Black holes with AdS asymptotics and holographic RG flows

Black holes with AdS asymptotics and holographic RG flows Anastasia Golubtsova 1 based on work with Irina Aref eva (MI RAS, Moscow) and Giuseppe Policastro (ENS, Paris) arxiv:1803.06764 (1) BLTP JINR, Dubna Supersymmetry in Integrable Systems (SIS 18) August 13-16 1 / 37

Outline 1 Introduction Holographic dictionary Exact holographic RG flows Set up How to integrate Vacuum solutions Non-vacuum solutions, black holes 3 RG equations at T = 0 4 RG-flow at finite T 5 Outlook 3 / 37

DW/CFT dualities Itzhaki et. al. 98, Boonstra et. al. 98;Skenderis 99 AdS DW, CFT QFT, AdS isometry group Poincaré isometry group of DW a restoration of the conformal symmetry only at UV and/or IR fixed points S = Mp d 1 d d x dr [ g R 1 ] ( φ) V (φ) + S Y H. The domain wall solution ds = e A(r) η ij dx i dx j + dr, φ = φ(r) The scale factor e A measures the field theory energy scale The scalar field e φ the running coupling λ The β-function β = dλ d log E = dφ da 4 / 37

Possibilities for the potential Improved holographic QCD Gursoy,Kiritsis 07, Gubser 08 For asymptotically AdS UV λ 0 V (λ) = V 0 + v 1 λ + v λ +... For confinement in the IR λ V (λ) λ Q (log λ) P The auxiliary scalar function W (φ) (aka superpotential) W (φ(u)) = (d 1) da dr, d 4(d 1) W + 1 ( φw ) = V. V (φ) from IHQCD model, Kiritsis et al 07 11 14 17 18 :) allows to reproduce the behaviour of β-function :( no exact solutions for the model Toy model V (φ) = e αφ :) has good behaviour in the IR-limit (can study conformal anomalies, apply to deconfined phase of QCD )Policastro 15 :( UV -fixed point is not the AdS 5 / 37

RG flow at finite temperature Thermal gas solution ds = e A(r) η ij dx i dx j + dr, φ = φ(r). The black hole ds = e A(r) ( f(r)dt + δ ij dx i dx j) + dr f(r), Gubser s bound for singular solutions (000) V (φ h ) < 0, V (φ h ) V (φ UV ). f(r) = 1 C λ 4(1 X ) 3X. 6 / 37

Set up The action reads S = 1 κ d 4 x du ( g R 4 ) 3 ( φ) + V (φ) 1κ d 4 x γ, V (φ) = C 1 e k1φ + C e kφ, C i, k i, i = 1, are some constants. V 4-3 - -1 1 ϕ - -4 Figure: The behaviour of the potential V (φ) for C 1 < 0, C > 0. 8 / 37

The ansatz for the metric ds = e A(u) dt + e B(u) The gauge C = A + 3B. The sigma-model 3 dyi + e C(u) du, i=1 L = 1 G MN ẋ M ẋ N V, V = 1 x 1 = A, x = B, x 3 = φ, x = C. (G MN ) = 0 3 0 3 6 0 4 0 0 3 s=1 C s e (x1 +3x +k sx 3), d du., M, N = 1,, 3. 9 / 37

(G MN ) minisuperspace metric on the target space M L = 1 ẋ, ẋ + C 1 e V,x + C e W,x. V time-like, W - spacelike vectors on M (the basis is (e 1, e, e 3 )) ( V, V = 3 k1 16 ) (, W, W = 3 k 16 ), V, W = 3 9 9 ( k 1 k 16 9 LET V, W = 0 k 1 k = 16 9, k 1 = k, k = 16 9k, 0 < k < 4/3. The new basis e 1 = V V, e = W W, e i, e j = η ij, (η ij ) = diag ( 1, 1, 1). X i = η ii e i, x, x i = 3 SjX i j, e j = j=1 3 Sje i i. S i j components of general Lorentz transformations. 10 / 37 i=1 ).

Root systems 11 / 37

The A 1 A 1 -mechanical model Let V and W vectors to root vectors of su() su() Lie algebra L = 1 E 0 = 1 3 i,j=1 3 i,j=1 η ij Ẋ i Ẋ j + C 1 eη11 V,V η ij Ẋ i Ẋ j C 1 eη11 V,V 1/ X 1 + C 1/ X 1 C Liouville equations for sl()-toda chains (sl() = su()) 1/ X eη W,W, 1/ X eη W,W. Ẍ s = R s, R s C s e ηss Rs,Rs 1/ X s, s = 1,, Ẍ 3 = 0, with R 1, R 1 = V, V, R, R = W, W. Gavrilov, Ivashchuk, Melnikov 9407019 Lü, Pope,960707, 9604058 Lü, Yang, 1307.305 1 / 37

The solution to the A 1 A 1 - mechanical model The solution reads with F s(u u 0s) = X 1 = V, V 1/ ln (F 1 (u u 01 )), X = W, W 1/ ln (F (u u 0 )), X 3 = p 3 u + q 3, [ E s sinh Cs Cs E s sin Cs E s cosh ] (u u 0s), η ssc s > 0, η sse s > 0, ], η ssc s > 0, η sse s < 0, E s R s,r s [ E s R s,r s (u u 0s) R s,r s C s (u u 0s), η ssc s > 0, E s = 0, [ ] E s R s,r s (u u 0s), η ssc s < 0, η sse s > 0, u 0s, E s, E s, p 3, q 3 are constants of integration. Lorenz transformations S1 i V i = V, V, W i 1/ Si = W, W, 1/ αi = S3p i 3, β i = S3q i 3 13 / 37

The general solution ds = F 8 9k 9k 16 (16 9k 1 F ) φ = 9k 9k ln 9k F1 + ln 16 9k F 16 ( e α1u dt + e 3 α1u d y ) + F 3 9k 18k 16 16 9k 1 F du C s sinh [µs(u u0s)], if ηsscs > 0, ηsses > 0, E s C s sin [µs(u u0s)], if ηsscs > 0, ηsses < 0, E s F s(u u 0s) = Cs µs(u u0s), if ηsscs > 0, Es = 0, C s cosh [µs(u u0s)], if ηsscs < 0, ηsses > 0, E s s = 1,, µ 1 = 3E1 (k 16 ( ( ) ) ), µ = 3E 16 1 9 9 k 16 9. 14 / 37

Constraints E 1 + E + (α1 ) 3 = 0. 1 α 1 = 0 Vacuum solutions, Poincaré invariant, E 1 = E α 1 0 Non-vacuum ones, no Poincaré invariance E 1 E Conditions from the V (φ): C 1 < 0, C > 0, 0 < k < 4/3. Constants of integration u 0 u 01 The degenerate case with u 01 = u 0 = u 0, left: u u 0. 15 / 37

Behaviour of solutions u 01 u 0, α 1 = 0 ds = F F 1 = 8 9k 9k 16 (16 9k 1 F ) ( dt + dy 1 + dy + dy 3) + F 3 9k 18k 16 16 9k 1 F du, C 1 E 1 sinh(µ C 1 u u 01 ), F = E sinh(µ u u 0 ), E 1 = E, E 1 < 0, E > 0, µ = 4 3k µ 1. The dilaton and its potential φ = 9k 9k 16 log F F 1 ( ) 18k ( ) V = C 1 e kφ + C e 3φ/(9k) F 9k 3 16 F 9k 16 = C 1 + C. F 1 F 1 16 / 37

Boundaries for u 01 u 0 0 The left solution u < u 0 (conformally flat) u ds z ( /3 dt + dy1 + dy + dy3 + dz ), z e 3µ 1 u 4+3k, φ 9k 16 9k (µ µ 1 )u log z u u 0 ɛ ds z 18k ( 64 9k dt + dy1 + dy + dy3 + dz ), z 64 9k 4(16 9k ) (u u 0) 64 9k 4(16 9k ), φ 36k 64 9k log z +. The middle solution u 0 u 01 (conformally flat ) u u 01 + ɛ the same as at u u 01 + ɛ u + ds z /3 ( dt + dy 1 + dy + dy 3 + dz ), φ log z 17 / 37

Boundaries: u 01 = u 0 = u 0, α 1 = 0 In the UV u u 0 we obtain the AdS-spacetime ds 1 z ( dt + dy 1 + dy + dy 3 + dz ), z 4u 1/4. The dilaton is constant in the UV 9k φ = 16 9k log 3k 4 + 9k (16 9k ) log C 1 C In the IR u + we obtain the conformally flat spacetime. ds z /3 ( dt + dy 1 + dy + dy 3 + dz ), z e 3µ 1 u 4+3k. The dilaton in the IR φ log z 18 / 37

ϕ 5 0 0 ϕ ϕ 0 4 6 8 10 1 14 u 15 10-5 10 5-8 -6-4 - u -5-10 A -10-0 0. 0.4 0.6 0.8 1.0 1.u B -10-15 -0-5 C Figure: Dilaton as a function of u: A) u u 01, u 01 = 1. For all u 01 = 1, u 0 = 0,E 1 = E = 1, C 1 = C = 1, k = 0.4, 1, 1.. ϕ -6-4 - 4 6 u -0.5-1.0-1.5 -.0 Figure: The behaviour of the dilaton (solid lines) and its asymptotics at infinity (dashed lines) for u 01 = u 0 = 0, C 1 = C = 1, E 1 = E = 1 and different values of k. From bottom to top k = 0.4, 1, 1.. 19 / 37

Non-vacuum solutions, black holes The metric ds = F 8 9k 9k 16 (16 9k 1 F ) The dilaton reads ( e α1u dt +e 3 α1 u 3 dyi )+F i=1 φ = 9k 9k 16 ln F 9k 1 + 9k 16 ln F, 3 9k 18k 16 16 9k 1 F C F 1 = 1 C sinh(µ1 E 1 u u 01 ), F = sinh(µ E u u 0 ), µ 1 = 3E1 16 9 k, µ = 3E 4 3k du. 16 9 k = 4 E µ 1. 3k E 1 E 1 + E + 3 (α1 ) = 0. 0 / 37

Dilaton at boundaries u 01 u 0, α 1 0 The left solution u < u 0 u φ u u u 0 ɛ φ u u0 ɛ 9k log 16 9k [ 9k (µ 16 9k µ 1) u + 1 log C E 1 [ C E 1 C 1 E µ ɛ sinh(µ 1 (u 01 u 0 )) C 1 E ] ] +. The middle solution u 0 u 01 u u 01 + ɛ the same as for the middle solution at u ] u 01 + ɛ u + φ u 9k (µ µ 1) u + 1 log C E 1. 16 9k [ µ 1 = µ, E = 6k (α 1 ) 16 9k. µ 1 > µ C 1 E 1 / 37

Dilaton at boundaries u 01 = u 0, α 1 0 φ u ± φ u u0 9k [ 9k ± (µ µ 1 ) u + 1 16 log C E 1 C 1 E ] 9k [ 16 9k log ( µ µ 1 ) + 1 log C 1E C E 1 µ 1 = µ, E = 6k (α 1 ) 16 9k. µ 1 > µ. ] / 37

Black hole, solutions with u [u 01, + ) ds = C X (u)e κu 3 α1 u ( e 8 3 α1u dt + d y + X (u) 3 C 3 e 3κ+ 3 α1u du ) X (u) = (1 e µ1(u u01) ) 8 16 9k (1 e µ(u u0) ) 8 ( κ E + 6 (16 9k ) 3 (α1 ) + 3 4 k E ), ( 1 C 1 ) 8 ( 9k C 16 1 C ) E 1 e µ1u01 E e µu0 9k (16 9k ) 9k (16 9k ) The absence of conic singularity κ 3 α1 = 0, E = 6k (α 1 ) 16 9k, µ = µ 1 4 α 1 β = π 3C 3/ Null geodesics ds = 0, for the light moving in the radial direction t t 0 = u u 0 dūc 3/ ( ) 1 +.... u. 3 / 37

Black hole solution ( ds = C X e 8 3 α1u dt + d y ) + C 4 X (u) 4 e 8 3 α1u du, 9k (16 9k ), X = (1 e µ(u u01) ) 8 16 9k (1 e µ(u u0) ) ( 1 C 1 ) 8 ( 9k C 16 1 C ) E 1 e µu01 E e µu0 [ ] 9k φ = 9k 16 log E 1 C sinh(µ(u u 0 )) E C 1. sinh(µ(u u 01 )) and near horizon ( ) 9k lim φ u + = (16 9k ) log E C 1 E 1 C. The Hawking temperature 9k (16 9k ) T = 3π α 1 C 3/ 4 / 37.

Special case: u 01 = u 0 = u 0 ( ) ds = C 1 e µ(u u 0) 1 ( e µu dt + d y ) ( ) +C 4 1 e µ(u u 0) e µu du, µ = 4 3 α1, C = ( ( ) 4 ) 1/ C1 e µu0 E 1 The dilaton The curvature φ = 9k 16 ( C 9k (16 9k ) log C 1 E C E 1. R = 5µ C 4. E ) 9k 4(16 9k ). ( z = z ) h 1 e µu 1 4, C = z h, ds = 1 f(z)dt + d y + dz f(z) ( ) 4. z f = 1 z h The saturation of the Gubser s bound V (φ(u h )) = V UV. z ( ), 5 / 37

Holographic RG equations The solution in the domain wall coordinates ds = dw + e A(w) ( dt + η ij dx i dx j). φ(w), λ = e φ the running coupling. The β-function β(λ) = dλ QF T d log E = dλ da The β-function satisfies the holographic RG eqs. Kiritsis et al. 081.079 dx dφ = 4 ( 1 X ) ( 1 + 3 ) d log V, 3 8X dφ where X(φ) is related with the β-function The energy scale X(φ) = β(λ) 3λ A = e A 7 / 37

RG equations at T = 0 The domain wall coordinates dw = F The running coupling λ = e φ = 16 9k 16 1 F ( F F 1 9k 16 9k ) 9k 9k 16. du. The energy scale A = e A = F 4 9k 9k 16 1 F 4(16 9k ). The X-function X = 1 3 ( F F 1 ) 9k 16 9k λ A. 8 / 37

X X 50 E=0. E=0.5 E=1 E= f s 0.5 1.0 1.5.0ϕ -4-4 ϕ E=0.1 E=0.8-1 E=4 E=10 - -50-100 E=0.01 X A -3 X B E=0.15 E=0.35 1.0 1.0 E=0.6 0.8 0.5 0.6 0.4 -.5 -.0-1.5-1.0-0.5 ϕ -0.5 C 0. -3.5-3.0 -.5 -.0-1.5-1.0-0.5 ϕ -0. D Figure: The behaviour of the X-function with the dependence on the dilaton plotted using the solutions for A. A)left B) middle C)right D) u 01 = u 0 9 / 37

X X 1 0 P X c ϕ s ϕ 1 0 P ϕ -1-1 - - -3 X c1-3 -4-4 -5 - -1 0 1 X A -5 1.0 - -1 0 1 X B 1 0 ϕ 0.5-1 0.0 P ϕ - -3-0.5-4 - -1 0 1 C -1.0-0.8-0.6-0.4-0. 0.0 D Figure: All solutions X with potential fixed as C 1 = C = and k = 0.4 30 / 37

The behaviour of the running coupling λ on the energy scale λ.5 λ 5 λ 1.0 E=0.001, u 0 =0,u 01 =0.001.0 4 E=0.1, u 0 =0,u 01 =1 0.8 E=0.01, u 0 =0,u 01 =1 E=0.15, u 0 =0,u 01 =1 E=0.8, u 0 =0,u 01 =1 E=0.35, u 0 =0,u 01 =1 E=4, u 0 =0,u 01 =1 E=0.6, u 0 =0,u 01 =1 1.5 3 E=10, u 0 =0,u 01 =1 0.6 1.0 0.4 0.5 E=0.1, u 0 =0,u 01 =0.1 E=0.5, u 0 =0,u 01 =0.5 E=3, u 0 =0,u 01 =1 E=5, u 0 =0,u 01 =1 0.0 A 0.0 0.5 1.0 1.5.0.5 A 1 0 0 1 3 4 5 A B 0. 0.0 A 0.0 0.5 1.0 1.5.0.5 3.0 3.5 Figure: λ on the energy A on the dilaton plotted using the solutions for A and φ.a) the left branch with u 0 > u, B) the middle branch u 0 u 01. For all plots k = 0.4, C 1 =, C =, different curves on the same plot corresponds to the different values of E 1 = E, labeled as E on the legends and different u 01 and u 01 also indicated on the legends. C 31 / 37

RG flow at finite temperature The black brane ds = e A(w) ( f(w)dt + δ ij dx i dx j) + dw f(w), Ex. The Chamblin-Reall solution f = 1 C λ 4(1 X ) 3X, λ = e φ. The Y -variable is defined through the function f g Y (φ) = 1, g = log f, 4 A dx = 4 ( 1 X + Y ) ( 1 + 3 dφ 3 8X dy = 4 ( 1 X + Y ) Y dφ 3 X. d log V dφ ), 33 / 37

The running coupling on the energy scale (u 01 0, u 0 0) λ 50 λ.0 λ.0 40 1.5 1.5 30 1.0 1.0 0 10 0.5 0.5 A 0 0.0 0.1 0. 0.3 0.4A 0 B 0 1 3 4 5 A 0.0 1 3 4 5 A 0.0 Figure: The dependence of λ on the energy scale A= e A at the left solution A), the middle solution B) and the right one C). α 1 = 0 (cyan), α 1 = 0.5 (blue), α 1 = 0.5 (green), α 1 = 0.8 (red), α 1 = 1 (brown). C 34 / 37

The bottom line Done Vacuum and non-vacuum holographic RG-flows were constructed Holographic running coupling mimic QCD Holographic RG flows can have AdS fixed points. A new solution with horizon was found Studies of the running coupling λ on the E scale not to deal with superpotential W? Careful studies of the behaviour λ = e φ on the energy scale at T 0 Analysis of confinement-deconfinement phase transition (Polchinski-Strassler model?). Holographic c-theorem? Full supergravity picture? 36 / 37

Thank you for attention! 37 / 37