Some analytic results from conformal bootstrap

Transcription

1 25 June, 2015 Some analytic results from conformal bootstrap Aninda Sinha Indian Institute of Science, Bangalore Strings 2015, ICTS, Bangalore Strings 2015, ICTS, Bangalore

2 Since the seminal work of Rattazzi, Rychkov, Tonni and Vichi in 2008, many interesting results have emerged using conformal bootstrap. Most of these impressive results rely heavily on (clever) numerics. I will talk about some analytic results that follow from conformal bootstrap.

3 Work done with Apratim Kaviraj, Kallol Sen arxiv: , and in progress. See Apratim s poster and cf Kallol s gong show talk Builds on work by Fitzpatrick, Kaplan, Poland, Simmons-Duffin; Komargodski, Zhiboedov. Related work by Alday, Bissi, Lukowski for N=4 SYM.

4 Summary of main results Given a (4d for most part) CFT with a scalar operator of dimension and a spin-2 (minimal) twist-2 operator there is an infinite sequence of large spin operators of dimension =2 +2n + ` + (n, `) Anomalous dim. ` n 1 (n, `) = 160 n 4 c T `2 n ` 1 (n, `) = 80 n 3 c T ` UNIVERSAL ht ab (x)t cd (x 0 )i =?UNIVERSAL? assume large N c T x x 0 2d I ab,cd(x x 0 )

5 For large N, we can think of these operators as double trace operators of the form Heemskerk, Penedones, Polchinski, Sully; El-Showk, Papadodimas However the CFT bootstrap analysis of course only yields conformal dimension, spin and the OPE coefficients and not the precise form of these operators.

6 Why is this interesting? Result is universal. Does not depend on lagrangian or the dimension of the seed operator. Just assumes twist gap of these operators from other operators in the spectrum. Anomalous dimension of double trace operators is related to bulk Shapiro time delay. Sign of anomalous dimension is related to causality. Interplay between unitarity of CFT and causality of bulk. Camanho, Edelstein, Maldacena, Zhiboedov Can be extended to arbitrary (eg. 3d) dimensions. May be relevant for 3d Ising model at criticality. El-Showk, Paulos, Poland, Rychkov, Simmons-Duffin, Vichi

7 Can compare with AdS/CFT. Two different ways to calculate the anomalous dimensions a) Eikonal approximation of 2-2 scattering. b) Energy shift in a black hole background. Cornalba, Costa, Penedones, Schiappa Fitzpatrick, Kaplan, Walters; Kaviraj, Sen, AS Turns out that the result matches exactly with the AdS/CFT prediction. Kaviraj, Sen, AS

8 Quick review of bootstrap s-channel t-channel

9 Quick review of bootstrap s-channel t-channel

10 Quick review of bootstrap s-channel t-channel even spin

11 Quick review of bootstrap 1+ X,` s-channel t-channel even spin P,`g,`(u, v) =( u v ) 1+ X,` P,`g,`(v, u)

12 Quick review of bootstrap 1+ X,` s-channel t-channel even spin P,`g,`(u, v) =( u v ) 1+ X,` P,`g,`(v, u)

13 Quick review of bootstrap Can only be reproduced upon considering large spin operators on the RHS 1+ X,` s-channel P,`g,`(u, v) =( u v ) t-channel even spin X 1+ P,`g,`(v, u),`

14 Quick review of bootstrap Can only be reproduced upon considering large spin operators on the RHS 1+ X,` s-channel P,`g,`(u, v) =( u v ) t-channel even spin X 1+ P,`g,`(v, u),` u = x2 12x 2 34 x 2 24 x2 13, v = x2 14x 2 23 x 2 24 x2 13 Conformal cross ratios

15 Quick review of bootstrap Can only be reproduced upon considering large spin operators on the RHS 1+ X,` s-channel P,`g,`(u, v) =( u v ) t-channel 1+ X,` Crossing u $ v even spin P,`g,`(v, u) u = x2 12x 2 34 x 2 24 x2 13, v = x2 14x 2 23 x 2 24 x2 13 Conformal cross ratios

16 Quick review of bootstrap Can only be reproduced upon considering large spin operators on the RHS 1+ X,` s-channel P,`g,`(u, v) =( u v ) t-channel 1+ X,` Crossing u $ v even spin P,`g,`(v, u) Twist = ` u = x2 12x 2 34 x 2 24 x2 13, v = x2 14x 2 23 x 2 24 x2 13 Conformal cross ratios

17 Quick review of bootstrap Can only be reproduced upon considering large spin operators on the RHS 1+ X,` s-channel P,`g,`(u, v) =( u v ) t-channel 1+ X,` Crossing u $ v even spin P,`g,`(v, u) u = x2 12x 2 34 x 2 24 x2 13, v = x2 14x 2 23 x 2 24 x2 13 Conformal cross ratios Twist = ` P,` OPE x OPE

18 Quick review of bootstrap Can only be reproduced upon considering large spin operators on the RHS 1+ X,` s-channel P,`g,`(u, v) =( u v ) t-channel 1+ X,` Crossing u $ v even spin P,`g,`(v, u) u = x2 12x 2 34 x 2 24 x2 13, v = x2 14x 2 23 x 2 24 x2 13 Conformal cross ratios Twist = ` P,` OPE x OPE g,`(u, v) Dolan, Osborn; Blocks

19

20 Closed form expressions for conformal Dolan, Osborn; blocks are known only in even dimensions.

21 Closed form expressions for conformal Dolan, Osborn; blocks are known only in even dimensions. However, simplifications occur in certain limits Fitzpatrick et al; Komargodski, Zhiboedov

22 Closed form expressions for conformal Dolan, Osborn; blocks are known only in even dimensions. However, simplifications occur in certain limits Fitzpatrick et al; Komargodski, Zhiboedov ` 1 In the crossed channel we u 1,v <1 interchange u, v

23 Closed form expressions for conformal Dolan, Osborn; blocks are known only in even dimensions. However, simplifications occur in certain limits Fitzpatrick et al; Komargodski, Zhiboedov ` 1 In the crossed channel we u 1,v <1 interchange u, v g (d),` (u, v) =u 2 (1 v)`2f 1 ( 2 + `, 2 + `, +2`, 1 v)f (d) (,u) factorizes (twist,spin,v) x (twist, u, d)

24 Closed form expressions for conformal Dolan, Osborn; blocks are known only in even dimensions. However, simplifications occur in certain limits Fitzpatrick et al; Komargodski, Zhiboedov ` 1 In the crossed channel we u 1,v <1 interchange u, v g (d),` (u, v) =u 2 (1 v)`2f 1 ( 2 + `, 2 + `, +2`, 1 v)f (d) (,u) factorizes (twist,spin,v) x (twist, u, d)

25

26 Recursion relations for blocks in any dimension

27 Recursion relations for blocks in any dimension z z 2 g (d) 2) (1 z)(1 z),`(v, u) =g(d 2,`(v, u) 4(` 2)(d + ` 3) (d +2` 4)(d +2` 2) apple 4(d 3)(d 2) d 2 2)(d 2 ) 2) g(d 2,`(v, u) ( + `) 2 2) g(d 16( + ` 1)( + ` + 1),`+2(v, u) (d + ` 4)(d + ` 3)(d + ` 2) 2 2) g(d 4(d +2` 4)(d +2` 2)(d + ` 3)(d + ` 1),` (v, u)

28 Crossed channel Recursion relations for blocks in any dimension z z 2 g (d) 2) (1 z)(1 z),`(v, u) =g(d 2,`(v, u) 4(` 2)(d + ` 3) (d +2` 4)(d +2` 2) apple 4(d 3)(d 2) d 2 2)(d 2 ) 2) g(d 2,`(v, u) ( + `) 2 2) g(d 16( + ` 1)( + ` + 1),`+2(v, u) (d + ` 4)(d + ` 3)(d + ` 2) 2 2) g(d 4(d +2` 4)(d +2` 2)(d + ` 3)(d + ` 1),` (v, u) Dolan, Osborn;

29 Crossed channel Recursion relations for blocks in any dimension z z 2 g (d) 2) (1 z)(1 z),`(v, u) =g(d 2,`(v, u) 4(` 2)(d + ` 3) (d +2` 4)(d +2` 2) apple 4(d 3)(d 2) d 2 2)(d 2 ) 2) g(d 2,`(v, u) ( + `) 2 2) g(d 16( + ` 1)( + ` + 1),`+2(v, u) (d + ` 4)(d + ` 3)(d + ` 2) 2 2) g(d 4(d +2` 4)(d +2` 2)(d + ` 3)(d + ` 1),` (v, u) Dolan, Osborn; Solution to recursion relations in closed form known only in even d.

30 Crossed channel Recursion relations for blocks in any dimension z z 2 g (d) 2) (1 z)(1 z),`(v, u) =g(d 2,`(v, u) 4(` 2)(d + ` 3) (d +2` 4)(d +2` 2) apple 4(d 3)(d 2) d 2 2)(d 2 ) 2) g(d 2,`(v, u) ( + `) 2 2) g(d 16( + ` 1)( + ` + 1),`+2(v, u) (d + ` 4)(d + ` 3)(d + ` 2) 2 2) g(d 4(d +2` 4)(d +2` 2)(d + ` 3)(d + ` 1),` (v, u) Dolan, Osborn; Solution to recursion relations in closed form known only in even d. In the large spin limit and relation simplifies. u 1,v <1 the recursion

31 Crossed channel Recursion relations for blocks in any dimension z z 2 g (d) 2) (1 z)(1 z),`(v, u) =g(d 2,`(v, u) 4(` 2)(d + ` 3) (d +2` 4)(d +2` 2) apple 4(d 3)(d 2) d 2 2)(d 2 ) 2) g(d 2,`(v, u) ( + `) 2 2) g(d 16( + ` 1)( + ` + 1),`+2(v, u) (d + ` 4)(d + ` 3)(d + ` 2) 2 2) g(d 4(d +2` 4)(d +2` 2)(d + ` 3)(d + ` 1),` (v, u) Dolan, Osborn; Solution to recursion relations in closed form known only in even d. In the large spin limit and relation simplifies. u 1,v <1 the recursion (1 v) 2 F (d) (,v) = 16F (d 2) ( 4,v) 2vF (d 2) ( 2,v)+ (d 2) 2 16(d 3)(d 1) v2 F (d 2) (,v).

32 New results from bootstrap

33 New results from bootstrap Gauss Hypergeometric

34 New results from bootstrap Gauss Hypergeometric F (d) (,v)= 2 (1 v) d 2 2 2F ( d + 2), 1 2 ( d + 2), ( d + 2),v Kaviraj, Sen, AS

35 New results from bootstrap Gauss Hypergeometric F (d) (,v)= 2 (1 v) d 2 2 2F ( d + 2), 1 2 ( d + 2), ( d + 2),v Kaviraj, Sen, AS Bootstrap equation demands at leading order

36 New results from bootstrap Gauss Hypergeometric F (d) (,v)= 2 (1 v) d 2 2 2F ( d + 2), 1 2 ( d + 2), ( d + 2),v Kaviraj, Sen, AS Bootstrap equation demands at leading order 1 (function of u) v /2 (1 v) F (d) (,v)

37 New results from bootstrap Gauss Hypergeometric F (d) (,v)= 2 (1 v) d 2 2 2F ( d + 2), 1 2 ( d + 2), ( d + 2),v Kaviraj, Sen, AS Bootstrap equation demands at leading order 1 (function of u) v /2 (1 v) F (d) (,v) Needs large

38 New results from bootstrap Gauss Hypergeometric F (d) (,v)= 2 (1 v) d 2 2 2F ( d + 2), 1 2 ( d + 2), ( d + 2),v Kaviraj, Sen, AS Bootstrap equation demands at leading order 1 (function of u) v /2 (1 v) F (d) (,v) To match powers of v, we Needs large must have

39 New results from bootstrap Gauss Hypergeometric F (d) (,v)= 2 (1 v) d 2 2 2F ( d + 2), 1 2 ( d + 2), ( d + 2),v Kaviraj, Sen, AS Bootstrap equation demands at leading order 1 (function of u) v /2 (1 v) F (d) (,v) To match powers of v, we Needs large must have Fitzpatrick et al; Komargodski, Zhiboedov =2 +2n

40 New results from bootstrap Gauss Hypergeometric F (d) (,v)= 2 (1 v) d 2 2 2F ( d + 2), 1 2 ( d + 2), ( d + 2),v Kaviraj, Sen, AS Bootstrap equation demands at leading order 1 (function of u) v /2 (1 v) F (d) (,v) To match powers of v, we must have Fitzpatrick et al; Komargodski, Zhiboedov =2 +2n Needs large Same as what appears in MFT. OPE s known.

41 We can go to subleading order It can be shown that the anomalous dimension at large spin goes like an inverse power of the spin for spacetime dimension>2. This means that we can treat the inverse spin as an expansion parameter and this result is true even for theories which do not have a large N. Our objective is to determine the n-dependence for the anomalous dimension.

42 After some clever detective work we find m Twist of exchanged operator (n, `)` m = nx m=0 C n,mb (d) m (d) nested sums : Since k is a positive integer, this is a polynomial.

43 Progress is possible in 4d (and similar techniques apply in even d)

44 Progress is possible in 4d (and similar techniques apply in even d)

45 Progress is possible in 4d (and similar techniques apply in even d) Use basic defn

46 Progress is possible in 4d (and similar techniques apply in even d) Use basic defn To reduce to one sum

47 Progress is possible in 4d (and similar techniques apply in even d) Use basic defn To reduce to one sum

48 Progress is possible in 4d (and similar techniques apply in even d) Use basic defn To reduce to one sum Using the following:

49 Progress is possible in 4d (and similar techniques apply in even d) Use basic defn To reduce to one sum Using the following:

50 Progress is possible in 4d (and similar techniques apply in even d) Use basic defn To reduce to one sum Using the following:

51 Progress is possible in 4d (and similar techniques apply in even d) Use basic defn To reduce to one sum Using the following:

52 So effectively we just need to do the integral d=4 which can be easily done by going to polar coordinates.

53 So effectively we just need to do the integral d=4 which can be easily done by going to polar coordinates.

54 So effectively we just need to do the integral d=4 which can be easily done by going to polar coordinates. This is negative and monotonically decreasing with n for any conformal dimension satisfying the unitarity bound

55 n( ) n =2 n =1 Always negative for n 3 n =0

56 n =2 Always negative for n 3 n( ) n =1 n =0 If unitarity bound is violated anomalous dimensions can be positive.

57 Comments on general dimensions Assume minimal twist for stress tensor exchange d-2 With some effort this can be derived analytically in all d. In terms of ct: For this to match with the AdS Eikonal calculation, we need P m = 16G N (d 1) (1 + d 2 )3 (d + 1) (d + 2) Exactly expected from AdS/CFT 2

58 Universality at large twist

59 Universality at large twist Plots in diverse spacetime dimensions for various conformal dimensions. Asymptotes indicate same intercept independent of conformal dimension.

60 Negativity violated if unitarity violated n d 1 < 0

61 Subleading terms (d=4) Starting with the differential equation one can also get the subleading terms in 1/` t-channel not in s- channel

62 (n, `) = 160 c T n 4 `2 (1 m n ` ) universal In principle we can extract order by order.

63 Role of higher spin exchange from CFT Introduce h = + n + `, h = + n Using saddle point methods and the other limit we find (n, `) / n2`m+ m 2 (` + n) 2 m `(` +2n) Agrees exactly with AdS/CFT results of Cornalba, Costa, Penedones, Schiappa 07 For Valid for large spin, twist (n, `) / n2`m 1 ` Depends only on spin! `m =0 allows small anom. dim

64 `m =2 (n, `) n3 ` ( 1 N 2 + # 2 +#n4 4 + ) From massive spin=2 This means that adding a finite set Comes from OPE coefficient of higher spin=4 exchange of higher spin modes will not change the sign of the anomalous dimension. Anom. dims. will not be small for some very large twist. To allow for perturbative unitarity cf Camanho, Edelstein, Maldacena, Zhiboedov we may need to add an infinite set of higher spin modes.

65 It will be very interesting to see what a consistent CFT spectrum can be which leads to perturbatively small anomalous dimensions in a theory with large N and a gap.

66 Holography

67 Holography Fitzpatrick, Kaplan and Walters suggested the following simple calculation.

68 Holography Fitzpatrick, Kaplan and Walters suggested the following simple calculation. The double trace operators can be thought of as two massive particles in AdS rotating around each other. The anomalous dimension arises due to the interacting energy of these particles.

69 Holography Fitzpatrick, Kaplan and Walters suggested the following simple calculation. The double trace operators can be thought of as two massive particles in AdS rotating around each other. The anomalous dimension arises due to the interacting energy of these particles. Essential idea is to do perturbation theory in inverse distance corresponding to a Newtonian approximation in AdS.

70 It has been shown that for n=0, the result of the calculation agrees with the bootstrap prediction. (Unlike Eikonal where both spin and n needed to be large) Non-zero n is quite hard. However, we have been able to make progress (barring overall constants) at large n, i.e., It turns out to give exactly the same universal behaviour predicted by bootstrap!

71 Non-renormalization apple from holography ` n 1 Higher derivative correction E d n,`orb = µ 2 Z r(1 + 0h r 2h )dr apple nx k, =0 E 2 n,` (1 + r 2 ) 2 k(r) (r)+@ r k (r)@ r (r) = I 1 + I 2, (5.9)

72 Non-renormalization apple from holography ` n 1 Higher derivative correction E d n,`orb = µ 2 Z r(1 + 0h r 2h )dr apple nx k, =0 E 2 n,` (1 + r 2 ) 2 k(r) (r)+@ r k (r)@ r (r) = I 1 + I 2, E 4 = µ(` +2n)2 (` +2+n) nx ( 1) k (k + ` + n + ) n,`orb 4 (` + 2) (n + 1) (` +2+k) (n +1 k) (2 + k + ` + ) (k + 1) k=0 apple (1 + ` + k) (1 + ) 3 F 2 n, k + ` +1, ` + n + ; ` +2, 2+k + ` + ;1 + 0h (1 + ` + k h) (1 + + h) 3 F 2 n, k + ` +1 h, ` + n + ; ` +2, 2+k + ` + ;1 (5.9). The spin dependence for the Einstein term can be shown to be derivative term gives while the higher. Thus no t Hooft coupling dependence! Prediction for susy bootstrap: N=4 t Hooft coupling shows up at

73 The ` n 1 result exactly agrees with the CFT calculation. It will be interesting to do the other limit to check non-universality due to higher derivative corrections and compare with causality constraints.

74 Summary We have derived certain interesting universal results using conformal bootstrap. We should understand the large twist limit better both from the CFT side (what is a consistent spectrum) and from the gravity side (role of higher derivative corrections).

75

76 Thank you for listening

77