Some Aspects of Large- N Vector Models and their Higher-Spin - PowerPoint PPT Presentation

Motivations In this talk: Vector models exhibit global and discrete symmetry breaking. The bosonic model O ( N ) ! O ( N � 1) . The fermionic model parity breaking. If there is holography without strings and branes, what is the bulk counterpart of the global O ( N ) boundary symmetry and its breaking pattern? 1. Emphasise the role of singletons for vector model holography: Using a bulk singleton deformation I will reproduce the boundary gap equation that describes the O ( N ) ! O ( N � 1) breaking. Using a further boundary deformation I will reproduce the known O (1 /N ) anomalous dimension of the elementary scalars in the boundary. The latter result raises the issue whether O ( N ) symmetry breaking is related to higher-spin symmetry breaking. I will discuss the singleton deformation in the fermionic model. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 6 / 54

Motivations 2. Discuss the OPE techniques in the vector models: I will review sketch the calculations of couplings, anomalous dimensions and ”central charges” in the bosonic and fermionic vector models. I will discuss sketch mention the problems extending such techniques to the bulk. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 7 / 54

Outline Motivations 1 O ( N ) vector models: review 2 O ( N ) ! O ( N � 1) symmetry breaking in the bosonic model The fermionic O ( N ) vector model: lightning review Anomalous dimensions O ( N ) /HS holography 3 The gap equations from holography The singleton deformation of higher-spin theory and boundary symmetry breaking Aspects of the OPE in O ( N ) vector models 4 The conformal partial waves: free field theory The skeleton graphs Summary and outlook 5 A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 8 / 54

O ( N ) vector models: review N -elementary (Euclidean) scalar fields φ a ( x ) subject to a constaint L = 1 Z d 3 x ∂ µ φ a ∂ µ φ a , φ a φ a = N g , a = 1 , 2 , ..N . 2 g ! 0 is the free field theory limit which lies in the UV. Introduce a Lagrange multiplier ρ and integrate the φ ’s to obtain Z ( D ρ ) e � NS eff ( ρ ) , S eff ( ρ ) = 1 Z d 3 x ρ 2Tr ln( � ∂ 2 + ρ ) � Z = 2 g gap equation The saddle point at large- N , with constant ρ 0 = m 2 , yields the d 3 p ∂ S eff ( ρ ) ρ 0 = 0 ) 1 Z 1 � g = � p 2 + ρ 0 (2 π ) 3 ∂ρ � The large- N expansion is obtained setting 1 ρ ( x ) = ρ 0 + p σ ( x ) , N A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 9 / 54

O ( N ) vector models: review The e ff ective action S N eff ( σ , ρ 0 ) for the real fluctuations σ is 1 2 g ( V ol ) 3 + 1 2Tr ln( � ∂ 2 + ρ 0 ) � ρ 0 N S N S eff ( ρ ) = eff ( σ , ρ 0 ) 1 Z S N eff ( σ , ρ 0 ) = σ ( x ) ∆ 3 ( x, y ; ρ 0 ) σ ( y ) 2 1 Z p + σ ( x ) σ ( y ) σ ( z ) P 3 ( x, y, z ; ρ 0 ) + .. 3! N The kernels ∆ 2 ( x, y ; ρ 0 ) , P 3 ( x, y, z ; ρ 0 ) .. are constructed using propagators of the φ ’s only. The generating functional W [ η ] for connected correlation functions of σ is Z e W [ η ] ⌘ ( D σ ) e � S N eff ( σ , ρ 0 )+ R ησ A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 10 / 54

O ( N ) vector models: review Using a UV cuto ff Λ the gap equation becomes ✓ 1 p ◆ | ρ 0 | � 1 1 Λ = + O ( ρ 0 / Λ ) , = 2 π 2 g ⇤ g 4 π g ⇤ p For g > g ⇤ , we are in the massive phase with m = | ρ 0 | 6 = 0 . For g = g ⇤ there is no mass scale left and we describe the critical O ( N ) vector model. For g < g ⇤ , ρ 0 = 0 but an arbitrary mass scale remains - the subtraction point of renormalisation - and we enter a symmetry broken phase. The O ( N ) symmetry is broken once we depart from the free theory, and it is restored at the nontrivial fixed point. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 11 / 54

O ( N ) vector models: review A clearer way to see the O ( N ) ! O ( N � 1) symmetry breaking pattern is to separate out the N ’th component of φ a ’s, which we denote as φ . Integrating over the remaining N � 1 elementary scalars we obtain Z [ D φ ][ D ρ ] e � ( N � 1) S eff ( ρ , φ ) Z = The e ff ective action is now defined as 1 Z d 3 x φ ( � ∂ 2 + ρ ) φ S N � 1 S eff ( φ , ρ ) = eff ( ρ ) + 2( N � 1) 1 N Z d 3 x ρ 2Tr ln( � ∂ 2 + ρ ) � S N � 1 eff ( ρ ) = ( N � 1) 2 g Apart from the di ff erent N scaling of the coupling constant g : the e ff ective action S N � 1 eff ( ρ ) is essentially the same as S eff ( ρ ) . A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 12 / 54

O ( N ) vector models: review The large- N expansion is now performed as 1 ρ ( x ) = ρ 0 + p N � 1 σ ( x ) , φ ( x ) = φ 0 + ϕ ( x ) . with ρ 0 , φ 0 determined by the modified gap equations φ 2 d 3 p ∂ S eff N 1 Z 1 � 0 ( φ 0 , ρ 0 ) = 0 ) N � 1 = g � � p 2 + ρ 0 (2 π ) 3 ∂ρ ( N � 1) � ∂ S eff � ( φ 0 , ρ 0 ) = 0 ) ρ 0 φ 0 = 0 � ∂φ � A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 13 / 54

O ( N ) vector models: review The resulting e ff ective action is then written as 1 N � 1 S N � 1 S eff ( φ , ρ ) = V eff ( φ 0 , ρ 0 ) + eff ( ϕ , σ ) eff ( σ , ρ 0 ) + 1 Z S N � 1 S N � 1 eff ( ϕ , σ ) = ϕ ( x ) D 0 ( x, y ; ρ 0 ) ϕ ( y ) 2 1 Z φ 0 Z σ ( x ) ϕ 2 ( x ) + + p p σ ( x ) ϕ ( x ) 2 N � 1 N � 1 O ( N ) ! O ( N � 1) symmetry breaking pattern The e ff ective action for the O ( N ) model the e ff ective action of the σϕ 2 and a R R O ( N � 1) model by integrating-in ϕ with a ϕσ interaction. At the critical point ρ 0 = φ 0 = 0 , one integrates-in a massless elementary scalar ϕ ( x ) with marginal interaction. This shifts N � 1 ! N . A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 14 / 54

O ( N ) vector models: review The modified gap equation is written φ 2 ✓ N 1 g � 1 ◆ + | m | 0 N � 1 = 4 π + · · · N � 1 g ⇤ φ 0 and | m | cannot be simultaneously nonzero and | m | < Λ . When g < Ng ⇤ / ( N � 1) , | m | = 0 but φ 0 6 = 0 ) O ( N ) is broken to O ( N � 1) . The N � 1 Goldstone bosons are the massless elementary scalars that were integrated out. When the coupling is tuned to N g = N � 1 g ⇤ > g ⇤ we have φ 0 = m = 0 and we arrive at the critical O ( N ) vector model. As the coupling increases to g > Ng ⇤ / ( N � 1) , we have φ 0 = 0 , but then we enter the O ( N ) symmetric phase with m = 2 Λ ✓ N g ⇤ ◆ 1 � , π N � 1 g A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 15 / 54

N − 1 N N + 1 0 g g ∗ N + 2 N + 1 g ∗ N + 1 g ∗ N N N − 1 g ∗ + ϕ + ϕ Figure: The phase diagram of the vector models. Stars denote the CFTs. The solid arrows denote marginal deformations towards the IR fixed point after the absorption of an elementary scalar ϕ . The dotted arrows denote irrelevant double-trace deformations leading to the UV fixed point of the symmetry enhanced theory. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 16 / 54

O ( N ) vector models: review When g < Ng ⇤ / ( N � 1) we assign the di ff erence φ 2 N 1 g � 1 0 = N � 1 6 = 0 N � 1 g ⇤ R to an expectation value of φ 0 . Then the linear interaction term φ 0 σϕ is nontrivial and we can shift the scalar fluctuation as φ 0 1 p ϕ = ˆ ϕ + � ∂ 2 σ , N � 1  φ 2 � S N � 1 Z eff ( σ , 0)+ 1 R 1 R ϕ 2 � 0 R � ∂ 2 σ 2 + .. 1 ϕ D 0 ˆ ˆ ϕ + σ ˆ � 2 p N � 1 2 2( N � 1) Z ⇠ e . The last term in the exponent is a nonlocal version of the irrelevant σ 2 which drives the theory in the UV where we R double-trace deformation expect to find the free O ( N ) model. A richer picture arises in d = 5 which can be compared to the recent results of [Giombi, Klebanov et. al. (14)] . A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 16 / 54

O ( N ) vector models: review The Euclidean action for the three-dimensional Majorana fermions Z  1 ∂ ψ i + G � ψ i / ψ = ψ T σ 2 , i = 1 , 2 , ..N. ¯ 4 N ( ¯ , ¯ d 3 x ψ i ψ i ) 2 S = � 2 G has dimensions of inverse mass, hence the G ! 0 free theory lies in the IR. ψ i ψ i signifies parity breaking. An expectation value for σ ⇠ G ¯ With respect to a critical coupling 1 = Λ π 2 , G ⇤ we have 1) for G < G ⇤ , σ = 0 and parity is unbroken, 2) for G = G ⇤ we are at the fermionic critical O ( N ) fixed point that lies in the UV and 3) for G > G ⇤ , σ 6 = 0 and hence parity is broken. We can also show that the critical O ( N ) GN model arises from the critical O ( N � 1) GN model by integrating-in elementary fermions with a marginal σ ¯ ψψ interaction. Starting from G < G ⇤ an additional double-trace relevant coupling is induced that drives the theory at its IR free fixed point. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 18 / 54

O ( N ) vector models: review The systematic 1 /N expansion leads to the calculation of anomalous dimensions [e.g. A. Vasiliev et. al. (81-81), Gracey (91-92), R¨ uhl et. al. (92-93), T. P. (94-96)] . From conformal invariance we have C φ C σ x 2 ∆ φ δ ab , h φ a ( x ) φ b (0) i = h σ ( x ) σ (0) i = x 2 ∆ σ We fix d = 3 and define three critical indices γ φ , κ and z of order O (1 /N ) as ∆ φ = 1 φ C σ = 1 C 2 2 + γ φ , ∆ σ = 2 � 2 γ φ � 2 κ , π 4 + z The two-point function of φ a is given by h φ a ( x ) φ b (0) i = 1 1  1 � 1 4 � 4 1 3 π 2 ln | x | 2 + ... δ ab ) γ φ = 4 π | x | N 3 π 2 N For the calculations of κ and ζ one needs to consider the 2-pt function of σ and also the renormalisation of the vertex σφ 2 . The most updated results are already a few decades old. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 20 / 54

The HS/ O ( N ) holography Practically, O ( N ) /HS correspondence proceeds by considering the bulk action 1 d 4 x p� g 1  ⇤ s � 1 � Φ ( s ) + O ( 1 Z L 2 ( s 2 � 2 s � 2) X 2 Φ ( s ) I HS = p ) N s =0 , 2 , 4 ,.. Φ ( s ) are symmetrized and double-traceless rank- s tensors, ⇤ s are generalized Pauli-Fierz operators on the fixed AdS 4 background metric g µ ν . There is also a ”mass” term necessary to maintain HS gauge invariance. The quadratic part of I HS yields the two-point functions of all free higher-spin currents normalized to O (1) . The free boundary theory is obtained by the alternative quantisation (AQ) of the conformally coupled scalar Φ (0) . The standard quantisation (SQ) gives the non-trivial fixed point. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 22 / 54

O ( N ) /HS holography AdS/CFT yields the renormalized boundary generating functional W en [ J ] , wherefrom we get the e ff ective action Γ [ h O i ] by a Legenrde transform. A Lagrangian deformation of the boundary field theory action by a functional f ( O ) corresponds - at least at large- N - to a simple deformation of the e ff ective action Γ f [ σ ] = Γ 0 [ σ ] + f ( σ ) , σ = h O i . Given such a deformation the boundary gap equation reads δ Γ f [ σ ] � σ = σ ⇤ = 0 � δσ � A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 23 / 54

O ( N ) /HS holography We extend the bulk theory by a second scalar with m 2 L 2 = � 2 d 4 x p� g 1  ⇤ + 2 � Z I extHS = I HS + Σ . 2 Σ L 2 We take Φ (0) ⌘ Φ in AQ, and Σ in SQ. Asymptotically, we have Φ ⇠ α z + β z 2 , Σ ⇠ η z + σ z 2 Φ yields a ∆ = 1 operator with vev α , Σ yields a ∆ = 2 operator with vev σ . We assume that these fields do not mix in the bulk. This means that the regularity conditions of the bulk equations yield α = α ( β ) and σ = σ ( η ) , and determine the boundary generating functional as Z Z I extHS ! W [ β , η ] = α ( β ) β � σ ( η ) η . the di ff erent relative signs arising due to the opposite quantizations used. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 24 / 54

O ( N ) /HS holography To get the boundary gap equation we introduce boundary terms that couple the two fields together i.e. a Lagrangian deformation of the form Z ✓ ασ + V ( σ ) � 1 ◆ , V ( σ ) = � λ 0 3 λ ( α � h ) 3 f ( α , σ ) = g σ . with λ and λ 0 dimensionless and h is a parameter with dimensions of mass. We then obtain Z ✓ 1 1 σ + σ ( α � λ 0 ◆ 2 α K 1 α � 1 g ) � 1 2 σ K � 1 3 λ ( α � h ) 3 Γ [ α , σ ] = where K 1 is an appropriate kernel. For constant α and σ , we obtain the gap equations λ 0 α = g λ ( α � h ) 2 σ = p The first equation corresponds to the model’s constraint and gives λ 0 = N . A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 25 / 54

O ( N ) /HS holography The second equation can be rewritten as p p p p λ = 16 π 2 p σ N N N N , h = ) = ± N g ⇤ g g ⇤ 4 π Keeping the minus sign, this coincides with the vector model’s gap equation. The free UV fixed point is reached taking g, λ ! 0 and the cuto ff to infinity, whereby σ decouples and only α (the ∆ = 1 operator) survives. The nontrivial IR fixed point arises when g ! g ⇤ . In this case, the introduction of the operator α is equivalent to a finite shift of the operator σ ) the operator α becomes redundant. ( α � h ) 3 corresponds to the classically marginal term ( φ a φ a ) 3 . h introduces relevant terms in order that the non-trivial fixed point is properly described and appears at a finite value of g . A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 26 / 54

O ( N ) /HS holography Next, we deform the higher-spin action by a singleton field S as d 4 x p� g 1 Z  5 � I dHS = I extHS + 2 S ⇤ + S , 4 L 2 with asymptotic behaviour S ⇠ ξ z 1 / 2 + φ z 5 / 2 . For S the only possible unitary quantization is to use AQ [e.g. Ohl and Uhlemann (12)] giving a free boundary operator of ∆ = 1 / 2 that decouples from the rest of the CFT. To force S interact with the other HS’s we deform the bulk theory by ˜ Z  V ( σ ) � λ 1 � λ 0 3 ( α � h ) 3 + ˜ ασ � ˜ ˜ λσφ 2 f d ( α , σ , φ ) = , V ( σ ) = g σ , where λ 0 ! ˜ λ 0 = N +1 N is required in order to absorb φ by suitably adjusting p the coupling 1 /g in the massive phase of the theory. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 28 / 54

O ( N ) /HS holography The gap equations are then N + 1 1 α + ˜ λφ 2 p = g N p ! 2 16 π 2 N = α � σ N g ⇤ ˜ = 0 λφσ The third equation is familiar from the σ -model: there are two phases, one in which φ = 0 (massive phase) and the other in which σ = 0 (broken phase). The first equation has an O ( N + 1) -invariant form if we interpret α ⇠ h φ a φ a i and φ ⇠ h φ N +1 i . Substituting then α we find p p p σ . λφ 2 = N + 1 1 N N ˜ p g � + 4 π 2 g ⇤ N p Setting ˜ λ = 1 / N this coincides exactly with field theory gap equation. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 29 / 54

O ( N ) /HS holography At the critical point α is redundant and the boundary term becomes 1 Z σφ 2 . f d ( σ , φ 2 ) = p N This is a simple marginal deformation of the extended higher-spin action and leads to a 1 /N expansion for the boundary two-point functions of φ and σ . For example, we obtain h φ ( x 1 ) φ ( x 2 ) i def = h φ ( x 1 ) φ ( x 2 ) i 0 + 1 Z h φ ( x 1 ) φ ( x 2 ) σ ( x ) φ 2 ( x ) σ ( y ) φ 2 ( y ) i 0 + · · · 2 N p where we have dropped the O (1 / N ) term whose contribution vanishes, as do all other fractional powers of 1 /N . The above is the same expansion as in the field theory analysis for φ , at least to leading order in 1 /N , and hence it gives the same anomalous dimension. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 30 / 54

O ( N ) /HS holography This is despite the fact that the deformation may be regarded as a marginal deformation of the IR O ( N ) fixed point in the presence of an additional scalar φ . Generally, the graphical expansion for φ and σ generated by the deformation above is the same as the graphical expansion for φ a and σ generated by the boundary field theory ! hence yields the same anomalous dimensions. The moral At least to leading order in 1 /N , the bulk HS theory is deformed by throwing in singletons ! these simply are pushed in the boundary, hence we have N ! N + 1 . We need, however, some nontrivial bulk interaction with the HSs in order to induce the necessary boundary terms to glue the extra field to the rest. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 31 / 54

Aspects of the OPE in O ( N ) vector models We will focus on the following 4-pt functions h φ a ( x 1 ) φ b ( x 2 ) φ c ( x 3 ) φ d ( x 4 ) i Φ abcd ( x 1 , x 2 , x 3 , x 4 ) ⌘ δ ab δ cd Φ S ( x 1 , x 2 , x 3 , x 4 ) = E [ ab,cd ] Φ A ( x 1 , x 2 , x 3 , x 4 ) + T ( ab,cd ) Φ st ( x 1 , x 2 , x 3 , x 4 ) + h φ a ( x 1 ) φ b ( x 2 ) O ( x 3 ) O ( x 4 ) i ⌘ δ ab Φ φ O ( x 1 , x 2 , x 3 , x 4 ) h O ( x 1 ) O ( x 2 ) O ( x 3 ) O ( x 4 ) i ⌘ Φ O ( x 1 , x 2 , x 3 , x 4 ) These will be functions of v and Y related to the usual conformal ratios as u = x 2 12 x 2 v = x 2 12 x 2 , Y = 1 � v 34 34 , x 2 13 x 2 x 2 14 x 2 u 24 23 with v, Y ! 0 as x 2 12 , x 2 34 ! 0 . A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 33 / 54

Aspects of the OPE in O ( N ) vector models We would need the following OPEs δ ab  � 1 + g φφ O s φ a ( x 1 ) φ b ( x 2 ) X = [ O s ( x 2 )] , 12 ) ∆ φ � 1 ( x 2 C O s 2 ∆ s ∆ s g φφ O [ cd ] E [ ab,cd ] X [ O [ cd ] + s ( x 2 )] s 12 ) ∆ φ � 1 ( x 2 2 ∆ 0 C O [ cd ] s ∆ 0 s s g φφ O ( cd ) T ( ab,cd ) X [ O ( cd ) + s ( x 2 )] , x 12 = x 1 � x 2 s 12 ) ∆ φ � 1 ( x 2 C O ( cd ) 2 ∆ 00 s ∆ 00 s s The [ O s ] ’s represent the full contributions (i.e. including descendants). The C O s ’s are the 2-pt function normalisation constants and the g φφ O s ’s are the corresponding 3-pt function couplings. We normalized to one the 2-pt function of the φ a ’s. The above OPE represents a converging series in the limit x 12 ! 0 limit. A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 34 / 54

Aspects of the OPE in O ( N ) vector models We would also need " # [ F a ( x 2 )] 1 g φφ O [ φ a ( x 2 )] + g φ O F φ a ( x 1 ) O ( x 2 ) = + .. ∆ φ + ∆ ∆ φ 12 ) � ∆ F C F ( x 2 ( x 2 ( x 2 12 ) 12 ) � 2 2 2 " # 1 C O + g O [ O ( x 2 )] 2 + g OO T C µ ν [ T µ ν ( x 2 )] O ( x 1 ) O ( x 2 ) = + .. x 2 ∆ 12 ) � ∆ 12 ) � d C O ( x 2 C T ( x 2 2 12 A. C. Petkou (AUTH) Vector Models and HS Holography GGI Florence, 16 March 2015 35 / 54

Some Aspects of Large- N Vector Models and their Higher-Spin - PowerPoint PPT Presentation

Some Aspects of Large- N Vector Models and their Higher-Spin Holography Anastasios C. Petkou Institute of Theoretical Physics Aristotle University of Thessaloniki Research co-financed by Greek national funds through the Operational Program

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