Bootstrap approach to CFT in D dimensions

Transcription

1 Bootstrap approach to CFT in D dimensions Slava Rychkov CERN & École Normale Supérieure (Paris) & Université Pierre et Marie Curie (Paris) Strings 2013, Seoul

2 Origins of Conformal Bootstrap, early 1970 s Raoul Gatto Sergio Ferrara Aurelio Grillo Alexander Polyakov 2 /17

3 Results from those early days primary operators + descendants [Mack, Salam 1969] unitarity bounds [Ferrara, Gatto, Grillo 1974, Mack 1977] conformally invariant OPE constraints on the correlation functions of primaries 3 /17

4 Results from those early days primary operators + descendants [Mack, Salam 1969] unitarity bounds [Ferrara, Gatto, Grillo 1974, Mack 1977] conformally invariant OPE constraints on the correlation functions of primaries They realized that: 1) Any CFT is characterized by conformal data = {primary operator dimensions Δi, OPE coefficients cijk} 2) OPE associativity: should fix the data conformal bootstrap Enter QCD... 3 /17

5 Conformal blocks D>2 discl... conf. blocks 4 /17

6 Conformal blocks D>2 discl... conf. blocks 4 /17

7 ...and coordinates for them z-coord: cut used to express conf. blocks in [Dolan, Osborn 2000,2003,2011] 5 /17

8 ...and coordinates for them z-coord: cut used to express conf. blocks in [Dolan, Osborn 2000,2003,2011] ρ-coord: [Pappadopulo, S.R., Espin, Rattazzi 2012, Hogervorst, S.R. 2013] 5 /17

9 ...and coordinates for them z-coord: cut used to express conf. blocks in [Dolan, Osborn 2000,2003,2011] ρ-coord: [Pappadopulo, S.R., Espin, Rattazzi 2012, Hogervorst, S.R. 2013] D=2, Al.Zamolodchikov,fractional... known coeffs. 5 /17

10 Convergence of conf. block decomposition [Pappadopulo, S.R., Espin, Rattazzi 2012] 6 /17

11 Convergence of conf. block decomposition [Pappadopulo, S.R., Espin, Rattazzi 2012] convergence for all r<1 + polynomial bound on weighted spectral density 6 /17

12 Convergence of conf. block decomposition [Pappadopulo, S.R., Espin, Rattazzi 2012] convergence for all r<1 + polynomial bound on weighted spectral density Cf. 6 /17

13 Simplest bootstrap equation crossing: Mathematically well-def d, S-matrix... 7 /17

14 Simplest bootstrap equation crossing: Mathematically well-def d, S-matrix... 7 /17

15 Simplest bootstrap equation crossing: convergence cuts Mathematically well-def d, S-matrix... 7 /17

16 Numerical exploration 1) Identifying swampland in the space of CFT data 2) Study of theories at the swampland boundary 8 /17

17 I. Charting out CFT swampland [Rattazzi, S.R, Tonni, Vichi, 2008] + many subsequent works Rule out large chunks of CFT data space which do not correspond to any CFT, because bootstrap equations do not allow a solution Keyword: linear programming (way to enforce ) 9 /17

18 I. Charting out CFT swampland [Rattazzi, S.R, Tonni, Vichi, 2008] + many subsequent works Rule out large chunks of CFT data space which do not correspond to any CFT, because bootstrap equations do not allow a solution Keyword: linear programming (way to enforce ) Roads to swampland: increase gaps in the spectrum CFT pump up OPE coefficients 9 /17

19 Example of a gap study Take any CFT with G SO(N) global symmetry fund. of SO(N) lowest dimension singlet and 10/17

20 Example of a gap study Take any CFT with G SO(N) global symmetry fund. of SO(N) lowest dimension singlet and D=4, G=SO(4) Any CFT is forced to live below these lines [Vichi 2011, Poland, Simmons-Duffin, Vichi, 2011] Analytic bounds from toy bootstrap : [Hogervorst, S.R. 2013] 10/17

21 A central charge lower bound Suppose know Δσ, can we say something about? 11/17

22 A central charge lower bound Suppose know Δσ, can we say something about? Hint: 11/17

23 A central charge lower bound Suppose know Δσ, can we say something about? Hint: min C T 2.0 Any CFT4 must live above the curve N C T 43 (free scalar) N d [Poland, Simmons-Duffin 2011, Rattazzi, S.R.,Vichi 2011] 11/17

24 II. Studying swampland boundary Example: 12/17

25 II. Studying swampland boundary Example: No sols to crossing 12/17

26 II. Studying swampland boundary Example: No sols to crossing Many sols to crossing 12/17

27 II. Studying swampland boundary Example: No sols to crossing unique solution to crossing physically meaningful? Many sols to crossing 12/17

28 2D and 3D gap study S.R., Vichi 2009;El-Showk, Paulos D=2, SL2(C) only D=3 El-Showk,Paulos,Poland,Simmons-Duffin, Ε S.R, Vichi Ising easy to increase precision... 13/17

29 2D and 3D gap study S.R., Vichi 2009;El-Showk, Paulos D=2, SL2(C) only D=3 El-Showk,Paulos,Poland,Simmons-Duffin, Ε S.R, Vichi Ising D Ising model easy to increase precision... 13/17

30 2D and 3D gap study S.R., Vichi 2009;El-Showk, Paulos D=2, SL2(C) only D=3 El-Showk,Paulos,Poland,Simmons-Duffin, Ε S.R, Vichi Ising D Ising model Ε D Ising Ising (from RG methods) easy to increase precision... 13/17

31 Other kinks same kink happens for any 2 D<4; its position agrees with ε-expansion for D 4 [El-Showk, S.R, Vichi, work in progress] same kink happens for O(N) model in D=3; its position agrees with 1/N expansion for N [Poland, Simmons-Duffin, work in progress] Kinks have something to do with operator decoupling... 14/17

32 Spectrum of σ x σ OPE in 3D Ising model Current knowledge (from RG methods): Operator Spin l ε (1) ε (4) ε (11) T µν 2 3 C µνκλ (12) 15/17

33 Spectrum of σ x σ OPE in 3D Ising model Current knowledge (from RG methods): Operator Spin l ε (1) ε (4) ε (11) T µν 2 3 C µνκλ (12) Assuming 3D Ising lives at the kink can determine all*) operators in σ x σ OPE + their OPE coeffs [work in progress] *) numerical work. In practice: all /17

34 Warmup study for 2D Ising Blue=exact Red=from SL2(C) bootstrap fixing El-Showk, Paulos 2012 operators in σ x σ OPE Δ 16/17

35 Other interesting developments Analytic results about l>>1spectrum from bootstrap near light cone Fitzpatrick,Kaplan,Poland,Simmons-Duffin 2012, Komargodski, Zhiboedov 2012 Bootstrap for conformal boundary conditions and defects [Liendo, Rastelli, van Rees 2012 Gaiotto, Paulos, work in progress] Bootstrap for <JJJJ> and <TTTT> [work in progress by Dymarsky] Bootstrap for SUSY theories -N=1 Poland,Simmons-Duffin subsequent work -N=4, N=2 Beem, Rastelli, van Rees work in progress 17/17