Emergent IR Dual 2d CFTs in Charged AdS 5 Black Holes Maria Johnstone (University of Edinburgh) Korea Institute for Advanced Study (KIAS) 20 th February 2012

arXiv: v1 Jan de BoerJan de Boer, M. M. Sheikh-Jabbari, Joan SimonM. M. Sheikh-Jabbari Joan Simon Emergent IR Dual 2d CFTs in Charged AdS 5 Black Holes

Plan Introduce the EVH/CFT proposal Take the near horizon limit of Near BPS near EVH R-Charged black holes Discuss the near horizon AdS 3 geometry and emergent CFT 2 Connection between CFT 4 dual to AdS 5 and the CFT 2 dual to AdS 3 Relationship between the EVH/CFT proposal and the Extremal Black Hole/CFT Correspondence (finite horizon)

? not well-understood seems to contain no dynamics

? not well-understood seems to contain no dynamics

AdS 5 xS 5 Near Horizon Geometry Near Horizon CFT 4 =SU(N) SYM Low energy Limit IR Limit

AdS 5 xS 5 Near Horizon Geometry Near Horizon CFT 4 =SU(N) SYM Low energy Limit IR Limit central charge of the CFT.

Near Horizon Geometry Near Horizon Low energy Limit IR Limit (No dynamics) CFT 4 =SU(N) SYM AdS 5 xS 5 central charge of the CFT.

Near Horizon Geometry Near Horizon Low energy Limit IR Limit (No dynamics) CFT 4 =SU(N) SYM AdS 5 xS 5 Probing the system at sufficiently low energies above the black hole, but below the gap, one expects no non-trivial dynamics Near horizon limit kills all the non-trivial dynamics in the IR regime unless together with the N.H. limit we also decrease the mass gap by taking This will generically take the entropy to infinity unless we take temperature to zero too… central charge of the CFT.

This generically takes the entropy to infinity unless we take temperature to zero too: For any thermodynamic system, and in particular a black hole, the entropy admits a low temperature expansion: in the IR limit, large N, Small T This is the property of a CFT2, Cardy formula: Extremal Black Hole: T=0, S blows up for large N dual IR k+1 dimensional CFT,

Extremal Vanishing Horizon (EVH) Black Holes Entropy remains finite in the EVH limit, only if we scale Newton’s constant in the same rate as the horizon area. So taking the large N limit sends both the 5d Newtons constant and the mass gap of the “UV CFT” to zero. In the N.H. limit we obtain an AdS 3 throat

Extremal Vanishing Horizon (EVH) Black Holes N.H. limit of EVH black holes contains an AdS3 throat. We will study the near horizon geometry of AdS 5 static near EVH black holes in certain large N limit, and the IR emergent 2d CFT.

Extremal Vanishing Horizon (EVH) Black Holes For asymptotic AdS 5 EVH black holes we have a UV CFT 4 and an IR 2d CFT. We would like to relate quantum numbers of UV CFT and the 2d IR CFT’s.

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10d Metric: RR 4-form potential: R Charged AdS 5 Black Holes

10d Metric: 5d Black Hole Metric R Charged AdS 5 Black Holes

10d Metric: R Charged AdS 5 Black Holes

Mass: R Charges: R Charged AdS 5 Black Holes

Interpretation as Giant Gravitons Distribution of giant gravitons One type of giant for each charge: one corresponding to each 3- cycle in the transverse 5 sphere Number of giants in each stack : Pairs of giants intersect on circles It has been suggested in the literature that an AdS 5 black hole with two R charges should allow a dual 2d CFT description defined on the circle where giants intersect, and that the central charge of this CFT is proportional to the total number of these intersections

Extremal Vanishing Horizon Condition: Entropy: Temperature: Extremal Vanishing Horizon Condition

There are two physically distinct EVH cases: Near-extremal near-BPS: -a dilute gas of giant gravitons in which the black hole temperature remains finite -total number of giant graviton intersections: Near-extremal non-BPS: temperature scales to zero.

Near Horizon Limit Near BPS: Near Extremal: Near Horizon Limit:

Near Horizon Geometry

BTZ geometry:

Near Horizon AdS 3 BTZ geometry:

Reducing 10d IIB SUGRA over S 3 ×M 4 : 3d Newton constant: 5d entropy:3d entropy: Near Horizon AdS 3

Each strip lies at Entropy density: Sum over entropies: Near Horizon AdS 3 Focus on Strip of S 5 :

Scaling of N Overall epsilon scaling: Decoupling limit of the AdS/CFT correspondence: send the string length to zero: Keeping the AdS 5 radius constant then requires that we scale N to infinity as

Scaling of N Although the bulk entropy vanishes, its entropy density is finite

The CFT 4 Taking the near-horizon limit on the gravity backgrounds corresponds to taking a low energy limit in the N = 4, d = 4 U(N) SYM theory. In the near-BPS limit it corresponds to focusing on specific sector in the N = 4 SYM which we identify.

The CFT 4 Scaling Dimension: R Charge: Deviation from BPS: finite

CFT 4 vs CFT 2

CFT 4 CFT 2

CFT 4 vs CFT 2 The vacuum of the 2d CFT:= 1/4 BPS black hole

Extremal Black Hole/CFT Correspondence “Low energy excitations around any finite horizon, extremal black hole is described by a chiral 2d CFT with a given central charge c, read from Asymptotic Symmetry Group (ASG) analysis in the N.H. geometry of the black hole, and at Frolov-Thorone temperature T”.

Extremal Black Hole/CFT Correspondence “Low energy excitations around any finite horizon, extremal black hole is described by a chiral 2d CFT with a given central charge c, read from Asymptotic Symmetry Group (ASG) analysis in the N.H. geometry of the black hole, and at Frolov-Thorone temperature T”.

Extremal Black Hole/CFT Correspondence

Central Charge:

For extremal R-charged AdS 5 we have three U(1) isometries, three charges, associated with three independent rotations in S 5 There exist three inequivalent chiral CFTs reproducing the black hole entropy. central charges are fixed by the “Cardy formula”. Extremal Black Hole/CFT Correspondence

To analyze EVH/CFT vs Kerr/CFT, we need to relate the AdS 3 /CFT 2 dictionary to the limiting values of the Kerr/CFT predictions. EVH/CFT vs Extremal Black Hole/CFT Correspondence Finite level and vanishing central charge From an AdS3 perspective this generates an infinite gap in the CFT. The CFT temperature scales inversely to the entropy

To analyze EVH/CFT vs Kerr/CFT, we need to relate the AdS 3 /CFT 2 dictionary to the limiting values of the Kerr/CFT predictions. Finite central charge & vanishing level AdS3 perspective: keep a finite gap in the CFT, but sending the level to zero (vacuum). CFT temperature scales like the entropy: EVH/CFT vs Extremal Black Hole/CFT Correspondence

Vanishing central charge and level. If α, β > 0, the system is pushed to its vacuum while generating an infinite gap. To analyze EVH/CFT vs Kerr/CFT, we need to relate the AdS 3 /CFT 2 dictionary to the limiting values of the Kerr/CFT predictions.

EVH/CFT vs Extremal Black Hole/CFT Correspondence Vanishing central charge and level. If α, β > 0, the system is pushed to its vacuum while generating an infinite gap. This should correspond to the chiral CFT based on the Virasoro extension of the isometry direction along which giant gravitons intersect. Thus, one expects c 1 and T 1 to have different scaling behaviour from the two remaining Kerr/CFT descriptions. To analyze EVH/CFT vs Kerr/CFT, we need to relate the AdS 3 /CFT 2 dictionary to the limiting values of the Kerr/CFT predictions.

The CFT temperatures in the near-BPS limit: Black hole entropy Central charges EVH/CFT vs Extremal Black Hole/CFT Correspondence For finite N, all central charges scale to zero, consistent with vanishing of the bulk entropy and dilute giant graviton approximation in which the number of physical degrees of freedom is being scaled to zero.

The CFT temperatures in the near-BPS limit EVH/CFT vs Extremal Black Hole/CFT Correspondence Central charge density: Considerbulk entropy density is finite.

The CFT temperatures in the near-BPS limit EVH/CFT vs Extremal Black Hole/CFT Correspondence The other two central charge densities indicate a breaking down of this effective description: Considerbulk entropy density is finite. This confirms our expectation that the surviving CFT is the one living on the intersection of giant gravitons.

Chiral CFT of Kerr/CFT description is given by (c1, T1): Extremal case: EVH/CFT vs Extremal Black Hole/CFT Correspondence

Extremal case: EVH/CFT vs Extremal Black Hole/CFT Correspondence Chiral CFT of Kerr/CFT description is given by (c1, T1):

Per Strip Total Comparison with the near horizon analysis: In our near horizon limit we focused on a strip of the transverse 5-sphere and hence the Bekenstein-Hawking entropy of N.H. BTZ did not match the full black hole entropy. The standard dictionary between AdS3 bulk data and 2d dual CFT quantities therefore implies the central charge so obtained will only capture the number of intersecting giant gravitons present in the focused strip. EVH/CFT vs Extremal Black Hole/CFT Correspondence

Brown Henneaux Per strip Cardy’s formula is consistent with Bekenstein-Hawking:

In the N.H. BPS EVH case we focused on strips, the issue which is not present in the Kerr/CFT setup. We also note that taking the N.H. limit and the (near)- EVH limits do not commute: If we first take N.H. limit of an extremal near EVH black hole we end up with a different geometry than taking near horizon limit of an extremal black hole and then taking the EVH limit. EVH/CFT vs Extremal Black Hole/CFT Correspondence

Summary Introduced and studied aspects Extremal Vanishing Horizon (EVH) black holes in the family of static R-charged AdS5 black holes. a generic black hole in this family can be understood as excitations above the EVH black hole. Generic EVH black holes are defined as black holes with A, T,G N → 0 while the ratios A/T and A/G N remain finite, where A is the horizon area, T is the Hawking temperature and GN is the Newton constant. N.H. limit of generic EVH black hole has an AdS 3 throat. The AdS 3 throat for a near- EVH black hole turns to a BTZ black hole. Appearance of near horizon AdS 3 throat motivated the EVH/CFT proposal: near-EVH black holes or low energy excitations around an EVH black hole is described by a 2d CFT. Dealing with asymptotic AdS 5 black holes, we also have a UV dual CFT (N = 4 SYM) description Based on the gravity picture we then proposed a relation between the IR 2d and UV 4d dual CFTs Connected EVH/CFT and Kerr/CFT proposals The EVH/CFT proposal seems not to be limited to cases discussed here, seems to be generic to all EVH black holes.

Future directions In the case of EVH KK black hole it was shown that the near horizon limit is indeed a decoupling limit (arXiv: , M.M. Sheikh-Jabbari, Hossein Yavartanoo). We would like to show that the near horizon limit for the case of R-charged EVH black holes is a decoupling limit.arXiv: M.M. Sheikh-JabbariHossein Yavartanoo Exploring and establishing further the connection between the UV and IR CFTs proposal is an open interesting question. The EVH black holes are not limited to static ones and can be stationary. Within the class of asymptotic AdS 5 black holes we have a more general family of EVH black holes which involve rotation as well as R-charge. This class of charged-rotating AdS 5 EVH black holes will be studied in a future publication (M.M. Sheikh-Jabbari, Hossein Yavartanoo, Joan Simon).M.M. Sheikh-JabbariHossein YavartanooJoan Simon There are many questions/theorems both in gravity side and in the CFT side which should be studied and clarified.

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