Generalized Weyl anomalies in higher-spin theory Wei Li - PowerPoint PPT Presentation

Generalized Weyl anomalies in higher-spin theory Wei Li (ITP-Chinese academy of sciences) Trieste ， Mar 24, 2016

Main question ✤ CFT has conformal anomaly ✤ CFT with higher-spin symmetry has generalized conformal anomalies (in addition to conformal anomaly) ✤ Today: compute boundary higher-spin conformal anomalies from bulk higher-spin theory

Reference ✤ Some aspects of holographic W-gravity JHEP 1508, 035 (2015) with Stefan Theisen .

Plan 1. Generalized conformal anomaly in CFT with higher- spin symmetry 2. Bulk computation of boundary anomaly 3. Discussion

Conformal anomaly in CFT Even dimensional CFT in curved background g ij , ϕ ijk , . . . T i W i classical: i = 0 ij = 0 . . . h T i h W i quantum mechanical: i i 6 = 0 ij i 6 = 0 . . . Capper Du ff ’73 Generating function of conformal anomalies I Integrate out CFT fields to obtain (non-local) e ff ective action Z e − W [ g ] = D Φ e − S CFT [ Φ ,g ] I Weyl transformation δ σ 2 g ij = 2 σ 2 g ij δ σ 2 ϕ ijk = 4 σ 2 ϕ ijk p g σ 2 ( x ) h T i Z δ σ 2 W [ g ] = i i 6 = 0 I Additional anomalous symmetry: W -Weyl transformation p g σ 3 ( x ) A 3 6 = 0 Z δ σ 3 W [ g, ϕ ] =

Conformal anomaly in CFT with higher-spin symmetry Even dimensional CFT in curved background g ij , ϕ ijk , . . . T i W i classical: i = 0 ij = 0 . . . h T i h W i quantum mechanical: i i 6 = 0 ij i 6 = 0 . . . Capper Du ff ’73 Generating function of conformal anomalies I Integrate out CFT fields to obtain (non-local) e ff ective action Z e − W [ g, ϕ ] = D Φ e − S CFT [ Φ ,g, ϕ ] I Weyl transformation δ σ 2 g ij = 2 σ 2 g ij δ σ 2 ϕ ijk = 4 σ 2 ϕ ijk p g σ 2 ( x ) A 2 6 = 0 Z δ σ 2 W [ g, ϕ ] = I Additional anomalous symmetry: W -Weyl transformation Z p g σ 3 ( x ) A 3 6 = 0 δ σ 3 W [ g, ϕ ] =

Weyl anomalies in 2D CFT (from 2-point function in flat background) Conserved currents T ij ( ⌘ W (2) W (3) W (4) ij ) . . . ijk ijkl Naively ∂ i T ij = 0 η ij T ij = 0 and Anomalous Ward Identity 1. Symmetry and conservation gives h T ij ( p ) T kl ( � p ) i = A ( p 2 ) p i p j � η ij p 2 �� p k p l � η kl p 2 � � 2. Incompatible with conformal symmetry: A ( p 2 ) = 0 η ij T ij = 0 = ) = ) h T ij ( p ) T kl ( � p ) i = 0 3. Give up conformal symmetry: A ( p 2 ) = c p 2

W -Weyl anomalies in 2D W -CFT (from 2-point function in flat background) Conserved currents T ij ( ⌘ W (2) W (3) W (4) ij ) . . . ijk ijkl Naively ∂ i W i ··· = 0 η ij W ij ··· = 0 and Anomalous Ward Identity 1. Symmetry and conservation gives h W ijk ( p ) W lmn ( � p ) i = A (3) ( p 2 ) p i p l � η il p 2 �� p j p m � η jm p 2 �� p k p n � η kn p 2 � ⇥� ⇤ + . . . 2. Incompatible with W -conformal symmetry: A (3) ( p 2 ) = 0 η ij W ijk = 0 = ) = ) h W ijk ( p ) W lmn ( � p ) i = 0 3. Give up W -conformal symmetry: A (3) ( p 2 ) = c (3) p 2

Weyl and W -Weyl anomalies from OPE OPE of holomorphic currents: c T ( z ) T ( w ) ∼ ( z − w ) 4 + . . . c (3) W ( z ) W ( w ) ∼ ( z − w ) 6 + . . . . . . c ( s ) W ( s ) ( z ) W ( s ) ( w ) ∼ ( z − w ) 2 s + . . . Each spin gives one W s -Weyl anomaly

Generating function of Weyl anomaly ✤ Without higher-spin fields, 2D effective action is uniquely given by Polyakov action R 1 Z W 2D [ g ] = ⇤ R ✤ Analogue of Polyakov action for other cases is not known 4D CFT ? Deser Schwimmer ’93; Deser ’96,’99 2D CFT with higher-spin symmetry ? Computing effective action from CFT is one-loop

Bndy Weyl anomaly from bulk Henningson Skenderis '98 go on-shell boundary action bulk action Weyl variation Weyl anomaly

Bndy Weyl anomaly from bulk Henningson Skenderis '98 go on-shell boundary action bulk action Weyl variation Weyl anomaly Bulk computation of boundary Weyl anomaly is Classical

Bndy Weyl anomaly from bulk original procedure Henningson Skenderis '98 go on-shell boundary action bulk action Weyl variation Weyl anomaly Weyl variation of on-shell bulk action gives Weyl anomaly

Bndy Weyl anomaly from bulk PBH procedure Imbimbo Schwimmer Theisen Yankielowicz '99 go on-shell boundary action bulk action Weyl variation PBH transf. go on-shell variation of Weyl anomaly bulk action

PBH procedure for pure gravity Step-1: Fe ff erman-Graham gauge of bulk metric (asympt. AdS 2 n +1 ) ds 2 = d ρ 2 ρ 2 + 1 ρ 2 g ij ( ρ , x ) dx i dx j (0) g ij ( x ) + ρ 2 (2) g ij ( x ) + ρ 4 (4) FG expansion : g ij ( ρ , x ) = g ij ( x ) + . . . Step-2: PBH transformation ≡ Bulk di ff eo ξ µ preserving FG gauge = Boundary Weyl Step-3: Weyl anomaly from PBH transformation on bulk action p Z ⇣ δ ξ S bulk ⌘ h i h T i d d x σ ( x ) i i = | on-shell = � 2 G L ( G ) | ρ =0 , on-shell ρ ∂ M Step-4: rewrite in terms of boundary data (i.e. go on-shell)

Bndy Weyl anomaly from bulk PBH procedure Imbimbo Schwimmer Theisen Yankielowicz '99 bulk action PBH transf. go on-shell variation of Weyl anomaly bulk action PBH procedure: a easy way to compute Weyl anomaly from bulk

What is higher-spin theory ? ✤ gravity theory coupled to higher-spin gauge symmetry ✤ dual CFT with higher-spin currents

Why higher-spin ? ✤ Higher-spin gravity (Vasiliev’s theory) is an interesting extension of Einstein gravity ✤ Holography with higher-spin symmetry is different from traditional Gauge/Gravity duality ✤ Can use higher-spin symmetry to study string theory

Problem: there is no covariant metric-like formulation of higher-spin theory

We want metric-like formulation of higher-spin theory pure gravity (metric formulation)

We want metric-like formulation of higher-spin theory pure gravity Diffeomorphism invariance Riemannian geometry (metric formulation)

We want metric-like formulation of higher-spin theory pure gravity Diffeomorphism invariance Riemannian geometry (metric formulation) Diffeo is coupled to higher-spin higher-spin gauge transformation (metric-like formulation) What is higher-spin geometry?

We want metric-like formulation of higher-spin theory pure gravity Diffeomorphism invariance Riemannian geometry (metric formulation) ? Diffeo is coupled to higher-spin higher-spin gauge transformation (metric-like formulation) What is higher-spin geometry?

We want metric-like formulation of higher-spin theory pure gravity Diffeomorphism invariance Riemannian geometry (metric formulation) ? Diffeo is coupled to higher-spin higher-spin gauge transformation (metric-like formulation) What is higher-spin geometry? Campoleoni Fredenhagen Pfenninger Theisen '12

Chern-Simons formulation of higher-spin theory pure gravity pure gravity (metric formulation) sl(2) Chern-Simons higher-spin sl(N) Chern-Simons

3D higher Spin theory in AdS 3 — Action Action: Tr[ AdA + 2 S CS [ A ] = k Z S = S CS [ A ] − S CS [ ˜ 3 A 3 ] A ] with 4 π M Lorentzian: A, ˜ A ∈ sl ( N, R ) Euclidean: A, ˜ A ∈ sl ( N, C ) and ˜ A = − A † Translation to metric-like formalism 1. Dreibein and spin connection e = A − ˜ ω = A + ˜ A A 2 2 2. metric and higher-spin fields G µ ν = Tr[ e µ e ν ] ϕ µ νρ = Tr[ e { µ e ν e ρ } ] . . .

3D higher Spin theory in AdS 3 — Spectrum Spectrum 1. Choose an sl (2) subalgebra that corresponds to spin-2: spin-2 : { L 1 , L − 1 } L 0 , 2. Decompose sl ( N ) in terms of irreps of the gravitonal sl (2) { W ( s ) spin-s : m } m = − s + 1 , . . . , s − 1 Principal embedding: 1 spin- s field for each s = 2 , ..., N L 1 L 0 L − 1 W 2 W 1 W 0 W − 1 W − 2 highest/lowest weight modes

3D higher Spin theory in AdS 3 — Spectrum Spectrum 1. Choose an sl (2) subalgebra that corresponds to spin-2: spin-2 : { L 1 , L − 1 } L 0 , 2. Decompose sl ( N ) in terms of irreps of the gravitonal sl (2) { W ( s ) spin-s : m } m = − s + 1 , . . . , s − 1 Principal embedding: 1 spin- s field for each s = 2 , ..., N L 1 L 0 L − 1 W 2 W 1 W 0 W − 1 W − 2 lowest weight/zero/highest weight modes

From Chern-Simons to metric-like formulation pure gravity pure gravity (metric formulation) sl(2) Chern-Simons ? higher-spin higher-spin (metric-like formulation) sl(N) Chern-Simons

From Chern-Simons to metric-like formulation pure gravity pure gravity (metric formulation) sl(2) Chern-Simons ? higher-spin higher-spin (metric-like formulation) sl(N) Chern-Simons not at the level of action

From Chern-Simons to metric-like formulation Li Theisen '15 PBH procedure in PBH procedure in pure gravity sl(2) Chern-Simons (metric formulation) PBH procedure in sl(N) Chern-Simons

From Chern-Simons to metric-like formulation Li Theisen '15 PBH procedure in PBH procedure in pure gravity sl(2) Chern-Simons (metric formulation) PBH procedure in sl(N) Chern-Simons Weyl and W-Weyl anomalies in sl(N) Chern-Simons

PBH procedure for sl (2) Step-1: Fe ff erman-Graham gauge of sl (2) (i.e. pure gravity)  = ρ L 0 a ( x ) ρ − L 0 − d ρ A ( ρ , x ) ρ L 0  ˜ = ρ − L 0 ˜ a ( x ) ρ L 0 + d ρ A ( ρ , x ) ρ L 0  with Tr[ L 0 ( a − ˜ a )] = 0 U ( ρ , x ) = ρ L 0 u ( x ) ρ − L 0 Step-2: PBH transformation for sl (2): with u 2 = σ 2 L 0 Step-3: Weyl anomalies from PBH transformation on bulk action z − ¯ ⇥ � �⇤ Weyl Anomaly = c Tr L 0 ∂ a ¯ ∂ a z Step-4: rewrite in terms of boundary data