Conformal Field Theories in more than Two Dimensions

Transcription

1 Conformal Field Theories in more than Two Dimensions Hugh Osborn February 2016 Inspired by

2 It is difficult to overstate the importance of conformal field theories (CFTs) Fitzpatrick, Kaplan, Khanker Poland, Simmons-Duffin 2014

3 Pre Modern, where did it start Contemporary Modern 1. When does conformal symmetry arise? 2. Conformal Kinematics, Null Cone 3. Quantum Fields 4. Three point functions, Operator Product Expansion 5. Four point functions 6. Bootstrap 7. Superconformal 8. Minkowski Space Methods Post Modern, where might it go

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5 What is Conformal Symmetry, Conformal transformations preserve angles x! x 0 dx 02 = (x) 2 dx 2 Two dimensions are different x µ = v µ µ v v µ =2 µ v µ (x) =a µ! µ x + applex µ + b µ x 2 2 x µ b x,! µ =! µ translation Lorentz scale v µ is a conformal Killing vector special conformal 1 2 (d + 1)(d + 2) parameters

6 Conformal field theories are obtained by RG flow to non trivial IR limit where the beta functions vanish For gauge theories this is expected for a restricted range of flavours defining the conformal window QCD SU(2) 3 6. N f apple 10 SU(3) 7 9. N f apple 16 The lower bound is a strong coupling problem depending on lattice calculations For large N f,n c,n f = 11 2 N c there is the weakly coupled Banks Zaks fixed point In N =1 SQCD by Seiberg duality conformal window is 3 2 N c <N f < 3N c weakly coupled magnetic theory N =4conformal for any g YM weakly coupled electric theory

7 Conserved current for any conformal Killing vector if J µ v = T µ µ T µ =0, µ T µ =0 In general in a QFT expect µ T µ = I O I µ J µ If I =0 there is a conserved current for scale transformations S µ = T µ x J µ S µ =0 but conformal invariance is broken Ways out: J µ L µ can redefine to make it traceless T µ or µ J µ = r I O I then get a CFT if B I = I + r I =0 This reflects potential arbitrariness in the definition of beta functions beyond a choice of scheme

8 Many authors have discussed whether there are scale but not conformal invariant theories It is very likely that for unitary theories scale invariance does Polchinksi Riva Cardy Dorigoni Rychkov Fortin Grinstein Stergiou Nakayama Jackiw Pi El-Showk Dymarsky Komargodski Schwimmer Theisen imply conformal symmetry For non unitary theories there are counterexamples In two dimensions it is a theorem

9 Conformal transformations are nonlinear (x y) 2! (x) (y)(x y) 2 Need 4 points to construct an invariant It is often simpler to use homogeneous coords X µ,x d,x d+1 X X 1 2 X X = µ X µ X +(X d ) 2 (X d+1 ) 2 =0 Conformal group defined by linear transformations preserving the null cone SO(d, 2) null projective cone (Dirac) x µ = Xµ X +, X+ = X d + X d+1, (x y) 2 = X Y X + Y + SO(d +1, 1)

10 AdS d+1 may be defined by X X = 1 and so has the conformal group as its isometry group The boundary of AdS d+1 is the projective null cone This is just the tip of the AdS/CFT correspondence The inversion operation plays a crucial role in CFTs x µ! x µ /x 2, X d+1! X d+1 like parity, parity with inversion part of connected conformal group

11 Algebra and Fields Conformal Generators M AB = M BA contain P µ M µ D K µ translation Lorentz scale special conformal so(d +1, 1) or so(d, 2) [D, P µ ]=P µ [D, K µ ]= K µ [M AB,M CD ]= AC M BD BC M AD AD M BC + BD M AC Radial Quantisation, treat D as the Hamiltonian evolving in = log x Fields are labelled by D eigenvalue and spin adjoint (x) = (X), ( X) = (X) (x) =(x 2 ) (x/x 2 )

12 states i = (0) 0i h = h0 (0) correspond to fields Conformal primary K µ i =0 D i = i h P µ =0 P µ = K µ Require the fields generate unitary positive energy representations for a unitary CFT D has positive eigenvalues, zero on the vacuum Descendants generated by action of momentum operators Q (Pµ ) n µ i n µ =0, 1, 2,... K µ i =0 ensures is bounded below, positive energy

13 For fields with spin µ 1...µ`(x) A 1...A`(X), X 2 =0 X A r A 1...A`(X) =0, r =1...` A 1...A`(X) A1...A`(X)+X Ar A 1...Âr...A`(X), r =1...` µ 1...µ`(x) =(x 2 ) Q r=1...` Iµ r r (x) 1... `(x/x 2 ) I µ (x) = µ 2 xµ x Inversion tensor x 2 µ1...µ`i = µ1...µ`(0) 0i h µ 1...µ` = h0 µ 1...µ`(0)

14 For unitarity there are constraints on and the spin Kµ,P =2 µ D +2M µ h K µ P i =2h (M µ + µ D) i =2 µ. K µ = P µ ) 0 =0 ) P µ i =0 At the next level h K K P µ P i = 4 ( + 1)( µ + µ ) µ 1 2 (d 2) = 1 2 (d 2) ) P 2 i =0 For symmetric tensor fields of rank ` ` + d 2 ` =1, 2,... Equality requires µ 1...µ` Mack Dobrev P µ µµ 1...µ` 1 i =0 conserved current

15 Two point functions define the normalisation of the fields h (x) 1 (y)i = (x y) 2 ) h i =1 h (X) 1 (Y )i = (X Y ) Generalisations to spin are straightforward and involve the inversion tensor

16 Three point functions for primary operators in CFTs are determined up to a finite number of coefficients Three points can be mapped to any three points by a conformal transformation Scalars h 1 (x 1 ) 2 (x 2 ) 3 (x 3 )i = c 123 x x x x 12 = x 1 x 2 3 = 1 2 ( ) h 1 2(x) 3i = c x =1 123 Generalisations to spin are a linear algebraic problem Number of independent terms for spins `1,`2,`3 equal to the number of on shell amplitudes in d +1dim Minkowski space

17 Conserved currents correspond to amplitudes for massless particles For parity conserving amplitudes number is min `i +1 Hence for the em tensor three point function Zhiboedov 2012!! ht µ (x 1 ) T (x 2 ) T (x 3 )i there are three independent terms These correspond to the number of free CFTs in four dimensions, scalars, fermions, vectors Three point functions for higher spin currents correspond to those of free theories in even dimensions

18 The spectrum of operators, scale dimensions and spins, and the three point functions determine a CFT through the operator product expansion (OPE) The product of any two conformal primary fields is given by an expansion in terms of conformal primaries and their descendants The expansion is convergent and is determined by reproducing the three point functions (x) (0) = X O c O 1 (x 2 ) 1 2 (2 +`) µ 1...µ`O µ1...µ`(0) C,`(x, 0) µ 1...µ` = x µ 1...x µ` for two scalars O symmetric traceless tensor scale dimension = P O O,`

19 The OPE applied to the four point functions gives non trivial constraints on the spectrum of operators h (x 1 ) (x 2 ) (x 3 ) (x 4 )i = 1 (x 122 x 342 ) u = x 12 2 x 2 34 v = x 14 2 x 2 23 x 132 x 2 24 x 132 x 2 24 F (u, v) two conformal invariants Crossing symmetry F (u, v) =F (u/v, 1/v) 1 $ 2 u = F (v, u) 2 $ 4 v

20 The OPE gives h (x 1 ) (x 2 ) (x 3 ) (x 4 )i c O 2 = X (x O 122 x 342 ) 1 2 (2 +`) C,`(x 12,@ 2 ) µ 1...µ`C,`(x 34,@ 4 ) 1... ` Oµ1...µ`(x 2 ) O 1... `(x 4 ) or F (u, v) =1+ X O identity a,` G,`(u, v) a,` = c O 2 > 0 conformal partial waves or conformal blocks converges as u! 0 v! 1 This is analogous to the partial wave expansion for S-matrix amplitudes, conformal blocks replaced by single variable Legendre polynomials or their generalisations The large order behaviour of as v! 0 u! 1 a,` is constrained by reproducing the behaviour Pappadopulo,Rychkov Espin, Rattazzi, 2012 In two dimensions for minimal models this can be restricted to a finite sum if the conformal blocks are extended to Virasoro conformal blocks, related to the infinite dimensional Virasoro algebra

21 New variables u = z z v =(1 z)(1 z) F (u, v) =F(z, z) =F( z,z) Restrict coords to a plane h (x 3 ) (x 2 ) i = 1 (z z) F(z, z) x 3 =(1, 1) x 2 =(z, z) x 2 2 = z z 2 dimensional complex plane Crossing u $ v z! 1 z u! u/v, v! 1/v z! z/(z 1)

22 Equations for conformal blocks Conformal blocks are non polynomial and rather non trivial functions Conformal generators M iab i =1, 2, 3, 4 M 1AB + M 2AB + M 3AB + M 4AB h (x 1 ) (x 2 ) (x 3 ) (x 4 )i =0 The Casimir operator plays a crucial role 1 2 M ABM BA O,` = C,` O,` C,` = ( d)+`(` + d 2) This implies 1 2 M 1AB + M 2AB M BA BA M 2 (x 122 x 342 ) 1 = C,` G,` (x 122 x 342 ) G,`

23 Conformal blocks are non polynomial and rather non trivial functions In principle they are solutions of second order and fourth order PDEs 1 2 M AB M BA O,` = C,` O,` 1 2 M AB M BC M CD M DA O,` = D,` O,` imply D 2 G,` = C,` G,` D 4 G,` = D,` G,` D 2, D 4 2nd, 4th order differential operators Boundary condition as u! 0, v! 1 G,`(u, v) =u 1 2 ( `) (1 v)` 1+O(u, 1 v) Conformal blocks were discussed by Ferrara, Gatto, Grillo, Parisi in the 1970 s who obtained results in particular limits. More general expressions were obtained quite recently but we still lack compact formulae for arbitrary d

24 In terms of u, v variables are complicated, simpler in terms of D 2, D 4 z z D 2 = D z + D z +(d 2) (1 z z, z z D z =(1 z)z 2 d2 dz 2 z 2 d dz For d =2 the equation separates D z g (z) = ( 1) g (z) g (z) =z F (, ;2 ; z) G,`(u, v) =G,`(z, z) just a hypergeometric function = g 1 2 ( +`) (z) g 1 2 ( `) ( z)+g 1 2 ( +`) ( z) g 1 2 ( `) (z) symmetry under z $ z u = z z, v =(1 z)(1 z) essential as are symmetric

25 In four dimensions there is also a nice solution in terms of the same single variable hypergeometric functions since D 2 = z z z z D z + D z 2 z z z z This ensures the four dimensional conformal block is just G,`(u, v) =G,`(z, z) = z z z z g 1 2 ( +`) (z) g 1 2 ( `) 1 ( z) g 1 2 ( +`) ( z) g 1 2 ( `) 1 (z) 1 2 ( + `)( 1 2 ( + `) 1) + ( 1 2 ( `) 1)( 1 2 ( `) 2) 2= 1 2 C,` d =4 Generalisations are possible in any even dimension

26 For general dimensions there are no simple results except in various limits G,` has poles in arising from singular vectors descendants of O µ1...µ`i A vector is singular if it is a conformal primary and a descendant P µ k+1...p µ` O µ1...µ`i k =0,...` 1 O = d + k 1 which is a conformal primary tensor rank `s = k s = d + ` ( s `s) 1 2 ( `) =m =1,...,` C O,` = C s,`s e.g. angular momentum J + j, ji =0, J 2j+1 j, ji is a singular vector if 2j =0, 1, 2,...

27 Two more sets of singular vectors ( O,`)=( 1 2 d m, `)! ( s,`s) =( 1 d + m, `) 2 ( O,`)=(1 ` m, `)! ( s,`s) =(1 `, ` + m) m =1, 2, ( s `s) 1( `) =m 2 poles of the form H,`(, ) =H 1,`(, ) + X singular vectors G,` c s iterating gives a rational approximation O c s O G s,`s known by solving the Casimir equation for large 2m H s,`s(, ) c s calculable analytic in well defined at infinity H,`(, ) = +`G,`(u, v) no branch cuts z = (1 + p 4 u = z z v =(1 z)(1 z) z = 1 z) 2 (1 + ) 2 Kos Poland Simmons-Duffin following Zamolodchikov in 2d

28 Baron von Bootstrap! Munchausen fantasist and hero of the romantic age the Baron escapes a swamp by pulling on his hair, in later myth his bootstraps as in The Surprising Adventures of Baron von Munchhausen, published in English in 1785

29 Bootstrap equation Combining crossing with the conformal block expansion proposed by Polyakov in the context of CFTs in 1971 revived by Rychkov, Rattazzi, Tonni, Vichi in 2008 mini Milner prize to Rychkov in 2013 v F (u, v) =u F (v, u) 1 = X O F,`(z, z) = v G,`(u, v) u G,`(v, u) v u a,` F,`(z, z) > 0 known function There is a region in the neighbourhood of the crossing symmetric point!! u = v = 1 z = z = = =3 2 p ! where this expansion converges.

30 Can truncate the expansion by considering a finite Taylor expansion around in powers of z + z 1, (z z) 2 z = z = 1 2 F,`(z, z)! F,`;n,m n m z F,`(z, z) z= z= 1 2 restrict m n0 m0 = P,` a,` F,`;n,m a,` 0 n, m + n even, truncate to m + n apple N Truncate the spectrum of operators, These equations do not have solutions unless there are restrictions or bounds on,` This is a problem in linear programming

31 Scalar field theory in 4 dimensions Rattazzi Rychkov Tonni Vichi 2008

32 OPE El-Showk Paulos Poland Rychkov Simmons-Duffin Vichi 2012 The x axis is the y axis is the lowest in the OPE 4 scalar theory 2 free theory d =3 = =1

33 Kos Poland Simmons-Duffin 2014 Z 2 symmetry Analyse = P O + O + O+ = P O O O h h i i = P O + O + O+ h i

34 Crucial assumption, no relevant! < 3 operators other than,

35 Simmons Duffin 2015

36 Energy Momentum Tensor, this plays a critical role in CFTs The normalisation is fixed by Ward identities S d T µ (x) O(0) d O d 1 1 (x 2 ) 1 2 d S d 2 T µ (x) T (0) = C T 1 x µ x x 2 1 (x 2 ) d Iµ, (x) inversion tensor d µ O(0) generalises to non zero spin C T is a measure of the numbers of degrees of freedom, C T > 0 for unitary theories d =2 C T =2c d =4 C T = 160c = 4 3 n S +4n W + 16 n A d =3 C T = 3 2 n S +3n F C T can be determined in the conformal bootstrap from the contribution of the ` =2, =d c Ising /c free = conformal block

37 Higher dimensional perturbative CFTs Non linear sigma model L = 1 i@ i i i =1 i =1,...,N This defines a CFT for any d in a 1/N expansion i N-vector = 1 (d 2) + O(1/N ) 2 singlet =2+O(1/N ) violates unitarity bound for d>6 For d even there is a renormalisable Lagrangian which gives equivalent results in an epsilon expansion with small couplings L 4 = 1 i@ i g i i L 6 = 1 i@ + g i i L 8 = 1 i@ i 2 + g i i Diab Fei Giombi Klebanov Tarnopolsky and Gracey

38 Results for CT d =4 C T = C T free scalar N d =6 C T = C T free scalar (N +1) d =8 C T = C T free scalar (N 4) +1 from dynamical sigma -4 from higher derivative sigma There may exist non trivial unitary CFTs for d=5 (at least for sufficiently large N) but also perhaps non unitary non trivial CFTs when d=7 Similar considerations apply to the Gross-Neveu model in the large N limit but here the sigma field has dimension 1 and the theory is non unitary for d>4

39 Superconformal Symmetry Extension of conformal group to include supersymmetry In addition to the usual fermionic charges there are additional charges S, S Q, Q {Q, Q} = P {S, S} = K {Q, S} = M + D + R Lorentz Scale R-charge R-symmetry is an essential part of the superconformal group N =1 U(1) R N =2 U(2) R N =4 SU(4) R For i a superconformal primary K i = S i = S i =0 Q Supermultiplet generated by n,j,k P n Q j Qk i

40 Shortening conditions One or more of the Q, Q acting on i may give zero This gives rise to short or semi-short n =2, 4, 8, 16 1 n -BPS multiplets Such multiplets are protected is determined in terms of ` and R-symmetry representation For short multiplets ` =0 and the R-symmetry representations are restricted For 4 point functions the conformal partial wave expansion extends to one in terms of superconformal blocks For N =1short multiplets in d=4 correspond to chiral superfields with = 3 2 r

41 N =4 O20 I =2 O I 1 20 (x 1) O I 2 20 (x 2) O I 3 20 (x 3) O I 4 20 (x 4) = A I 1I 2 I 3 I 4 (u, v)! F (u, v) v 2 F (u, v) u 2 F (v, u) v 2 u 2 =1+ half BPS 20 dim short supermultiplet 1 a(u + v) 1 (x 122 x 342 ) 2 AI 1I 2 I 3 I 4 (u, v) superconformal Ward identity protected short multiplet contribution bootstrap equation a central charge Beem Rastelli van Rees 2013 c = a = 1 4 dim G G gauge group Expand F (u, v) over the sum of protected and unprotected superconformal blocks with positive coefficients Get constraints on potential `

42 Conjecture: the corner of the cube corresponds to a self dual point under Sl(2, Z) = i g 2 YM = e 1 2 i,e 1 3 i Such self dual points are difficult to access by other methods

43 Minkowski approach (x) (0) = X O c O C,`(x, 0) µ 1...µ` = x µ 1...x µ` 1 (x 2 ) 1 2 (2 +`) µ 1...µ`O µ1...µ`(0) In the light cone limit x 2! 0 the OPE is essentially an expansion in which operators of equal twist contribute equally ` = This limit is relevant for deep inelastic scattering where the twist controls the approach to scaling in the deep inelastic limit Positivity requires that the twist is a convex function `3 `3 `1 `1 apple `2 `2 `1 `1 ` `1 <`2 <`3 Nachtmann 1973

44 Such results have been refined using a bootstrap type approach relating s and t channel expansions in the light cone limit ho 1 (x 1 ) O 2 (x 2 ) Ō1(x 3 ) Ō2(x 4 )i has an s channel expansion in terms of operators with leading twist multi trace operators ` = O1 + O2 C ` min min is the minimum twist of operators appearing in the t channel expansion Fitzpatrick Kaplan Poland Simmons-Duffin Komargodski Zhiboedov 2012(Dec) d>2 large ` C is also calculable

45 Emergent symmetries In d=3 the Ising model appears to be the unique CFT with two relevant fields, and Z 2 symmetry Does Z 2 emerge in the RG at the fixed point? no 3 interaction In d=4 Lagrangian theories with IR fixed points defining a CFT require gauge fields and large numbers of fermions and so should have a large flavour symmetry group In d=6 a conjecture would be that all non trivial CFTs have superconformal symmetries. d=6 superconformal theories have no relevant scalar

46 1. Understand the origin of kinks in bootstrap bounds in terms of decoupling of particular states. Why are there islands? Maybe there is a more analytic approach. 2. Show there are no non trivial unitary CFTs for d>6. This is true for SCFTs for algebraic, representation theory reasons (Nahm). 3. Are there CFTs with large anomalous dimensions or which are not a deformation of a free Lagrangian theory? This might be relevant to extending ideas of naturalness. 4. Construct conformal blocks with external spins. No succinct formulae exist as yet, but there are now bootstrap calculations for fermions in d=3 and significant results for d=4. Echeverri Elkhidir Karateev Serone Bootstrap < T T T T>, very hard.

47 N =1 bootstrap, chiral superfields Apply bootstrap to h i free theory point 2 =0 Poland Stergiou 2015 SCFT N =1 d =4 24c = n C +3n V SU(N c ) N f C T = 160c c free = 1 24 c interacting 1 9 flavours SQCD 24c =2N c N f + 3(N c 2 1) 3 2 N c apple N f apple 3N c Difficult to accommodate with gauge fields

48 Two dimensional CFTs are not always a good guide to higher dimensions. We have no idea as to any classification in d=3,4 Perhaps we will have a new yellow book, the bible for 2dim CFTs, but not yet I would like to remember Francis Dolan with nearly all my work on CFTs was done