Anomaly-induced Thermodynamics in Higher Dimensional AdS/CFT. Gim Seng Ng McGill University

Transcription

1 Anomaly-induced hermodynamics in Higher Dimensional AdS/C Gim Seng Ng McGill University CRM Montreal, 21 Oct 2015

2 Based on : , and (with. Azeyanagi, R. Loganayagam and M. J. Rodriguez)

3 Motivations BHs have entropy! Entropy-matching Often extremal or/and susy How about non-extremal/ non-susy finite temperature entropy matching? In 2d C, BZ entropy is reproduced by the universal Cardy s formula Higher-dimensional generalizations of Cardy s formula? Higher-dimensional AdS/C Cardy entropy-matching?

4 Outline Chiral Half of Cardy s ormula in 2n Replacement Rule from 2n+1 2n

5 chiral half of Cardy Consider a 2d C on a circle of radius R Let us put it at finite temperature and rotation/boost At high temperature: 1/R S Cardy 2πR c R + c L 1 R 2 Ω 2 (4π) + 2πR2 Ω c R c L 24 1 R 2 Ω 2 (4π) 24 ime R Weyl Anomaly Gravitational Anomaly (``Chiral Half /anomalous-part)

6 chiral HAL of Cardy S C2,anom 2πR 2 Ω c R c L 1 R 2 Ω 2 24 (4π) Generalizations to higher-d C (on sphere) Replacement rule [Surowka, Loganayagam, Jensen, Yarom, ] o understand the replacement rule, we need to review the following two things: =0 anomalies Anomalous hydrodynamics

7 ANOMALY INLOW Anomalies are captured by Chern-Simons terms Append an extra auxiliary direction. he (2n+1) theory is anomaly-free, but with Chern-Simons terms I CS Non-conservation of the (2n)-theory is captured by the ``inflow of charges into the extra auxiliary direction: J Q2n j j P anom P anom = di CS j Q 2n

8 anomalous hydro Hydro: effective long-wavelengths descriptions: { u α,,,...} A α, g αβ,... Hydro derivative expansion: J α = qu α +...+(J α ) anom +... αβ = Eu α u β + p g αβ + u α u β +...+( αβ ) anom +... Leading anomalous contribution: parity-odd vorticity : V α αβ... u β ( u) n 1...

9 anomalous hydro J α = qu α +...+(J α ) anom +... αβ = Eu α u β + p g αβ + u α u β +...+( αβ ) anom +... (J α ) anom = [, ] ( αβ ) anom = V α +... u α V β + V α u β +... is like an anomalous Gibbs free energy Recall that standard relations: Q = G, E = G G G, S = G

10 anomalous hydro (J α ) anom = [, ] ( αβ ) anom = V α +... u α V β + V α u β +... Anomaly-induced entropy: S anom = CV dx = i (2πR 2 Ω i ) Question: What is?

11 chiral HAL of Cardy Revisit 2d C Cardy s formula S C2,anom 2πR 2 Ω c R c L 24 (4π) S anom = (2πR2 Ω) = c R c L 2(2π)24 2(2π)2 = c g [R 2 ] trr 2 2(2π) 2 anomaly polynomial for 2d grav. anomaly

12 Replacement rule OR chiral half of cardy s formula [Surowka, Loganayagam, Jensen, Yarom, ] V α ε αβ... [, ] Vα +... (Jα )anom = (αβ )anom = Sanom = uβ ( u) n 1 uα V β + Vα u β +... b +2 C V dx = = Panom [, R],i`[R2k ] (2πR2 Ωi ) i 2(2π)2k

13 REPLACEMEN RULE 2d: Panom = ca + cg i`[r ], = ca + cg 2(2π) P = c + c i`[r ], A M 4d: anom 6d: Panom cg i`[r4 ], (Jα )anom = (αβ )anom = Sanom = = ca 3 + cm = cg 2(2π)4 [, ] Vα +... b +2 C V dx uα V β + Vα u β +... = (2πR2 Ωi ) i 2 2(2π)2

14 Outline Chiral Half of Cardy s ormula in 2n Replacement Rule from 2n+1 2n (Q: Bulk dual of replacement rule?)

15 A holographer s recipe 1. ind your favorite AdS bulk theories Issue 1: Need gauge-gravitational Chern-Simons terms 2. ind the relevant AdS BH solutions Issue 2: Need AdS-Kerr-Newman+CS solutions (with all rotations/charges turned on) 3. Calculate responses, charges, thermodynamics Issue 3: Holographic renormalization/charges / entropy for Chern-Simons terms 4. Match/predict C results

16 step 1: favorite Ads gravity setup oy model: D=2n+1 Einstein-Maxwell+negative c.c. +CS terms Ics Equations of motion: R ab 1 2 (R 2Λ) g ab = 8πG N ( M ) ab +( H ) ab b ab = g 2 YM (J H ) a Maxwell contribution: ( M ) ab CS/``Hall contributions: J H = P anom ( H ) ab = c Σ (ab)c Σ (ab)c = 2 P anom R a b P anom = di CS

17 step 2: fluid/gravity BH solution (no anomalies) Gravity dual of charged rotating fluid (for now, without anomalies) ds 2 = 2u dx dr + r 2 f(r, m, q)u u ν + P ν dx dx ν + A = Φ(r, q)u dx + Non-trivial bulk radial dependence: P ν = g ν + u u ν Φ(r, q) = Horizon values: q r 2n 2 f(r, m, q) =1 m r 2n κ q Φ(r H )=, Φ (r H )=2π Φ (r) q 2 r2(2n 1) r 2 2 df dr

18 step 2: fluid/gravity BH solution (with anomalies) Gravity dual of anomalous charged rotating fluid ds2 = f(r, m, q)u uν + Pν dx dxν + km/ Q`/2` 2u dx dr + r2 +gv (r, m, q)(u Vν + uν V )dx dxν +... A = Φ(r, q)u dx + km/ Q`/2` Rab +av (r, m, q)v dx (R 2 2Λ) gab = 8πGN (M )ab + (H )ab ab 2 b = gym (JH ) Leading contributions from the CS-terms : gv, av Bulk replacement rule: JH G r Panom G Φ H 2 r Φ, i`[r2k ] G Φ 2Φ2k a

19 step 3&4: currents and replacement rule Bulk replacement rule metric corrections boundary stress-tensor/currents αβ anom = αβ (J α ) anom g α ( r ) anom = anom = G Φ G Φ Φ G Φ r=r H u α V β + u β V α G(r = r H )=[, [R 2k ] 2(2π) 2k ] G Φ r=rh V α Bulk replacement rule-> boundary replacement rule!!

20 step 3&4: entropy achikawa ormula (see also Solodukhin and Bonora-Cvitan-Prester-Pallua-Smolic) (note: NO Wald s formula applied to Ics) S CS = (8πk)Γ N R 2k 2 N k=1 P anom [R 2k ] P anom = di cs, Γ N, R N = Applying this to the fluid/gravity metric we found gives S anom = CV dx = i (2πR 2 Ω i ) Computations opaque, long and tedious (what bulk structures imply the replacement for entropy?)

21 summary, current and future work Higher-dimensional chiral-half of Cardy s formula (Replacement rule) Construct (anomalous) fluid/gravity solutions in the bulk for the Einstein-Maxwell-CS theory ound ``bulk replacement rule which implies the boundary replacement rule Current/uture: Anomaly-induced entanglement entropy ime-dependence (non-stationary) and/or higher-order? More realistic theories add matter and etc (shouldn t change the conclusions) [Azeyanagi-Loganayagam-Ng, ]