ˆ q 3 L 2 After the limit the metric takes the form + L 2 � � ds 2 = ǫ R 2 ds 2 rot.BTZ + d Ω 2 ds 2 � � S 3 C 4 R 2 S – p. 34/119
Perturbative Addition of the third charge, Cont’d where R 4 q 3 and ds 2 S = ˆ q 2 ˆ rot.BTZ is the metric for a rotating BTZ black hole in the AdS 3 background of unit radius, with mass and angular momentum � M BTZ = M + 2ˆ q 1 = ˆ µ + 2ˆ q 1 q 1 (ˆ ˆ µ + ˆ q 1 ) − 1 , J BTZ = 2 . µ 2 µ c ˆ µ c ˆ ˆ c Again there are two µ 1 ∼ µ 0 1 � = 1 and µ 1 ≃ 1 cases. As in the previous case, for µ 1 ≃ µ 0 1 , R 4 q 3 ( µ 0 1 ) 2 . S = ˆ q 2 ˆ The physical angular momentum of the original 10 d black-brane (or electric charge of the 5 d black hole) corresponding to q 1 charge, J 1 , is related to J BTZ as J 1 = N 2 ǫ 2 µ c ˆ L 2 J BTZ . 4 – p. 35/119
De tour to rotating BTZ black holes All stationary solutions to R µν = − 2 R 2 g µν , which are locally AdS 3 space-times, are of the form � � 2 � r 2 dφ + a 2 + − a 2 − F ( r ) � ds 2 = R 2 r 2 dt 2 + F ( r ) dr 2 + r 2 − dt , r 2 where φ ∈ [0 , 2 π ] and F ( r ) = r 4 + 2( a 2 − ) r 2 + ( a 2 + + a 2 + − a 2 − ) 2 . It is useful to introduce two other parameters + = − M + J − = − M − J a 2 a 2 , , 4 4 We can always assume a 2 + ≤ a 2 − , i.e. J ≥ 0 and J ∈ Z . We are then left with three possibilities. – p. 36/119
De tour to rotating BTZ black holes Conical Singularity: a 2 + , a 2 − > 0 , or M < − J. a + = a − = 1 / 2 corresponds to a global AdS 3 . For the generic case a + = a − = γ/ 2 , corresponding to J = 0 , the conic space has the same line element as a global AdS 3 but now φ ∈ [0 , 2 πγ ] . In string theory for rational values of γ and only when γ < 1 the conical singularity can be resolved. For the general a + � = a − case, the conical space can be resolved only when a 2 − is a rational number and 0 ≤ a 2 − ≤ 1 / 4 . In terms of M, J that is − 1 ≤ M − J ≡ − γ 2 < − 2 J , γ ∈ Q , J ∈ Z . – p. 37/119
De tour to rotating BTZ black holes a 2 + < 0 , a 2 − > 0 , corresponding to − J < M < J . The geometry is ill-defined and not sensible in string theory. Rotating BTZ Black hole: a 2 + , a 2 − ≤ 0 , or M ≥ J ≥ 0 This rotating BTZ black hole of mass M and angular momentum J has temperature √ M 2 − J 2 � √ √ ρ h = 1 � T BTZ = , M + J + M − J . 2 πρ h 2 Static BTZ: Special case of a − = a + ( i.e. J = 0 ). extremal rotating BTZ: Special case of a − = 0 ( M = J ), which has zero temperature. Massless BTZ black hole: Very special case of a − = a + = 0 ( M = J = 0 ). – p. 38/119
De tour to rotating BTZ black holes To summarize the above, the cases with integer-valued J and when M − J ≥ − 1 are those which are sensible geometries in string theory. For the − 1 < M − J < 0 resolution of conical singularity in string theory also √ demands J − M to be a rational number. Among the above cases M ≤ − J for any M, J and M = J, M ≥ 0 can be supersymmetrized. For the M ≤ − J case, the conic spaces, the solution becomes supersymmetric in a 3 d gauged supergravity which has at least two U (1) gauge fields. – p. 39/119
De tour to rotating BTZ black holes Supersymmetry.... To maintain supersymmetry we should turn on the Wilson lines of both of the U (1) (flat-connection) gauge fields. The two gauge fields which make the above metric supersymmetric are A (1) = a + ( dt + dφ ) , A (2) = a − ( dt − dφ ) , A (1) , A (2) are the flat connections of the two U (1) ’s. For M = J, M ≥ 0 , the extremal rotating BTZ black hole, no gauge fields are needed to keep supersymmetry. – p. 40/119
De tour to rotating BTZ black holes Among the supersymmetric configurations the global AdS 3 , that is when a + = a − = 1 / 2 , keeps the maximum supersymmetry the 3 d theory has, with anti-periodic boundary conditions for fermions on the φ direction. The massless BTZ, that is when a + = a − = 0 , as well as the extremal BTZ, corresponding to a 2 + = a 2 − > 0 , keep half of the maximal supersymmetry but with periodic boundary conditions for fermions on the φ direction. The conical spaces also keep half of maximal supersymmetry. – p. 41/119
Perturbative Addition of the third charge, Cont’d This metric is a rotating black hole only when M BTZ ≥ J BTZ (extremality bound) and also φ ∈ [0 , 2 π ] . In terms of our parameters the extremality bound is M 2 ≥ 4ˆ q 3 /L 2 . q 1 ˆ q 2 ˆ Note that M can be positive or negative. The (Hawking) temperature of our rotating BTZ is M 2 − 4ˆ � q 3 /L 4 q 1 ˆ q 2 ˆ T BTZ = � � � M 2 − 4ˆ � q 3 /L 4 π 2ˆ µ c M + 2ˆ q 1 + q 1 ˆ q 2 ˆ For the special case of M 2 = 4ˆ q 3 /L 2 we have an q 1 ˆ q 2 ˆ extremal rotating BTZ black hole which has T BTZ = 0 . – p. 42/119
Perturbative Addition of the third charge, Cont’d When M BTZ ≤ − J BTZ ≤ 0 , we have a sensible conical singularity only if � q 3 /L 2 ) , M ≤ − 2 Max (ˆ q 1 , q 1 ˆ ˆ q 2 ˆ q 1 ≤ 0 and if γ , γ 2 ≡ J BTZ − M BTZ , is a while M + 2ˆ rational number. In sum, to have a sensible string theory description we should have M BTZ − J BTZ + 1 ≥ 0 , and if 0 ≤ J BTZ − M BTZ ≡ γ 2 ≤ 1 , γ should be rational. – p. 43/119
Perturbative Addition of the third charge, the near-extremal case We may turn on the third charge q 1 “perturbatively”, with the scaling q 1 = ǫ 4 ˆ q 1 . After taking the above limit the metric takes the form + 1 ds 2 = µ 1 R 2 AdS ds 2 rot. BTZ + R 2 S d Ω 2 d M 2 � � 3 4 µ 1 where R 4 S = q 2 q 3 , R 2 AdS = R 2 S /f 0 and rot. BTZ = − N ( ρ ) dτ 2 + dρ 2 ds 2 N ( ρ ) + ρ 2 ( dϕ − N ϕ dτ ) 2 in which N ( ρ ) = ρ 2 − M BTZ + J 2 N ϕ = J BTZ BTZ 4 ρ 2 , , 2 ρ 2 – p. 44/119
Perturbative Addition of the third charge, the near-extremal case, Con with � M BTZ = M f 0 ˆ q 1 , µ c = q 2 q 3 /L 2 , f 0 = 1 + q 2 + q 3 , J BTZ = 2 . L 2 µ c µ c Note as in the two-charge case, in the above rotating BTZ the angular coordinate ϕ ∈ [0 , 2 πǫ ] . The above geometry has the interpretation of rotating BTZ only when the extremality bound is satisfied M 2 ≥ 4 µ c f 0 ˆ q 1 . The horizon radius, where N ( ρ ) vanishes, is ρ h = 1 �� � � M BTZ + J BTZ + M BTZ − J BTZ . 2 – p. 45/119
The Near-horizon Geometries as solutions to 6 d SUGRAs Questions: Are the AdS 3 × S 3 geometries solutions to some six-dimensional (super) gravities? Is there a consistent reduction of 10 IIB theory leading to these possible 6 d (supergravity) theories? If yes, Do these AdS 3 × S 3 near-horizon limit of a 6 d black string solution? – p. 46/119
The Near-horizon Geometries as solutions to 6 d SUGRAs, Cont’d Answers: As we will see the answer to first question is affirmative and we present the corresponding 6 d gravity theories. We also give the consistent reduction relating these 6 d theories to the 10 d IIB. As for the last question, for the near-BPS case the answer is affirmative, but for the near-extremal it is yet under construction. – p. 47/119
The 6 d SUGRA corresponding to the near-BPS geometry It is readily seen that the AdS 3 × S 3 coming as near-horizon limit of the 10 d near-BPS solution, which has equal AdS 3 and S 3 radii is a solution to � � 1 � R (6) − ( ∂ Φ) 2 − 1 d 6 x � − g (6) 3 e 2Φ F µνρ F µνρ S = , 16 πG (6) N The three-form F µνρ = ( dB 2 ) µνρ . The two-form is not self-dual. The above action is made into a consistent 6 d N = (1 , 1) SUGRA if besides the metric, two-form B 2 and the scalar Φ we also add two U (1) gauge fields. – p. 48/119
The 6 d SUGRA corresponding to the near-BPS geometry, Cont’d The two U (1) fields are not gauged, i.e. it is not a gauged SUGRA . The action for these gauge fields are � µν ) 2 + e − 2Φ ( F 2 e 2Φ ( F 1 µν ) 2 . S gauge = It is evident that the above 6 d theory can be obtained from the reduction of 10 d IIB theory on T 4 , or C 4 . The AdS 3 × S 3 is a solution to this 6 d theory with vanishing gauge fields, constant Φ and q 2 units of electric and q 3 units of magnetic three-form flux over the S 3 . – p. 49/119
The 6 d SUGRA corresponding to the near-BPS geometry, Cont’d The AdS 3 × S 3 also appears in the near-horizon over near-BPS black string, which is a marginal bound state of q 2 electric and q 3 magnetic strings. This 6 d strings, both of the electrically and magnetically charged ones, are 10 d three-brane giants wrapping two different two-cycles on C 4 . The tension of the 6 d string, the electric or magnetic ones both, is Nǫ s | Near BPS = πǫL 2 · T 3 = T (6) 2 πL 2 . – p. 50/119
The 6 d SUGRA corresponding to the near-BPS geometry, Cont’d The 6 d Newton constant is then N = G (10) (2 π ) 2 L 4 µ 0 2 µ 0 3 ǫ 2 µ 1 ∼ µ 0 G (6) 1 N , V ol C 4 = (2 π ) 2 L 4 ǫ 2 V ol C 4 µ 1 ∼ 1 Recalling that L 4 = 4 πg s Nl 4 G (10) = 8 π 6 g 2 s l 8 s , N s N = π 2 L 4 1 G (6) 8 · . µ 0 2 µ 0 N 2 ǫ 2 3 Note that to obtain the above for the µ 1 ∼ µ 0 1 , we have scaled the 6 d metric by a factor of ǫµ 0 1 so that, S = √ ˆ R 2 q 3 for both the µ 0 1 = 1 , and µ 0 q 2 ˆ 1 � = 1 cases . – p. 51/119
The 6 d SUGRA corresponding to the near-extremal geometry One can check that that the AdS 3 × S 3 coming as near-horizon limit of the 10 d near-extremal solution, which has unequal AdS 3 and S 3 radii is a solution to 1 R (6) − ( ∂ Φ) 2 + 8 L 2 cosh Φ − 1 � d 6 x � − g (6) 3 e 2Φ ( F 3 ) 2 � � S = , 16 πG (6) N The three-form F 3 = dB 2 . The two-form is not self-dual. Difference of this action with the previous one is in the potential term for scalar Φ . – p. 52/119
The 6 d SUGRA corresponding to the near-extremal geometry, Cont’d The AdS 3 × S 3 is a solution to this 6 d theory constant Φ and ˜ q 2 units of electric and ˜ q 3 units of magnetic three-form flux over the S 3 . The value of constant Φ is completely determined in terms of the charges ˜ q 2 , ˜ q 3 . The above 6 d action can be obtained from consistent reduction of IIB theory with the metric reduction ansatz µν dx µ dx ν + 1 ds 2 10 = µ 1 g (6) ds 2 M 4 µ 1 where M 4 = L 2 ds 2 e Φ ( dµ 2 2 + µ 2 2 dψ 2 2 ) + e − Φ ( dµ 2 3 + µ 2 3 dψ 2 � � 3 ) . R 2 S – p. 53/119
The 6 d SUGRA corresponding to the near-extremal geometry, Cont’d The two-form B 2 is coming from the reduction of the self-dual five-form: F 5 = 1 2 ∧ dχ 2 ∧ dx µ ∧ dx ν ∧ dx ρ 3! F 3 µνρ dµ 2 + 1 3 ∧ dχ 3 ∧ dx µ ∧ dx ν ∧ dx ρ , 3! e 2Φ ( ∗ F 3 ) µνρ dµ 2 The five-form equation of motion, dF 5 = 0 implies the equations of motion for the three-form: d ( e 2Φ ∗ F 3 ) = 0 . dF 3 = 0 , The 6 d Newton constant is then N = G (10) 2 L 4 = π 2 L 4 G (6) N 2 . π 2 – p. 54/119
The 6 d SUGRA corresponding to the near-extremal geometry, Cont’d Unlike the ungauged 6 d SUGRA, electric and magnetic string solutions to this 6 d gravity are not mutually BPS. The electrically and magnetically charged 6 d strings are both three-brane giants which are wrapping different two-cycles on M 4 . The tension of the 6 d strings are N 1 T (6) = T 3 ( πL 2 ) = 2 πL 2 = . s � G (6) 2 N These strings form a (p,q)-string type bound states . The mass of the bound state is the square root of the sum of the squares of mass of individual electric or magnetic strings. – p. 55/119
The Black Hole entropy Analyses To argue that our near-horizon limits are indeed decoupling limits we first compute the Bekenstein-Hawking entropy of the original 5 d black holes and compare it with the entropy of the 3 d (or 6 d ) black holes. As we will show these entropies match for both of the near-BPS and near-extremal cases. This matching is a strong evidence in support of the fact that in our decoupling limits we have not lost any degrees of freedom. – p. 56/119
The Black Hole entropy, the 5 d Analysis The 5 d Bekenstein-Hawking entropy is S BH = A (5) h . 4 G (5) N where A (5) = 2 π 2 r 3 h ( H 1 H 2 H 3 ) 1 / 2 | r = r h . h Recalling that N = G (10) L 4 = 4 πg s Nl 4 G (10) G (5) = 8 π 6 g 2 s l 8 N s , π 3 L 5 , s , N we obtain 2 πN 2 · A (5) S BH = 1 h L 3 . – p. 57/119
The 5 d black hole entropy analysis, the near-BPS case In the near-BPS limit the horizon is located at r 2 h = µ − µ c and hence = πγ ˆ µ c L 2 N 2 ǫ 2 , S Near BPS BH where γ 2 = µ − µ c µ c = µ c /ǫ 2 , ˆ µ c Once the third charge is also added perturbatively, the above is replaced with = π ˆ µ c L 2 ρ h N 2 ǫ 2 , S Near BPS BH where 1 � � M 2 − 4ˆ � ρ 2 q 3 /L 4 h = M + 2ˆ q 1 + q 1 ˆ q 2 ˆ . 2ˆ µ c – p. 58/119
The 5 d black hole entropy analysis, the near-BPS case The validity of classical gravity analysis demands that All curvature components should remain small in string units l s and the entropy, should be large: S BH ≫ 1 All curvature components scale as 1 /ǫ (in units of L − 2 ). The large entropy condition implies that together with ǫ → 0 , N → ∞ , e.g. as N ∼ ǫ − α , α ≥ 2 . This consideration is not strong enough to fix α . – p. 59/119
The 5 d black hole entropy analysis, the near-BPS case Noting the form of metric, that it has a factor of ǫ in front and that one expects the string scale to be the shortest physical length leads to N ∼ ǫ − 2 . ǫ ∼ l 2 ⇒ s Once the above scaling of ǫ and N is considered, S BH ∼ N ∼ ǫ − 2 → ∞ . In sum, our complete near-horizon, near-BPS limit is defined as an α ′ = l 2 s ∼ ǫ → 0 limit together with scaling q 2 , q 3 ∼ ǫ ; q 1 , µ ∼ ǫ 2 , while keeping L 4 ∼ Nl 4 s fixed. – p. 60/119
The 5 d black hole entropy analysis, the near-extremal case In the near-extremal limit to order ǫ , we have h = µ − µ c r 2 + O ( ǫ 4 ) . f 0 Therefore = π µ c L 2 · ρ 0 S Near Extremal N 2 ǫ. √ f 0 BH With the perturbative addition of the third charge 1 µ c L 2 N 2 ǫ , S BH = πρ h √ f 0 where ρ h = 1 �� � � M BTZ + J BTZ + M BTZ − J BTZ 2 � M BTZ = M f 0 ˆ q 1 , J BTZ = 2 µ c µ c – p. 61/119
The 5 d black hole entropy analysis, the near-extremal case To ensure the validity of the classical gravity analysis, one should also send N → ∞ while keeping ρ 0 and µ c /L 2 finite. This is done if we scale N ∼ ǫ − β , β ≥ 1 2 . The validity considerations does not fix β . As we will show, however, β = 1 is giving the appropriate choice, N ∼ ǫ − 1 → ∞ . In sum, we keep L, g s , q i /L 2 and ρ 0 finite while taking s ∼ N − 1 ∼ ǫ → 0 . l 4 In this case, as in the near-BPS case, S BH ∼ N → ∞ . – p. 62/119
The Black Hole Entropy, the 3 d Analysis The rotating BTZ × S 3 obtained in the near-horizon limit is also a solutions to 6 d (super)gravity theory. One can further reduce this 6 d theory on the S 3 to obtain a 3 d gravity theory. The rotating BTZ solution is then a black hole solution to this 3 d theory. What we are going to do here is to compute the BH entropy of this 3 d black holes, which is obtained from BH = A (3) S (3) 4 G (3) N where A (3) is the area of horizon for the BTZ black hole. – p. 63/119
The 3 d black hole entropy analysis, the near-BPS case The 3 d Newton constant is related to the 6 d one as G (6) L 4 1 1 G (3) N N = = · . 2 π 2 R 3 16 R 3 µ 0 2 µ 0 N 2 ǫ 2 3 S S The 3 d entropy for any value of µ 0 2 and µ 0 3 is hence BH = 8 π ˆ µ c L 2 ρ h N 2 ǫ 2 µ 0 s (3) 2 µ 0 3 , with the ρ h taking the same value as in the 5 d case. The total entropy to be compared against the 5 d entropy is integral of s (3) BH over values of µ 0 2 , µ 0 3 , yielding BH = π ˆ µ c L 2 ρ h N 2 ǫ 2 , S (3) This exactly matches the the entropy of the 5 d black hole after taking the near-BPS decoupling limit. – p. 64/119
The 3 d black hole entropy analysis, the near-extremal case For the near-extremal case that is A (3) = 2 πǫR AdS 3 ρ 0 . The 2 πǫ comes from the fact that ϕ ∈ [0 , 2 πǫ ] . The 3 d Newton constant is G (6) = L 4 · 1 G (3) N N = N 2 . 2 π 2 R 3 2 R 3 S S Therefore, BH = πR AdS R 3 S (3) S ρ 0 N 2 ǫ , L 4 The above is the same as the 5 d black hole entropy in the near-horizon near-extremal limit, recalling � µ c = R 4 S /L 2 . R AdS = R S / f 0 , – p. 65/119
Dual Field Theory Descriptions So far we have shown that one can take specific near-horizon, near-extremal limits over 10 d type IIB solutions which are asymptotically AdS 5 . As such one would expect that these solutions, the limiting procedure and the resulting geometry after the limit should have a dual description via AdS 5 /CFT 4 . On the other hand, after the limit we obtain a space which contains AdS 3 × S 3 , and hence there should also be another dual description in terms of a 2 d CFT. – p. 66/119
Dual Field Theory Descriptions So far we have shown that one can take specific near-horizon, near-extremal limits over 10 d type IIB solutions which are asymptotically AdS 5 . As such one would expect that these solutions, the limiting procedure and the resulting geometry after the limit should have a dual description via AdS 5 /CFT 4 . On the other hand, after the limit we obtain a space which contains AdS 3 × S 3 , and hence there should also be another dual description in terms of a 2 d CFT. – p. 67/119
Dual Field Theory Descriptions, the 4 d SYM Here we translate what taking the near-horizon limits on the gravity backgrounds corresponds to in the N = 4 , d = 4 U ( N ) SYM theory. We argue that taking the near-horizon near-BPS and near-extremal limits correspond to focusing on specific sectors in the N = 4 SYM which we identify. We argue that the decoupling in the gravity corresponds to the fact that these sectors are closed under SYM dynamics. The idea here is somewhat like that of BMN and almost-BPS operators there.... – p. 68/119
Dual Field Theory Descriptions, the 4 d SYM The operators of N = 4 , d = 4 U ( N ) SYM theory are specified by their SO (4 , 2) × SO (6) quantum numbers. The scaling dimension of operators ∆ and their R -charge J i respectively correspond to the ADM mass and angular momentum of the objects in the gravity. Explicitly, for the two-charge case of our interest, with the perturbative addition of the third charge, the operators are specified by four quantum numbers ∆ = L · M ADM = N 2 2 L 2 (3 2 µ + q 1 + q 2 + q 3 ) , q i = N 2 J i = πL q i ˜ ˜ L 2 , 4 G 5 2 – p. 69/119
Dual Field Theory Descriptions, the 4 d SYM and are singlets of SO (4) ∈ SO (4 , 2) . If µ and q i are finite, ∆ and J i scale as N 2 . In both of the near-BPS and near-extremal limits we are taking the ’t Hooft coupling, λ = L 4 /l 4 s to infinity. Despite of the large ’t Hooft coupling, we may have a perturbative description. Recall the BMN case, where the effective expansion parameters of the 4 d gauge theory is different in sectors of large R -charges and we have finite effective (or “dressed”) ’t Hooft coupling and the genus expansion parameter. – p. 70/119
The 4 d N = 4 SYM description, the near-BPS case In the near-BPS limit case together with some of the coordinates we also scale µ and q i as ǫ . Moreover, we need to also scale N ∼ ǫ − 2 . Therefore, the sector of the N = 4 U ( N ) SYM operators corresponding to the geometries in question have large scaling dimension and R -charge ∆ = N 2 ǫ q 3 + O ( ǫ )) /L 2 ∼ N 3 / 2 → ∞ (ˆ q 2 + ˆ 2 J i = N 2 ǫ q i + O ( ǫ )) /L 2 ∼ N 3 / 2 . 2 (ˆ – p. 71/119
The 4 d N = 4 SYM description, the near-BPS case In the same spirit as the BMN limit, one can find certain combinations of ∆ and J i which are finite and describe physics of the operators after the limit. In order that recall the way the limit was taken: iL ∂ ∂τ = iL ∂ ∂ � � ∂t + i = ∆ − J i ∂φ i i =2 , 3 i =2 , 3 − i ∂ = − i ∂ = J i ∂ψ i ∂φ i Up to leading order we have J i = N 2 ǫ 2 J i = N 2 ǫ µ ˆ q i ˆ � ∆ − L 2 , L 2 . 4 2 i =2 , 3 – p. 72/119
The 4 d N = 4 SYM description, the near-BPS case ∆ − � J i ∼ N 2 · N − 1 = N → ∞ , while J i ∼ N 3 / 2 . The “BPS deviation parameter”: η i ≡ ∆ − � i J i ∼ ǫ ∼ N − 1 / 2 → 0 , J i and hence we are dealing with an “almost-BPS” sector. It is instructive to make parallels with the BMN sector, where we deal with operators with ∆ ∼ J ∼ N 1 / 2 , while ∆ − J = finite, implying that, similarly to our case, η BMN ∼ N − 1 / 2 → 0 . Note that, ∆ − � J i is linearly proportional to µ and S BH ∼ ∆ − � J i ∼ N . non-extremality parameter ˆ – p. 73/119
The 4 d N = 4 SYM description, the near-BPS case In sum, the sector we are dealing with is composed of “almost 1/4 BPS” operators of U ( N ) SYM with ∆ ∼ J i ∼ N 3 / 2 , λ = g 2 Y M N ∼ N → ∞ N 3 / 2 ≡ ˆ J i q i J i ) · 1 N = ˆ µ � L 2 = fixed, (∆ − L 2 = fixed. i =2 , 3 The dimensionless physical quantities that describe µ/L 2 and g Y M . q i /L 2 , ˆ this sector are therefore ˆ To completely specify the sector, the basis used to contract N × N gauge indices should also be specified. This could be done by giving the (approximate) shape of the corresponding Young tableaux. – p. 74/119
The 4 d N = 4 SYM description, the near-BPS case To this end we recall the interpretation of the original 10 d geometry in terms of the back-reaction of the intersecting giant gravitons and that giant gravitons and their open string fluctuations are described by (sub)determinant operators. Here we are dealing with a system of intersecting multi giants. The “number of giants” in each stack in the near-BPS, near-horizon limit is N i = Nǫ · ˆ L 2 = 2 N 1 / 2 ˆ q i q i L 2 , µ ˆ i J i = N 2 N 3 Therefore, ∆ − � µ c . 4 ˆ – p. 75/119
The 4 d N = 4 SYM description, the near-BPS case Finally, let us consider addition of the third charge, where besides J 2 , J 3 we have also turned on J 1 , J 1 = N 2 ǫ 2 · 1 � q 1 (ˆ ˆ q 1 + ˆ µ ) . L 2 2 i =2 , 3 J i ∼ J 1 ∼ N 2 ǫ 2 ∼ N → ∞ . As we see ∆ − � In this case instead of ∆ − � i =2 , 3 J i it is more appropriate to define another positive definite quantity: 3 � q 1 ) 2 − ˆ � � µ 2 µ + 2ˆ ˆ q 1 − (ˆ µ + 2ˆ � ∆ − J i = N · ≥ 0 . L 2 i =1 – p. 76/119
The 4 d N = 4 SYM description, the near-BPS case It is remarkable that the above BPS bound is exactly the same as the bound in which the generic rotating BTZ metric could be made sense of. This bound is more general than just the extremality bound of the rotating BTZ black hole M BTZ − J BTZ ≥ 0 . This bound besides the rotating black hole cases also includes the case in which we have a conical singularity which could be resolved in string theory. End of the near-BPS case – p. 77/119
The 4 d N = 4 SYM description, the near-extremal case In the near-horizon, near-extremal limit we do not scale µ and q i ’s. Therefore, we deal with a sector of N = 4 SYM in which ∆ ∼ J i ∼ N 2 and, as noted N ∼ ǫ − 1 . To deduce the correct “BMN-type” combination of ∆ and J i , we recall the way the limit has been taken: R S t φ i = ψ i + ˜ q i R AdS 3 τ τ = ǫ L, ǫ , i = 2 , 3 . R AdS 3 q i R S Therefore, − i ∂ ∂ψ i = − i ∂ ∂φ i = J i and E ≡ − i ∂ ∂τ = − R AdS 3 iL ∂ q i ˜ ∂ � � � ∂t + i ǫ R S q i ∂φ i i =2 , 3 ∆ − 2 L 2 J 2 = − R AdS 3 � � � i N 2 ǫ R S q i i =2 , 3 – p. 78/119
The 4 d N = 4 SYM description, the near-extremal case Intuitive way of understanding E : In the near-extremal case we deal with massive giant gravitons which are far from being BPS and hence are behaving like non-relativistic objects which are rotating with angular momentum J i over i = L 2 circles with radii R i , R 2 S q i . R 2 Therefore, the kinetic energy of this rotating branes is proportional to � J 2 i /q i . In our limit ǫ ∼ 1 /N which for convenience we choose ǫ = 4 N . – p. 79/119
The 4 d N = 4 SYM description, the near-extremal case Recalling that ∆ is measuring the “total” energy of the system, n E should have two parts: the rest mass of the system of giants and the energy of “internal” excitations of the branes. To see this explicitly we note that · N 2 E = R AdS 3 4 ǫ · µ L 2 = E 0 + R AdS 3 · (2 πT (6) s M ) R S R S where have used µ = µ c + ǫ 2 M ( M is related to the mass of BTZ black hole), and E 0 = R AdS 3 R 3 S · N 3 . 16 L 4 – p. 80/119
The 4 d N = 4 SYM description, the near-extremal case E 0 which is basically E evaluated at µ = µ c , is the rest mass of the brane system. E − E 0 corresponds to the fluctuations of the giants about the extremal point. E − E 0 is proportional to T (6) s M , indicating that it can be recognized as fluctuations of a 6 d string. Recall also that from the 10 d viewpoint, the 6 d strings are uplifted to three-brane giants with two legs along the M 4 directions. Therefore, E − E 0 corresponds to (three) brane-type fluctuations of the original “intersecting giants”. – p. 81/119
De tour, The 4 d N = 4 SYM description, the near-extremal case At the extremal point the system is not BPS and the “rest mass” of the giants system is not simply sum of the masses of individual stacks of giants and contains their “binding energy” (stored in the deformation of the giant shape from the spherical shape). Nonetheless, it should still be proportional to the number of giants times mass of a single giant. In the 6 d language, as suggested previously, this corresponds to formation of a 6 d ( Q e , Q m ) -string. – p. 82/119
De tour, The 4 d N = 4 SYM description, the near-extremal case Inspired by the expression for the 10 d five-form flux and recalling that the IIB five-form is self-dual, the system of giants we start with, may also be interpreted as spherical three-branes wrapping S 3 ∈ AdS 5 while rotating on S 5 , the dual giants. In terms of dual giants, after the limit, we are dealing with a system of dual giants wrapping S 3 ∈ AdS 3 × S 3 which has radius R S . – p. 83/119
De tour, The 4 d N = 4 SYM description, the near-extremal case The mass of a single such dual giant m 0 (as measured in R AdS 3 units and also noting the scaling of AdS 5 time with respect to AdS 3 time) is then S ) = R 3 m 0 R AdS 3 /ǫ = T 3 (2 π 2 R 3 S L 4 · N. The number of dual giants is again proportional to N and hence one expects the total “rest mass” of the system m 0 to be proportional to N 3 R 3 S . End of De Tour to Dual Giants and their mass. – p. 84/119
The 4 d N = 4 SYM description, the near-extremal case In sum, from the U ( N ) SYM theory viewpoint the sector describing the near-extremal, near-horizon limit consists of operators specified with ∆ ∼ J i ∼ N 2 , λ ∼ N → ∞ , J i q i ˜ E − E 0 N 2 ≡ 2 L 2 = fixed, = fixed , N where as discussed, E , E 0 are defined in terms of ∆ , J i . – p. 85/119
The 4 d N = 4 SYM description, the near-extremal case As discussed, one may obtain a rotating BTZ if we turn on the third R -charge in a perturbative manner. In the 4 d gauge theory language this is considering the operators which besides the above E − E 0 and J i carry the third R -charge J 1 , J 1 ∼ N 2 ǫ 2 ∼ 1 : J 1 = N 2 2 L 2 ǫ 2 � q 1 µ c ˆ In terms of the AdS 3 parameters, since ϕ = ǫφ , then � ǫ = N 2 ǫ J ≡ − i ∂ ∂ϕ = − i 1 ∂φ = J 1 ∂ µ c q 1 ˆ ǫ 2 L 2 µ c – p. 86/119
The 4 d N = 4 SYM description, the near-extremal case As we see J , similarly to E − E 0 , is also scaling like N 2 ǫ ∼ N in our decoupling limit. When J 1 is turned on the expressions for ∆ and hence E are modified, receiving contributions from q 1 . These corrections, recalling that q 1 scales as ǫ 4 , vanish in the leading order. However, one may still define physically interesting combinations like E − E 0 ± J . End of the 4 d SYM descriptions – p. 87/119
Description in terms of 1 + 1 dim. dual theory In either of the near-BPS or near-extremal near-horizon limits we obtain a space-time which has an AdS 3 × S 3 factor. In both cases the AdS 3 factor is in global coordinates. This, within the AdS/CFT ideology, is suggesting that (type IIB) string theory on the corresponding geometries should have a dual 1 + 1 CFT description. – p. 88/119
Description in terms of 2 d dual theory, the near-BPS case In the near-BPS case metric takes the same form as the near-horizon limit of a D1-D5 system, though the AdS 3 is obtained to be in global coordinates. This could be understood noting that the two-charge geometry corresponds to a system of smeared giant D3-branes intersecting on a circle. In the near-horizon limit we take the radius of the giants to be very large (or equivalently focus on a very small region on the worldvolume of the spherical brane) while keeping the radius of the intersection circle to be finite (in string units). – p. 89/119
Description in terms of 2 d dual theory, the near-BPS case Therefore, upon two T-dualities on the D3-branes along the C 4 directions the system goes over to a D1-D5 system but now the D1 and D5 are lying on the circle (D5 has its other four directions along C 4 ). Here we give the dictionary from our conventions and notations to that of the usual D1-D5 system, and discuss the similarities and difference. Number of D-strings Q 1 and number of D5-branes Q 5 are respectively equal to the number of giants in each stack N 2 and N 3 . – p. 90/119
Description in terms of 2 d dual theory, the near-BPS case The degrees of freedom are coming from four DN modes of open strings stretched between intersecting giants which are in ( N 2 , ¯ N 3 ) representation of U ( N 2 ) × U ( N 3 ) . In taking the near-horizon, near-BPS limit we are focusing on a narrow strip in µ 2 , µ 3 directions and hence our BTZ × S 3 × C 4 geometry and in this sense the corresponding 2 d CFT description is only describing the narrow strips on the original 5 d black hole. – p. 91/119
Description in terms of 2 d dual theory, the near-BPS case Therefore, our 5 d black hole is described in terms of not a single 2 d CFT, but a collection of (infinitely many of) them. The only property which is different among these 2 d CFT’s is their central charge. The “metric” on the space of these 2 d CFT’s is exactly the same as the metric on C 4 . As far as the entropy and the overall (total) number of degrees of freedom are concerned, one can define an effective central charge of the theory which is the integral over the central charge of the theory corresponding to each strip. – p. 92/119
Description in terms of 2 d dual theory, the near-BPS case For the central charge we use the Brown-Henneaux central charge formula, c = 3 R AdS 2 G (3) N and recall that for each strip L 4 1 G (3) N 2 ǫ 2 µ 0 2 µ 0 R AdS = R S , N = · 3 16 R 3 S The effective total central charge is obtained by integrating strip-wise c over the C 4 . Noting that µ 2 µ 3 dµ 2 dµ 3 = 1 � 8 , µ 2 2 + µ 2 3 ≤ 1 – p. 93/119
Description in terms of 2 d dual theory, the near-BPS case The effective central charge of the system is c L = c R = c = 3 N 2 N 3 = 12 N · ˆ µ c L 2 . Compare this with the central charge of the usual D1-D5 system is given by 6 Q 1 Q 5 . In near-BPS case c ∼ N → ∞ , as opposed to N 2 because in our case the entropy scales as N 2 ǫ 2 and that ǫ 2 ∼ 1 /N . The 2 d CFT is described by L 0 , ¯ L 0 which are related to the BTZ black hole mass and angular momentum L 0 = 6 cN L = 1 L 0 = 6 cN R = 1 4( M BTZ − J BTZ ) , ¯ 4( M BTZ + J BTZ ) . – p. 94/119
Description in terms of 2 d dual theory, the near-BPS case Note that L 0 , ¯ L 0 are equal to the left and right excitation number of the 2 d CFT N L and N R , divided by N 2 N 3 . The above expressions for L 0 , ¯ L 0 are given for M BTZ − J BTZ ≥ 0 when we have a black hole description. When − 1 ≤ M BTZ − J BTZ < 0 , we need to replace + , ¯ them with L 0 = − c L 0 = − c 24 a 2 24 a 2 − . In the special case of global AdS 3 background, where a + = a − = 1 / 2 formally corresponding to M BTZ = − 1 , J BTZ = 0 , the ground state is describing an NSNS vacuum of the 2 d CFT. – p. 95/119
Description in terms of 2 d dual theory, the near-BPS case With the above identification, the Cardy formula for the entropy of a 2 d CFT gives �� � � S 2 d CFT = 2 π cN L / 6 + cN R / 6 = π �� � � 6 c M BTZ − J BTZ + M BTZ + J BTZ This exactly reproduces the expressions for the entropy we got in the 5 d and 3 d descriptions. Although the entropy and the energy of the system (which are both proportional to the central charge) grow like N and go to infinity the temperature and the horizon size remain finite. – p. 96/119
Description in terms of 2 d dual theory, the near-BPS case It is also instructive to directly connect the 4 d and the 2 d field theory descriptions. Comparing the expressions for M BTZ , J BTZ and ∆ − � i =2 , 3 J i , J 1 , we see that they match; explicitly J i = c J 1 = c � ∆ − 12( M BTZ + 1) , 12 J BTZ . i =2 , 3 The 4 d gauge theory BPS bound, ∆ − � i =1 , 2 , 3 J i ≥ 0 now translates into the bound M BTZ − J BTZ ≥ − 1 . This means that the 4 d gauge theory, besides being able to describe the rotating BTZ black holes, can also describe the conical spaces. – p. 97/119
Description in terms of 2 d dual theory, the near-BPS case In other words, ∆ − � 3 i =1 J i = 0 and N ˆ µ c L 2 respectively correspond to global AdS 3 and massless BTZ cases and when 3 J i < c 12 = N ˆ µ c � 0 < ∆ − L 2 , i =1 4 d gauge theory describes a conical space, provided γ , � 3 � γ 2 ≡ 12 � ∆ − J i − 1 , c i =1 is a rational number. – p. 98/119
Description in terms of 2 d dual theory, the near-BPS case This is of course expected if the dual gauge theory description is indeed describing string theory on the conical space background. One should also keep in mind that entropy and temperature are sensible only when ∆ − � 3 c i =1 J i ≥ 12 ; For smaller values the degeneracy of the operators in the 4 d gauge theory is not large enough to form a horizon of finite size (in 3 d Planck units). End of the 2 d CFT description of the near-BPS case. – p. 99/119
Description in terms of 2 d dual theory, the near-extremal case In the near-horizon limit of a near-extremal two-charge black hole we obtain an AdS 3 × S 3 in which the AdS 3 and S 3 factors have different radii. Although locally AdS 3 , the coordinate parameterizing S 1 ∈ AdS 3 is ranging over [0 , 2 πǫ ] = [0 , 8 π/N ] . As such, and recalling that the AdS 3 × S 3 is not supersymmetric, one expects the dual 2 d CFT description to have somewhat different properties than the standard D1-D5 system. – p. 100/119
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