Stress tensor correlators from holography

Transcription

1 Stress tensor correlators from holography Aninda Sinha Indian Institute of Science, Bangalore 1

2 Mainly based on with Kallol Sen. partly on with Shamik Banerjee, Arpan Bhattacharyya, Apratim Kaviraj and Kallol Sen 2

3 Outline Motivation Review of relative entropy for ht a1 b 1 i ht a1 b...t for n=1,2,3 for theories with 1 an b i n arbitrary curvature invariants. Hofman-Maldacena bounds The inevitability of string theory--camanho, Edelstein, Maldacena, Zhiboedov 3

4 Motivation T a1 b 1 (x)t a2 b 2 (0) + 3X k=1 c k t (k) a 1 b 1 a 2 b 2 a 3 b 3 (x)t a 3b 3 (0) + OPE coefficients Are the OPE coefficients bounded? Reasons (a) Conformal bootstrap (b)positive energy fluxes. 4

5 Motivation Conformal bootstrap intuition arises from work of Caracciolo-Rychkov, Rattazzi et al. Using scalar four point function bootstrap arguments, OPE coefficients appearing via h O,l i are bounded. May be similar feature for conserved currents (namely replace scalar with conserved currents and ask if hjjo,l i OPEs are bounded--very hard and till now open question). 5

6 Motivation Positive energy flux argument due to Hofman and Maldacena Z h0 T a1 b 1 (x 1 )T tt (x 2 )T a3 b 3 (x 3 ) 0i > 0 Leads to inequalities for OPE coefficients. 6

7 How does one calculate ht a1 b 1 (x 1 ) T an b n (x n )i? For this talk we focus on the holographic dual side of the story. 3pt functions fixed by conformal symmetry upto numbers. We will focus on how to get these numbers. I will explain how to calculate n=1,2,3 for any theory L(g ab,r abcd, r e R abcd ) 7

8 Even the one point function appears very difficult. Reason is that we usually need to start with an action which includes the surface term and counterterms. Not known except for very special theories like Lovelock theories. 8

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10 How does one systematically proceed? 9

11 How does one systematically proceed? Step 1, we need a general and useful procedure to write down the holographic stress tensor for an arbitrary higher derivative gravity. 9

12 How does one systematically proceed? Step 1, we need a general and useful procedure to write down the holographic stress tensor for an arbitrary higher derivative gravity. Step 2, we need a general and useful procedure to compute stress tensor correlation functions in arbitrary higher derivative gravity. 9

13 How does one systematically proceed? Step 1, we need a general and useful procedure to write down the holographic stress tensor for an arbitrary higher derivative gravity. Step 2, we need a general and useful procedure to compute stress tensor correlation functions in arbitrary higher derivative gravity. Step 3, we need an algorithm which is feasible within the life-span of a graduate student! 9

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15 To accomplish step 1, we appeal to entanglement entropy. Recently Faulkner et al showed that for spherical entangling surfaces, using positivity of relative entropy, one recovers the linearized (general) gravity equations. As a by product they obtain a simple way to compute the renormalized holographic stress tensor for a general theory of gravity.[faulkner, Guica, Hartman, Myers, Raamsdonk, 2013] 11

16 Relative entropy Relative Entropy follows From Klein's Inequality S (ρ σ ) = Δ H Δ S S (ρ σ ) 0 Change in modular hamiltonian H = e H /tr(e H ) Change in EE We get Δ H Δ S 1st law of entanglement when saturated From Holography we can compute Δ S Δ S = S (ρ ) Area functonal S (σ ) Area functional 12

17 We can compute holographically S (ρ σ ) [Blanco, Casini, Hung, Myers, 2013] holographically provided we know H We know H in some special cases. For a spherical entangling surface H = 2π x < R d d 1 x R2 r 2 2 R T ( x) 00 If we knew how to calculate ΔS we would be able to infer the stress tensor. Sphere radius Can compute using holography can compute H 13

18 Faulkner et al, The causal development of a ball is mapped to the evolution generated by ordinary H in the hyperboloid. [Casini, Huerta, Myers] EE gets mapped to thermal entropy. [Myers, AS; Casini, Huerta, Myers] Can use Wald entropy to calculate the latter.[myers, AS] 14

19 Holographic stress tensor from ΔS=ΔH More precisely we have renormalized stress tensor: want to know Can calculate using linearization of Wald entropy functional. Small sphere limit makes integral local. 15

20 If we think about it, calculation of correlation functions will be simpler if we do a background field expansion of the bulk lagrangian such that on the AdS background ΔRiemann is zero. [Sen, AS] 16

21 So schematically we have the bulk lagrangian to have terms like (ΔRiemann) n This suggests that the bulk action can be rearranged in terms of stress tensor correlation functions. For instance if we wanted n-point functions we only consider upto power n. The coefficients of these terms themselves would involve ALL higher derivative terms in the bulk lagrangian we started with. 17

22 So schematically we have the bulk lagrangian to have terms like (ΔRiemann) n This suggests that the bulk action can be rearranged in terms of stress tensor correlation functions. For instance if we wanted n-point functions we only consider upto power n. The coefficients of these terms themselves would involve ALL higher derivative terms in the bulk lagrangian we started with. 17

23 Full metric R abcd = 1 L2 (g acg bd g ad g bc ) R abcd = R abcd Rabcd There are some immediate advantages of doing this background field expansion. In TT gauge: ( R abcd ) AdS =0 ( R ab ) linearized =0 ( R) linearized =0 18

24 Consider (Δ R)2 or (Δ Rab) 2 in the action. To compute the effect of these terms on the n- point function, we have to expand this around AdS upto n th order in the perturbation. This immediately tells us that these will start contributing only to 4-point functions. This is the reason for the simple expressions we will find for one, two and three point functions. 19

25 Background field Lagrangian from L(g ab,r cdef ) Sen, AS 2014 Two point function C T =2 d +1 d 1 [d + 1] d/2 [d/2] L d 1 [c 1 + 2(d 2)c 6 ] Simple!! Stress tensor and geometry ht µ i = d/2 d 1 2 L 2 d +1 [d/2] [d] C T h (d) µ [Fefferman, Graham; de Haro, Skenderis, Solodukhin] Trace anomaly ht a a i A d E d {z} Euler + X i B i I i {z} W eyl Deser, Schwimmer; Henningson- Skenderis; Miao 20

26 Background field Lagrangian from L(g ab,r cdef ) Sen, AS 2014 A-type anomaly: a-theorems?? Two point function C T =2 d +1 d 1 [d + 1] d/2 [d/2] L d 1 [c 1 + 2(d 2)c 6 ] Simple!! Stress tensor and geometry ht µ i = d/2 d 1 2 L 2 d +1 [d/2] [d] C T h (d) µ [Fefferman, Graham; de Haro, Skenderis, Solodukhin] Trace anomaly ht a a i A d E d {z} Euler + X i B i I i {z} W eyl Deser, Schwimmer; Henningson- Skenderis; Miao 20

27 Background field Lagrangian from L(g ab,r cdef ) Sen, AS 2014 A-type anomaly: a-theorems?? Two point function C T =2 d +1 d 1 B-type anomaly [d + 1] d/2 [d/2] L d 1 [c 1 + 2(d 2)c 6 ] Simple!! Stress tensor and geometry ht µ i = d/2 d 1 2 L 2 d +1 [d/2] [d] C T h (d) µ [Fefferman, Graham; de Haro, Skenderis, Solodukhin] Trace anomaly ht a a i A d E d {z} Euler + X i B i I i {z} W eyl Deser, Schwimmer; Henningson- Skenderis; Miao 20

28 Example New massive gravity (Bergshoeff, Hohm, Townsend) L D=3 = R + 2 L 2 +4L2 (R ab R ab 3 6 R = R L 2 2 R ac = R ac L 2 g ac c 1 = 1 2`P (1 + 2 L2 L 2 ) 8 R2 ) This is the expected answer. 21

29 In even dimensions ct is related to a B-anomaly coefficient. Let me sketch the argument. RG equation gives [Erdmenger, Osborn; Osborn, Petkou] Quantum effective action Hit twice more with g µ 22

30 Since RHS is to be computed around flat space, it is easy to see that we need terms in the anomaly that have at most 2 Riemann curvatures. In 4 dimensions an explicit calculation picks out the c-anomaly coefficient. In 6 dimensions the only anomaly term with 2 Riemann s is B3. Hence this I 3 C 2 C abcd coefficient is picked out. All this we have explicitly check directly in holography. 23

31 Since RHS is to be computed around flat space, it is easy to see that we need terms in the anomaly that have at most 2 Riemann curvatures. In 4 dimensions an explicit calculation picks out the c-anomaly coefficient. In 6 dimensions the only anomaly term with 2 Riemann s is B3. Hence this I 3 C 2 C abcd coefficient is picked out. All this we have explicitly check directly in holography. We can also compute 3 point functions! 23

32 slide: R.Myers 24

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34 t 4 10 d =4 5 t 2 absent in 3d 0 t

35 Lovelock--de Boer, Kulaxizi, Parnachev; Buchel, Escobedo, Myers, Paulos, AS, Smolkin. Quasi-top--Myers, Paulos, AS When t4=0 (& c7=c8=0) ( s 1 4 )s / c 6 Application Thus KSS bound is violated whenever we have t2>0 in a perturbative expansion. For a general four derivative theory (non-unitary) we find /s! 0 27

36 HM bounds from entanglement see Arpan s talk z Smooth Singular 1X c i (z z h ) Banerjee, Bhattacharyya, Kaviraj, Sen, AS, ; Ogawa- Takayanagi i=0 0 < < 1 c 0 2 Reals 28

37 Hofman-Maldacena, Buchel- Myers, Shenker et al Banerjee, Bhattacharyya, Kaviraj, Sen, AS, Demanding a smooth surface puts bound on GB coupling. 29

38 Is string completion necessary? In IIB supergravity where the leading correction begins at 8-derivative order, both t2 and t4 are zero. t4 needs six derivative terms which are absent in supersymmetric theories t2 could be present in supersymmetric theories. Shapiro time delay calculation by Camanho, Edelstein, Maldacena and Zhiboedov show that causality is always violated in a truncated higher derivative theory!! 30

39 So for instance Gauss-Bonnet gravity in spite of having two derivative eoms will always lead to a time advance. To fix such problems in theories with nonzero t2/t4 it seems that we will either need an infinite set of higher derivative corrections or we need to invoke some new physics involving an infinite set of higher spin massive particles. (String theory is inevitable!) 31

40 Allowed t2,t4 shrinks as we go inside the bulk. Disappears for a finite value of the coordinate Hofman-Maldacena t Shapiro time delay t 2 32

41 Allowed t2,t4 region shrinks to zero size at some value of the radial coordinate (impact parameter). c p 2 3 Fix causality problem by adding more higher derivative terms? This is not possible since each factor of Riemann introduces (1 ).Easy to see using our background field approach. Thus will need new physics at 2, 3 1 g Shapiro time delay gets mapped to anomalous dimensions of operators in N=4 CFT. Starting with SUGRA spectrum Alday et al found that the conformal dimension could violate the unitarity bound unless the spectrum includes higher spin operators. 33

42 Having an infinite tower of higher spin massive operators appears inevitable for non-zero t2/t4. Points at the fact that string theory may be necessary to talk about the duals of CFTs which have unequal c,a. 34

43 courtesy: A.Kaviraj AdS/CFT Gamow-Ivanenko-Landau physics cube--holographic version 35

44 courtesy: A.Kaviraj AdS/CFT Is Interstellar possible? Gamow-Ivanenko-Landau physics cube--holographic version 35

45 Take home message: If string theory is right Interstellar is impossible! 36