Holographic Wilsonian RG and BH Sliding Membrane

Transcription

1 Holographic Wilsonian RG and BH Sliding Membrane based on by SJS,YZ (JHEP05:030,2011) related references: by N.Iqbal and H.Liu by Heemskerk and J.Polchinski by Faulkner,Liu and Rangamani July 7, 2011 ICTS-USTC,Hefei ang Zhou based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

2 Outline 1 Introduction: Membrane paradigm 2 Holographic Wilsonian RG (HWRG) 3 Equivalence 4 Double Trace Flow based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

3 Motivations Introduction: Membrane paradigm Map the Hamilton-Jacobi flow and Exact Wilson RG of Quantum Field Theory based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

4 Introduction: Membrane paradigm Motivations Map the Hamilton-Jacobi flow and Exact Wilson RG of Quantum Field Theory Interpolating physical quantities evaluating at horizon and at the boundary based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

5 Formulations Introduction: Membrane paradigm Many approaches to Holographic RG after 1997 (AdS=CFT, Maldacena) which I apologize for not listing here. (E.T.Akhmedov,V.Balasubramanian eta, Susskind and Witten,J.d.Boer and Verlinde 2 ) based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

6 Introduction: Membrane paradigm Formulations Many approaches to Holographic RG after 1997 (AdS=CFT, Maldacena) which I apologize for not listing here. (E.T.Akhmedov,V.Balasubramanian eta, Susskind and Witten,J.d.Boer and Verlinde 2 ) Sliding membrane approach (2008, N.Iqbal and H.Liu) based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

7 Introduction: Membrane paradigm Formulations Many approaches to Holographic RG after 1997 (AdS=CFT, Maldacena) which I apologize for not listing here. (E.T.Akhmedov,V.Balasubramanian eta, Susskind and Witten,J.d.Boer and Verlinde 2 ) Sliding membrane approach (2008, N.Iqbal and H.Liu) Wilsonian fluid/gravity (June.2010, Bredberg,Keeler,Lysov and Strominger.) based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

8 Introduction: Membrane paradigm Formulations Many approaches to Holographic RG after 1997 (AdS=CFT, Maldacena) which I apologize for not listing here. (E.T.Akhmedov,V.Balasubramanian eta, Susskind and Witten,J.d.Boer and Verlinde 2 ) Sliding membrane approach (2008, N.Iqbal and H.Liu) Wilsonian fluid/gravity (June.2010, Bredberg,Keeler,Lysov and Strominger.) Holographic Wilsonian approach (Oct.2010,I.heemskerk and J.Polchinski; Oct.2010, T.Faulkner,M.Rangamani and H.Liu) based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

9 Membrane paradigm Introduction: Membrane paradigm Black hole stretched horizon a fictitious membrane which means, to an external observer, a black hole appears to behave exactly like a dynamical fluid membrane. based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

10 Introduction: Membrane paradigm Membrane paradigm Black hole stretched horizon a fictitious membrane which means, to an external observer, a black hole appears to behave exactly like a dynamical fluid membrane. Viscous response: η = 1 16πG N (Damour,1978) (1) based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

11 Introduction: Membrane paradigm Membrane paradigm Black hole stretched horizon a fictitious membrane which means, to an external observer, a black hole appears to behave exactly like a dynamical fluid membrane. Viscous response: Electric response: η = 1 16πG N (Damour,1978) (1) σ = 1 e 2 (Znajek,1978; Damour,1978) (2) based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

12 Introduction: Membrane paradigm The idea is: In-falling boundary conditions can be viewed as physical properties of membrane located at the stretched horizon. based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

13 Introduction: Membrane paradigm The idea is: In-falling boundary conditions can be viewed as physical properties of membrane located at the stretched horizon. An action for membrane (Parikh and Wilczek 1998) is a surface term which cancel the boundary term action at the horizon. based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

14 Introduction: Membrane paradigm The idea is: In-falling boundary conditions can be viewed as physical properties of membrane located at the stretched horizon. An action for membrane (Parikh and Wilczek 1998) is a surface term which cancel the boundary term action at the horizon. For FFO to find a nonsingular field φ, near the horizon we have φ(r, t, x) = φ(τ, x) (3) where τ is the Eddington-Finklestein coordinate grr dτ = dt + dr. (4) g tt This implies in-falling condition r φ = grr g tt t φ at the horizon. based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

15 Introduction: Membrane paradigm For a massless scalar probe with action S scalar = 1 d d+1 x g 1 2 r>r h q(r) ( φ)2 (5) the membrane action is S surf = d d x γ( Π(r h, x) )φ(r h, x), Π mb Π(r h, x) g = rr r φ Σ γ γ q(r) (6) together with in-falling condition r φ = grr g tt t φ, give (orthonormal basis) ξ mb Π mb ˆt φ(r h) = 1 q(r h ) = 1 (ShearViscosity!) (7) 16πG N based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

16 Introduction: Membrane paradigm For a Maxwell field probe: S Maxwell = d d+1 x 1 g r>r h 4g(r) 2 F 2 (8) the membrane action is S surf = d d x γ( j µ )A µ, J µ γ mb j µ (9) γ Σ with in-falling condition F ri = grr g tt F ti, give σ mb = 1 g 2 (r h ) (MembraneConductivity!) (10) based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

17 Sliding membrane Introduction: Membrane paradigm The idea is: View a cutoff surface at finite r c as a sliding membrane where the disturb and response can be defined. based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

18 Introduction: Membrane paradigm Sliding membrane The idea is: View a cutoff surface at finite r c as a sliding membrane where the disturb and response can be defined. For a massless scalar probe: ξ(r c, k µ ) = Π(r c, k µ ) iωφ(r c, k µ ) (11) based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

19 Introduction: Membrane paradigm Sliding membrane The idea is: View a cutoff surface at finite r c as a sliding membrane where the disturb and response can be defined. For a massless scalar probe: ξ(r c, k µ ) = Π(r c, k µ ) iωφ(r c, k µ ) (11) For a Maxwell field probe: σ(r c, k µ ) = j i (r c, k µ ) F it (r c, k µ ) (12) based on by SJS,YZ (JHEP05:030,2011) Holographic Wilsonian related RG and references: BH Sliding Membrane July 7, by 2011 N.Iqbal ICTS-USTC,Hefei and H.Liu by/ Heem 28

20 Flow involving EOM Introduction: Membrane paradigm Using the definition, one can represent EOM: [ grr σ 2 r σ = iω (1 k2 g ii ) ] g tt Σ(r) ω 2 g tt Σ(r) (13) where Σ(r) = gg ii grr g tt (14) July 7, 2011 ICTS-USTC,Hefei 10 /

21 Flow involving EOM Introduction: Membrane paradigm Using the definition, one can represent EOM: [ grr σ 2 r σ = iω (1 k2 g ii ) ] g tt Σ(r) ω 2 g tt Σ(r) (13) where Σ(r) = gg ii grr g tt (14) Regular condition of (13) at the horizon infalling boundary condition. July 7, 2011 ICTS-USTC,Hefei 10 /

22 Flow involving EOM Introduction: Membrane paradigm Using the definition, one can represent EOM: [ grr σ 2 r σ = iω (1 k2 g ii ) ] g tt Σ(r) ω 2 g tt Σ(r) (13) where Σ(r) = gg ii grr g tt (14) Regular condition of (13) at the horizon infalling boundary condition. Smooth interpolation between Horizon value and Boundary value July 7, 2011 ICTS-USTC,Hefei 10 /

23 Flow involving EOM Introduction: Membrane paradigm Using the definition, one can represent EOM: [ grr σ 2 r σ = iω (1 k2 g ii ) ] g tt Σ(r) ω 2 g tt Σ(r) (13) where Σ(r) = gg ii grr g tt (14) Regular condition of (13) at the horizon infalling boundary condition. Smooth interpolation between Horizon value and Boundary value Even without any RG formulism, one does have flow equation for σ(r c )! July 7, 2011 ICTS-USTC,Hefei 10 /

24 Holographic Wilsonian RG (HWRG) Split path integral in the bulk Boundary Split: Z = DM kδ<1 DM kδ>1 e S = boundary boundary DM kδ<1 e S(δ) := exp( s(δ)) (15) ang Zhou July 7, 2011 ICTS-USTC,Hefei 11 /

25 Holographic Wilsonian RG (HWRG) Split path integral in the bulk Boundary Split: Z = DM kδ<1 DM kδ>1 e S = boundary boundary DM kδ<1 e S(δ) := exp( s(δ)) (15) Bulk Split: Z = Dφ z>ɛ Dφ z=ɛ Dφ z<ɛ e S IR(z>ɛ) S UV (z<ɛ) bulk = D φψ IR (ɛ, φ)ψ UV (ɛ, φ) (16) bulk July 7, 2011 ICTS-USTC,Hefei 11 /

26 Holographic Wilsonian RG (HWRG) Map between Boundary and Bulk Natural Map: Ψ IR (ɛ, φ) = DM kδ<1 exp { S 0 + 1κ 2 } d d x φ i (x)o i (x) (17) July 7, 2011 ICTS-USTC,Hefei 12 /

27 Holographic Wilsonian RG (HWRG) Map between Boundary and Bulk Natural Map: Ψ IR (ɛ, φ) = DM kδ<1 exp { S 0 + 1κ 2 } d d x φ i (x)o i (x) (17) s(δ) and S B : e s(δ) = D φe d d x φ(x)o(x) e S B (18) with S B = log Ψ UV (ɛ, φ z=ɛ ) (19) July 7, 2011 ICTS-USTC,Hefei 12 /

28 Holographic Wilsonian RG (HWRG) Holographic Wilsonian RGE Cutoff independent of full amplitude: 0 = d dɛ Z = d dɛ < e s(δ) > (20) July 7, 2011 ICTS-USTC,Hefei 13 /

29 Holographic Wilsonian RG (HWRG) Holographic Wilsonian RGE Cutoff independent of full amplitude: 0 = d dɛ Z = d dɛ < e s(δ) > (20) In the bulk level, IR dynamics plus a boundary action S B should be useful, since Wilson told us we need to integrate out the UV region. July 7, 2011 ICTS-USTC,Hefei 13 /

30 Holographic Wilsonian RG (HWRG) Holographic Wilsonian RGE Cutoff independent of full amplitude: 0 = d dɛ Z = d dɛ < e s(δ) > (20) In the bulk level, IR dynamics plus a boundary action S B should be useful, since Wilson told us we need to integrate out the UV region. Bulk HWRG equation then becomes: which gives. [ zh ] ɛ L + S B = 0 (21) ɛ ɛ S B = H IR (22) July 7, 2011 ICTS-USTC,Hefei 13 /

31 Holographic Wilsonian RG (HWRG) Holographic Wilsonian RGE ang Zhou Cutoff independent of full amplitude: 0 = d dɛ Z = d dɛ < e s(δ) > (20) In the bulk level, IR dynamics plus a boundary action S B should be useful, since Wilson told us we need to integrate out the UV region. Bulk HWRG equation then becomes: which gives. What is the power of above HWRG? [ zh ] ɛ L + S B = 0 (21) ɛ ɛ S B = H IR (22) July 7, 2011 ICTS-USTC,Hefei 13 /

32 Holographic Wilsonian RG (HWRG) Holographic Wilsionian RG flow One way to determine S B is taking use of ɛ S B [A µ, ɛ] + H IR [A µ, δs B δa µ ] = 0 (23) July 7, 2011 ICTS-USTC,Hefei 14 /

33 Holographic Wilsonian RG (HWRG) Holographic Wilsionian RG flow One way to determine S B is taking use of ɛ S B [A µ, ɛ] + H IR [A µ, δs B δa µ ] = 0 (23) For the Maxwell: S B = 1 d d k ( ) 2 (2π) d G 00 (k, ɛ)â 0 (k)â 0 ( k) + G ii (k, ɛ)â i (k)â i ( k) (24) July 7, 2011 ICTS-USTC,Hefei 14 /

34 Holographic Wilsonian RG (HWRG) Holographic Wilsionian RG flow One way to determine S B is taking use of ɛ S B [A µ, ɛ] + H IR [A µ, δs B δa µ ] = 0 (23) For the Maxwell: S B = 1 d d k ( ) 2 (2π) d G 00 (k, ɛ)â 0 (k)â 0 ( k) + G ii (k, ɛ)â i (k)â i ( k) (24) Using flow equation, we have flow equations for coefficients: ɛ G 00 = (G 00 ) 2 gg tt g zz + gg tt g ii k 2 ɛ G ii = (G ii ) 2 gg ii g zz gg tt g ii ω 2. (25) July 7, 2011 ICTS-USTC,Hefei 14 /

35 RG Solutions Holographic Wilsonian RG (HWRG) we want to find solutions for the RG equations (25). We start by assuming and 1 G 00 = 1 G 00 (0) ɛ G 00 (0) + k 2 1 G(1) 00 +, 1 G ii = 1 G(0) ii = (G 00 (0) )2 gg tt g zz, ɛg ii (0) + ω 2 1 G(1) ii + (26) (G ii (0) = )2 gg ii. (27) g zz We have the following solutions by imposing simple vanishing boundary conditions for G(1) 00, G (1) ii 1 z gg tt G(1) 00 (z) = g ii 1 z gg tt 0 G(0) 00, G(1) ii (z) = g ii 0 G(0) ii We see that k 2 and ω 2 correction are controlled by zero order solutions in the low frequency approximation.. (28) July 7, 2011 ICTS-USTC,Hefei 15 /

36 Holographic Wilsonian RG (HWRG) Flow for transport coefficient Define a transport coefficient: σ Ji E i = With the definition of current J i 0  i + i  0. (29) J µ δs B δa µ (30) July 7, 2011 ICTS-USTC,Hefei 16 /

37 Holographic Wilsonian RG (HWRG) Flow for transport coefficient Define a transport coefficient: σ Ji E i = With the definition of current From S B we have J i 0  i + i  0. (29) J µ δs B δa µ (30) Express σ by G 1 Â0 = G 00 J 0, 1 G ii = Âi J i. (31) 1 σ = i ( ) ω 2 ω G ii k2 G 00. (32) July 7, 2011 ICTS-USTC,Hefei 16 /

38 Conductivity Flow Holographic Wilsonian RG (HWRG) Using the flow equations for G coefficients we obtain ( ) ɛσ (σ iω = 2 1 k2 1 gg ii g zz ω 2 gg tt g zz ) gg tt g ii, (33) This is what we have in the sliding membrane paradigm! July 7, 2011 ICTS-USTC,Hefei 17 /

39 Conductivity Flow Holographic Wilsonian RG (HWRG) Using the flow equations for G coefficients we obtain ( ) ɛσ (σ iω = 2 1 k2 1 gg ii g zz ω 2 gg tt g zz ) gg tt g ii, (33) This is what we have in the sliding membrane paradigm! Under coordinate transformation z = 1/r, one has ɛ = r 2 r, g (z) g tt g ii = g (r) g tt g ii r 2, (34) 1 g (z) g zz g ii = 1 g (r) g rr g ii r 2. (35) July 7, 2011 ICTS-USTC,Hefei 17 /

40 Holographic Wilsonian RG (HWRG) Re Σ Im Σ Figure: r flow of AC conductivity, with d = 4 AdS-black hole background and ω = 2, 1.5, 1, from up to down in the left Figure and inversely in the right one. For d > 4 AdS-black hole, the behavior of solutions are similar. July 7, 2011 ICTS-USTC,Hefei 18 /

41 Equivalence Equivalence between HWRG and Sliding Membrane There are two RG flows. One is the flow given by the classical equation of motion, and the other is the flow from integrating out the geometry. Here we want to prove the equivalence of the two. July 7, 2011 ICTS-USTC,Hefei 19 /

42 Equivalence Equivalence between HWRG and Sliding Membrane There are two RG flows. One is the flow given by the classical equation of motion, and the other is the flow from integrating out the geometry. Here we want to prove the equivalence of the two. No split amplitude: Z 1 [φ 0 ] = Dφe S exp( S[φ c ]) (36) Split amplitude: Z 2 = bulk bulk D φψ IR (ɛ, φ)ψ UV (ɛ, φ), (37) ang Zhou July 7, 2011 ICTS-USTC,Hefei 19 /

43 Equivalence Equivalence between HWRG and Sliding Membrane There are two RG flows. One is the flow given by the classical equation of motion, and the other is the flow from integrating out the geometry. Here we want to prove the equivalence of the two. No split amplitude: Z 1 [φ 0 ] = Dφe S exp( S[φ c ]) (36) Split amplitude: Z 2 = Classical approximation: Ψ IR (ɛ, φ) = Ψ UV (ɛ, φ) = bulk bulk D φψ IR (ɛ, φ)ψ UV (ɛ, φ), (37) bulk bulk Dφ z>ɛ e S(z>ɛ) = e S IR[φ IR c ] (38) Dφ z<ɛ e S(z<ɛ) = e S UV [φ UV c ], (39) July 7, 2011 ICTS-USTC,Hefei 19 /

44 Sufficient: Equivalence Explicitly we write S ɛ [φ H, φ, φ 0 ] = S IR [φ IR c [φ H, φ]] + S UV [φ UV c [ φ, φ 0 ]]. (40) ang Zhou July 7, 2011 ICTS-USTC,Hefei 20 /

45 Sufficient: Equivalence Explicitly we write S ɛ [φ H, φ, φ 0 ] = S IR [φ IR c [φ H, φ]] + S UV [φ UV c [ φ, φ 0 ]]. (40) Z 2 = D φe Sɛ[ φ] = e Sɛ[φ H, φ,φ 0 ] (41) ang Zhou July 7, 2011 ICTS-USTC,Hefei 20 /

46 Sufficient: Equivalence Explicitly we write S ɛ [φ H, φ, φ 0 ] = S IR [φ IR c [φ H, φ]] + S UV [φ UV c [ φ, φ 0 ]]. (40) Z 2 = where φ is a solution for δs ɛ δ φ = δs UV δ φ D φe Sɛ[ φ] = e Sɛ[φ H, φ,φ 0 ] + δs IR δ φ (41) = 0, (42) which is nothing but Π IR = Π UV. Momentum continuity is ESSENTIAL for bulk classical calculation! ang Zhou July 7, 2011 ICTS-USTC,Hefei 20 /

47 Sufficient: Equivalence Explicitly we write S ɛ [φ H, φ, φ 0 ] = S IR [φ IR c [φ H, φ]] + S UV [φ UV c [ φ, φ 0 ]]. (40) Z 2 = where φ is a solution for δs ɛ δ φ = δs UV δ φ D φe Sɛ[ φ] = e Sɛ[φ H, φ,φ 0 ] + δs IR δ φ (41) = 0, (42) which is nothing but Π IR = Π UV. Momentum continuity is ESSENTIAL for bulk classical calculation! In order for Z 1 = Z 2, it is SUFFICIENT to have φ = φ c (z) z=ɛ so that φ = φ c. (43) We should also notice that it guarantees the ɛ independence of the Z 2 and S ɛ, i.e, dsɛ dɛ = 0! July 7, 2011 ICTS-USTC,Hefei 20 /

48 / :.-)*+, "#$%&' (! }~ klmnopqrstuvwxyz{j ^_àbcdefghi ]\ Necessary (on-shell) Equivalence What about the necessary part? ~ ϕ ϕh C A B ϕ0 ZH ε Z=0 Figure: Flows with (A) and without (C) the momentum continuity. The uniqueness of the smooth solution forbids a solution like B. July 7, 2011 ICTS-USTC,Hefei 21 /

49 / :.-)*+, "#$%&' (! }~ klmnopqrstuvwxyz{j ^_àbcdefghi ]\ Necessary (on-shell) Equivalence What about the necessary part? On-shell: ~ ϕ ϕh C A B ϕ0 ZH ε Z=0 Figure: Flows with (A) and without (C) the momentum continuity. The uniqueness of the smooth solution forbids a solution like B. July 7, 2011 ICTS-USTC,Hefei 21 /

50 Necessary (off-shell) Equivalence What about the off-shell case? July 7, 2011 ICTS-USTC,Hefei 22 /

51 Necessary (off-shell) Equivalence What about the off-shell case? Look at the RG again: ɛ S B [ φ] + H IR [ φ, δs B[ φ] ] = 0. (44) δ φ July 7, 2011 ICTS-USTC,Hefei 22 /

52 Necessary (off-shell) Equivalence What about the off-shell case? Look at the RG again: ɛ S B [ φ] + H IR [ φ, δs B[ φ] ] = 0. (44) δ φ Taking the derivative of eq. (??) with respect to φ we get ɛ Π φ = δh IR δ φ. (45) July 7, 2011 ICTS-USTC,Hefei 22 /

53 Necessary (off-shell) Equivalence What about the off-shell case? Look at the RG again: ɛ S B [ φ] + H IR [ φ, δs B[ φ] ] = 0. (44) δ φ Taking the derivative of eq. (??) with respect to φ we get ɛ Π φ = δh IR δ φ. (45) Momentum Continuity gives the ɛ dependence of φ Π φ = gg zz z φ(z) ɛ = gg zz ɛ ɛ φ (46) July 7, 2011 ICTS-USTC,Hefei 22 /

54 Necessary (off-shell) Equivalence What about the off-shell case? Look at the RG again: ɛ S B [ φ] + H IR [ φ, δs B[ φ] ] = 0. (44) δ φ Taking the derivative of eq. (??) with respect to φ we get ɛ Π φ = δh IR δ φ. (45) Momentum Continuity gives the ɛ dependence of φ Π φ = gg zz z φ(z) ɛ = gg zz ɛ ɛ φ (46) We recover the full EOM for φ! The equations (??) and (??) together with φ 0, φ H as the boundary condition of φ repeat the classical solution in whole bulk. July 7, 2011 ICTS-USTC,Hefei 22 /

55 Equivalence So far, we conclude that, in the bulk classical level, HWRG and sliding membrane are exactly equivalent. Basically, all bulk classical flows involve EOM. Still, the important question: how does the flow look like in boundary field theory language? ang Zhou July 7, 2011 ICTS-USTC,Hefei 23 /

56 Double Trace Flow Double Trace Flow We will obtain S B by computing on-shell UV action. Maxwell equations are written by z [ g ( z A µ µ A z ) ] + ν [ g ( ν A µ µ A ν ) ] = 0. (47) We assume A µ (z, k) = A (0) µ (z) + k 2 A (1) µ (z) + k µ k ν A (2) ν (z) + (48) and A z is independent on k. In the small momentum 0 limit, the first term will dominate and the above equation can be solved by ɛ z 0 C µ 1 = 1 Â (0) µ (z) A (0) µ (z 0 ) = z z 0 µ C1 gg zz dz. (49) g µµ (Â(0) µ (ɛ) A (0) µ,z 0 ) := f µ (Â(0) µ (ɛ) A (0) µ,z 0 ) (50) µµ dz gg zz g 1 ɛ 1 = f µ z 0 gg zz dz (51) g µµ July 7, 2011 ICTS-USTC,Hefei 24 /

57 Double Trace Flow Now, we want to integrate out z in region z 0 < z < ɛ. For the standard quadratic Maxwell action, we obtain the on-shell action as boundary term using the equations of motion: S on shell [z 0,ɛ] = 1 2 where we used z  (0) µ = d d x gg zz g µµ  c µ z  c ɛ µ = 1 z 0 2 d d x C µ ɛ 1 Âc µ. z 0 (52) C µ 1 gg zz g µµ. ϕ(x µ, z) = z z 0 A z dz, the gauge invariant field  (0) µ = A (0) µ µ ϕ. Using solution (??), we finally obtain the zero order on shell action S on shell [z 0,ɛ] = 1 d d x f µ ( µ (ɛ) A µ,z0 )( µ (ɛ) A µ,z0 ). (53) 2 µ This is S B at low frequency limit! If we set z 0 = 0. July 7, 2011 ICTS-USTC,Hefei 25 /

58 Double Trace Flow From the boundary point of view, we have deformations due to S B. Naturally, the deformed theory should be defined at ɛ, and the source should be A µ (ɛ) and the current should comes from J µ ɛ δs B δa µ,ɛ (54) July 7, 2011 ICTS-USTC,Hefei 26 /

59 Double Trace Flow From the boundary point of view, we have deformations due to S B. Naturally, the deformed theory should be defined at ɛ, and the source should be A µ (ɛ) and the current should comes from J µ ɛ δs B δa µ,ɛ (54) The Green function in linear level can be defined as G µµ ɛ = Jµ ɛ A µ,ɛ. (55) July 7, 2011 ICTS-USTC,Hefei 26 /

60 Double Trace Flow From the boundary point of view, we have deformations due to S B. Naturally, the deformed theory should be defined at ɛ, and the source should be A µ (ɛ) and the current should comes from J µ ɛ δs B δa µ,ɛ (54) The Green function in linear level can be defined as G µµ ɛ = Jµ ɛ A µ,ɛ. (55) The field theory effective action can be derived from which is given by S eff = d d x µ δs eff = 1 G κ δj J = 1 G ɛ µµ δj µ ɛ J µ ɛ. (56) [ 1 2 (Jµ ) 2 1 ] J µ (A µ,0 + µ ϕ) f µ. (57) July 7, 2011 ICTS-USTC,Hefei 26 /

61 Double Trace Flow ang Zhou From the boundary point of view, we have deformations due to S B. Naturally, the deformed theory should be defined at ɛ, and the source should be A µ (ɛ) and the current should comes from J µ ɛ δs B δa µ,ɛ (54) The Green function in linear level can be defined as G µµ ɛ = Jµ ɛ A µ,ɛ. (55) The field theory effective action can be derived from which is given by S eff = d d x µ δs eff = 1 G κ δj J = 1 G ɛ µµ δj µ ɛ J µ ɛ. (56) [ 1 2 (Jµ ) 2 1 ] J µ (A µ,0 + µ ϕ) f µ This is nothing but the Legendre transformation of S B!. (57) July 7, 2011 ICTS-USTC,Hefei 26 /

62 Double Trace Flow Double Trace Coupling flow Compare with Witten description: S B contains double trace deformation! For 4d AdS black hole 1 f i = 1 2 ɛ2, 1 = 1 ( ) 1 + ɛ 2 f 0 4 ln 1 ɛ 2 (58) July 7, 2011 ICTS-USTC,Hefei 27 /

63 Double Trace Flow Double Trace Coupling flow Compare with Witten description: S B contains double trace deformation! For 4d AdS black hole 1 f i = 1 2 ɛ2, 1 = 1 ( ) 1 + ɛ 2 f 0 4 ln 1 ɛ 2 (58) Double trace deformed Green function in linear level: J µ ɛ = G µµ ɛ A µ,ɛ, where ϕ(x µ, z) = z z 0 A z dz. 1 G µµ ɛ 1 G µµ z 0 = 1 f µ + µϕ J µ. (59) July 7, 2011 ICTS-USTC,Hefei 27 /

64 Double Trace Flow Double Trace Coupling flow Compare with Witten description: S B contains double trace deformation! For 4d AdS black hole 1 f i = 1 2 ɛ2, 1 = 1 ( ) 1 + ɛ 2 f 0 4 ln 1 ɛ 2 (58) Double trace deformed Green function in linear level: J µ ɛ = G µµ ɛ A µ,ɛ, where ϕ(x µ, z) = z z 0 A z dz. when ɛ 0, G ɛ G 0. 1 G µµ ɛ 1 G µµ z 0 = 1 f µ + µϕ J µ. (59) ang Zhou July 7, 2011 ICTS-USTC,Hefei 27 /

65 Double Trace Flow Conclusions and Future Questions HWRG and Sliding membrane flow are equivalent. July 7, 2011 ICTS-USTC,Hefei 28 /

66 Double Trace Flow Conclusions and Future Questions HWRG and Sliding membrane flow are equivalent. Double trace flow comes from integrating out UV bulk geometry. July 7, 2011 ICTS-USTC,Hefei 28 /

67 Double Trace Flow Conclusions and Future Questions HWRG and Sliding membrane flow are equivalent. Double trace flow comes from integrating out UV bulk geometry. Our Green function flow from Membrane is equivalent to Legendre transformation of S B in (I.heemskerk and J.Polchinski) July 7, 2011 ICTS-USTC,Hefei 28 /

68 Double Trace Flow Conclusions and Future Questions HWRG and Sliding membrane flow are equivalent. Double trace flow comes from integrating out UV bulk geometry. Our Green function flow from Membrane is equivalent to Legendre transformation of S B in (I.heemskerk and J.Polchinski) Open Question: what about the possible flow for non-local operator? Is it possible to establish the precise map between H-J flow and exact Wilson RG in field theory? July 7, 2011 ICTS-USTC,Hefei 28 /

69 Double Trace Flow Conclusions and Future Questions HWRG and Sliding membrane flow are equivalent. Double trace flow comes from integrating out UV bulk geometry. Our Green function flow from Membrane is equivalent to Legendre transformation of S B in (I.heemskerk and J.Polchinski) Open Question: what about the possible flow for non-local operator? Is it possible to establish the precise map between H-J flow and exact Wilson RG in field theory? Quick Question: Does this double trace flow mean anything in AdS/Nuclear or AdS/CMT? Running of everything we obtained before? Dispersion relation? Transport Coefficients? July 7, 2011 ICTS-USTC,Hefei 28 /