Strings and Fields 2019 @ YITP, Kyoto 20 Aug 2019 Thermodynamics of AdS 3 gravity: extremal CFTs vs. semi-classical gravity Yasunari Kurita Kanagawa Inst. Tech. 1
AdS 3 pure gravity Asymptotic symmetry is Virasoro sym. with central charge (Brown-Henneaux 1986) In terms of AdS/CFT, quantum gravity is boundary CFT. ⇒ Find the CFT! At T =0, the boundary is cylinder. For T > 0, it is 2-dim. Torus. Author: Polytope24 https://en.wikipedia.org/wiki/AdS/CFT_correspondence From now on, we consider CFT on torus 2
Witten(2007) • Assumption ： holomorphic factorization • Witten’s conjecture (2007) ： quantum theory of AdS 3 pure gravity is extremal CFTs with c = 24𝑙 (𝑙 = 1,2,3, … ) and it’s anti -holomorphic pair ℓ 𝐻 = 16𝑙 is quantized. For each k, one CFT possibly exists. Note 1 ： Note 2 ： 「 large k 」～「 small G 」 （ semi-classical ） 「 large k 」～「 large c 」 （ thermodynamic limit ） This has nice microscopic description for BTZ entropy! 3
BTZ entropy Witten 2007 • In extremal CFTs, for large k correction ？ Bekenstein-Hawking # of primaries which create BTZ 4
Appendix: Extremal CFT For c =24 k, extremal CFT is a CFT whose lowest dimension of nontrivial primary is k + １ (its maximum). For c =24 k, it is known that the lowest dimension of nontrivial primary is equal to or less than k +1. Note 1 ： large mass gap Extremal CFT spectrum AdS 3 classical gravity 𝑀 0 M Primary and its decendants massive BTZ 1 Nontrivial first primary 0 massless BTZ （ not black hole ） massgap massgap (conical singularity) (Virasoro decendants − 1 of the ground state) Global AdS 3 −𝑙 Ground state 8𝐻 Interpretation: nontrivial primary fields make BTZ black hole 5
Appendix2: Extremal CFT Note 2 ： the partition function is uniquely determined! 𝜐 is moduli of the boundary torus Contributions from Ground state and its Virasoro descendants primaries ( 𝑀 0 ≥ 1 ) (these determine the pole structure at q= 0.) Mathematical fact ： holomorphic ＆ modular inv. ⇒ “ 𝑎(𝜐) is a polynomial of J - function” Determine the polynomial of J to have the same pole Klein ’s structure with Z , then the partition function is uniquely j-invariant determined . 6
This work • investigates thermodynamic quantities obtained from extremal CFT partition functions Z k for several k . → We find usual Hawking-Page transition (AdS 3 ↔ BTZ) • For rapidly rotating spacetime, we also find several new phases which do not appear in the usual Hawking-Page transition. This is consistent with the phase diagram obtained by Maloney-Witten 2007. • compare these (quantum) results with those of semi-classical gravity. 7
3-dim. Hawking-Page Semi-classical Free energy entropy S AdS 3 phase BTZ phase ＢＴＺ AdS 3 T T c AdS 3 ＢＴＺ HP critical temperature Non-rotating Im 𝜐 𝜐 -plane Limit of Ω E Limit of Ω E AdS 3 BTZ conical? Moduli parameter complex ？ 8 Re 𝜐
Partition functions of extremal CFTs • For first several k, obtained by FLM(‘84) having Monster symmetry Indices are k Given k, the partition function is computable! For example ： k = 10 9
Extremal CFT Non-rotating case • k =20 case • k =1 case S S BTZ BTZ T T AdS 3 AdS 3 For large k, the transition (at T HP ) becomes sharp! The sharp transition appears in the semi-classical limit. One can see that quantum theory for AdS 3 gravity might be a sequence of extremal CFTs. 10
Rotating case For small k, one finds smooth transition between AdS 3 and BTZ in many parameter region, as non-rotating case. However, a t some points, singular behavior appears. For example : k =1 ， Ω 𝐹 ℓ = 0.2375 27 ・・・ entropy Free energy Angular momentum This singular transition appears at T HP The transition appears even for small k, Zeroes of Z k that is, not in the thermodynamic limit. 11
Ω 𝐹 = 0.2375 ・・ 𝐾 → −∞ Zeroes of Z 1 𝐾 → ∞ Ω 𝐹 = −0.2375 ・・ 𝜐 -plane At the zeroes of the partition function, the free energy diverges. 𝑎 1 =0 𝑎 1 =0 The zeroes (shown as in the right figure) are on the unit circle | 𝜐| = 1 , which corresponds to the Hawking-Page critical temperature. ｰ 1 ｰ 1/2 1/2 0 1 Along the pink line, J function takes real value. 12
Spin up • At Ω 𝐹 ℓ ≃ 0.790295 （ for the case of k =1 ） J entropy Free energy AdS 3 phase BTZ New phase These two critical temperatures are different from T HP 13
Zeroes of Z 1 again （ k =1 case ） 𝜐 平面 S and T transformations 𝜐 → − 1 𝜐 → 𝜐 + 1, Ω 𝐹 ℓ = −0. 79 𝜐 Ω 𝐹 ℓ = 0. 79 and their combinations move zeroes Z 1 = 0 to other points which are not on the circle | 𝜐 |=1. ⇒ The transition occurs at 𝑈 ≠ 𝑈 𝐼𝑄 ｰ 1 ｰ 1/2 0 1/2 1 The appearance of new phase might be a prediction from 3-dim. Quantum gravity (extremal CFT). 14
Semi-classical limit ( 𝑙 → ∞ ) • For large k, the # of zeroes of 𝑎 𝑙 increases and the zeroes 𝑎 1 =0 condense into the red line. ⇒ phase boundary The condensation of zeroes has been proved by Maloney and Witten (2007). ｰ 1 ｰ 1/2 1/2 0 1 Inconsistent with the semi-classical result! 15
Phase diagram Inverse temp. 𝛾 = 𝑈 −1 𝛾Ω 𝐹 extremal CFT Semi-classical T = T HP AdS 3 AdS 3 Ω 𝐹 ℓ = −1 Ω 𝐹 ℓ = 1 BTZ BTZ Unknown? Conical? Complex? Maloney- Witten(‘07) Phase diagrams are different ！ 16
Discussion (summary) • There is an inconsistency in the semi-classical limt ( 𝑙 → ∞ ). Possibility 1 ： unknown classical solution that corresponds to the new phase? Possibility 2 ： Witten conjecture might get some correction at least for large k : t he quantum theory for pure AdS 3 gravity might not be the sequence of extremal CFTs. • conformal bootstrap ： for 𝑙 ≥ 20 , non exisistence of extremal CFTs! Bae, Lee, Lee 2016 • The new phase at k =1 (the FLM model does exist) might be a new prediction from quantum gravity. In order to obtain semi-classical phase diagram, it seems that modular invariance has to be broken in large k limit. Is it correct? How? （ It also seems to be consistent with Honda-Iizuka-Tanaka-Terashima 2015 ） 17
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