The double-trace spectrum of planar N = 4 SYM: an unexpected 10d - - PowerPoint PPT Presentation

The double-trace spectrum of planar N = 4 SYM: an unexpected 10d conformal symmetry

[arXiv: 1706.08456, 1802.06889]

• F. Aprile
• J. M. Drummond
• P. Heslop
• H. P.

Tuesday Seminar, University of Southampton, 20/11/2018

Outline

Motivation & General Setup Preliminaries Half-BPS operators Operator Product Expansion The double-trace spectrum Unexpected 10d conformal symmetry Conclusions

Motivation

(i) Perturbative Quantum Gravity in AdS ◮ Want to study quantum corrections to supergravity: consider loops in AdS5 ◮ Even tree level computations in AdS are hard: supergravity correlation functions only recently computed in full generality

[Rastelli-Zhou’16, Arutyunov-Klabbers-Savin’18]

◮ Loops in AdS are even harder: so far only few examples of loop diagrams explicitly computed (φ3, φ4-theory)

[Aharony-Alday-Bissi-Perlmutter’16] [Yuan’17’18, Cardona’17, Ghosh’18, ...]

◮ Consistency checks of AdS/CFT correspondence

Motivation

(ii) Spectrum of (S)CFT’s ◮ Study spectrum of operators and their three-point functions − → recent interest from ”bootstrap program”

[Rattazzi-Rychkov-Tonni-Vichi’08 ...]

◮ Bootstrap approach to ”solving” N = 4 SYM in a 1/N2 expansion ◮ N = 4 SYM: connections to integrable systems away from the planar limit

[Bargheer-Caetano-Fleury-Komatsu-Vieira’17]

General Setup

AdS/CFT correspondence N = 4 SYM with ⇐ ⇒ Supergravity on AdS5×S5 gauge group SU(N) 4pt correlation functions ⇐ ⇒ AdS amplitudes (Witten diagrams) strong coupling (λ → ∞) ⇐ ⇒ weak coupling →

1 N2 expansion

◮ Interested in loop corrections to supergravity in AdS5 − → mixing problem of double-trace operators

General Setup

single-trace operator O2(x) ⇐ ⇒ graviton multiplet higher charge operators Op(x) ⇐ ⇒ Kaluza-Klein modes

H = + + + + +...

↓

• ↓
• ↓

N0 1 N2 1 N4 free field classical supergravity quantum corrections theory

• to SUGRA

Preliminaries

Simplest operators to consider: 1

2-BPS single-trace operators

Op(x, t) = y R1 · · · y Rp Tr

• ϕR1(x) · · · ϕRp(x)
• ,

y 2 = 0 ◮ 2pt- and 3pt-functions are protected by supersymmetry ◮ First non-trivial dynamics in 4pt-correlators: p1p2p3p4 := Op1(x1, y1)Op2(x2, y2)Op3(x3, y3)Op4(x4, y4)

Preliminaries

Simplest operators to consider: 1

2-BPS single-trace operators

Op(x, t) = y R1 · · · y Rp Tr

• ϕR1(x) · · · ϕRp(x)
• ,

y 2 = 0 ◮ 2pt- and 3pt-functions are protected by supersymmetry ◮ First non-trivial dynamics in 4pt-correlators: p1p2p3p4 := Op1(x1, y1)Op2(x2, y2)Op3(x3, y3)Op4(x4, y4) Strategy: use OPE Op1(x1) Op3(x3) Op2(x2) Op4(x4) O∆,ℓ ◮ Exchanged operators O∆,ℓ can be unprotected

Preliminaries

Analyse CFT data in large N expansion ∆ = ∆(0) + 1 N2 · η(1) + . . . C∆,ℓ = C (0)

∆,ℓ + 1

N2 · C (1)

∆,ℓ + . . .

Preliminaries

Analyse CFT data in large N expansion ∆ = ∆(0) + 1 N2 · η(1) + . . . C∆,ℓ = C (0)

∆,ℓ + 1

N2 · C (1)

∆,ℓ + . . .

Which operators contribute? Remember: N = 4 SYM at λ → ∞ ⇐ ⇒ supergravity limit ⇒ long single-trace operators (’string states’) decouple ⇒ remaining spectrum up to this order: double-trace operators ∼ Opn∂ℓOq

The double-trace spectrum

Problem: double-trace operators are degenerate! Ot,ℓ − → Oi

t,ℓ

At twist t ≡ ∆(0) − ℓ we have t

2 − 1 operators:

Oi

t,ℓ =

• (O2

t 2−2∂ℓO2), (O3 t 2−3∂ℓO3), . . . , (O t 2 0∂ℓO t 2 )

The double-trace spectrum

Problem: double-trace operators are degenerate! Ot,ℓ − → Oi

t,ℓ

At twist t ≡ ∆(0) − ℓ we have t

2 − 1 operators:

Oi

t,ℓ =

• (O2

t 2−2∂ℓO2), (O3 t 2−3∂ℓO3), . . . , (O t 2 0∂ℓO t 2 )

• Unmixing equations take the form:

◮ Disconnected free field theory:

• i

(C i

t,ℓ)2

◮ Tree-level supergravity:

• i

(C i

t,ℓ)2 · ηi t,ℓ

The system of equations

Approach: use data from mixed correlators OpOpOqOq

[Rastelli-Zhou’16]

This gives us exactly as many equations as unknowns! → we can solve the mixing problem

The system of equations

Approach: use data from mixed correlators OpOpOqOq

[Rastelli-Zhou’16]

This gives us exactly as many equations as unknowns! → we can solve the mixing problem The solution in the [0, 0, 0] channel takes the form ηi

t,ℓ = −2(t − 1)4(t + ℓ)4

(ℓ + 2i − 1)6 → − 1 ℓ2

[arXiv:1706.08456]

The system of equations

Approach: use data from mixed correlators OpOpOqOq

[Rastelli-Zhou’16]

This gives us exactly as many equations as unknowns! → we can solve the mixing problem Considering cases for different SU(4) channels [a, b, a] lead to a conjecture for the general solution: ηpq|[a,b,a] = − 2MtMt+ℓ+1

• ℓ + 2p − 2 − a − 1+(−1)a+ℓ

2

• 6

Mt = (t − 1)(t + a)(t + a + b + 1)(t + 2a + b + 2)

[arXiv:1802.06889]

Last talk in February

Open questions: ◮ Residual degeneracy for ηpq|[a,b,a] : Is it lifted at higher orders? Or protected by some symmetry? UPDATE: (work in progress) setting up computation to study the first example of residual degeneracy: [0, 2, 0] channel at twist 8

• −

504(ℓ + 7)(ℓ + 8) (ℓ + 1)(ℓ + 2)(ℓ + 5)(ℓ + 6) , − 504 (ℓ + 5)(ℓ + 6) , − 504 (ℓ + 5)(ℓ + 6) , − 504(ℓ + 3)(ℓ + 4) (ℓ + 5)(ℓ + 6)(ℓ + 9)(ℓ + 10)

Last talk in February

Open questions: ◮ Residual degeneracy for ηpq|[a,b,a] : Is it lifted at higher orders? Or protected by some symmetry? UPDATE: (work in progress) setting up computation to study the first example of residual degeneracy: [0, 2, 0] channel at twist 8

• −

504(ℓ + 7)(ℓ + 8) (ℓ + 1)(ℓ + 2)(ℓ + 5)(ℓ + 6) , − 504 (ℓ + 5)(ℓ + 6) , − 504 (ℓ + 5)(ℓ + 6) , − 504(ℓ + 3)(ℓ + 4) (ℓ + 5)(ℓ + 6)(ℓ + 9)(ℓ + 10)

• Observation by [CaronHuot-Trinh’18]:

Residual degeneracy explained by 10d conformal symmetry

• f tree level amplitudes!

[arXiv:1809.09173]

10d conformal symmetry

[CaronHuot-Trinh’18] made the following observations: ◮ AdS5×S5 metric is conformally equivalent to flat space ◮ The four-point tree amplitude of identical complex axi-dilatons in IIB supergravity is conformally invariant: Atree

10 = 8πGNδ16(Q)

stu , with Kµ · 1 stu = 0 ◮ After identifying 8πGN = π5L8

c

(where c = N2−1

4

), our conjecture for ηt,ℓ looks like partial wave coefficients of Atree

10 :

Atree

10

⇒ Aℓ(s) = 1 + i π c (L√s/2)8 (ℓ + 1)6 compared to e−iπηt,ℓ = 1 + i π c ∆8 (ℓeff + 1)6

10d conformal symmetry

Taking this coincidence seriously, the 10d conformal group SO(10, 2) should relate different supergravity correlators. ◮ SO(10, 2) ⊃ SO(4, 2) × SO(6) ↔ isometries of AdS5×S5. ◮ expect natural action of SO(10, 2) on 12-vectors wi: wi ≡ (Xi, yi) where Xi ↔ xµ

i space-time point of SO(4, 2)

yi ↔ parametrisation of SO(6) R-symmetry Conjecture: all tree-level supergravity four-point correlators arise from a single SO(10, 2)-invariant object φ(w1)φ(w2)φ(w3)φ(w4)10 ≡ G10(u10, v10)

• (x2

12 − y 2 12)2(x2 34 − y 2 34)22

10d conformal symmetry

Consequences for tree-level supergravity amplitudes: F(1)

p1p2p3p4(u, v) = Dp1p2p3p4G10(u, v),

where Dp1p2p3p4 is a differential operator obtained by Taylor- expanding φ(w1)φ(w2)φ(w3)φ(w4)10 in the yi’s of SO(6) Conjecture was checked against ◮ old results in the literature

[Arutyunov, Eden, Frolov, Petkou, Sokatchev,...]

◮ result for all supergravity correlators in Mellin-space

[Rastelli-Zhou’16]

Implications for 4d anomalous dimensions: Double-trace operators Opn∂ℓOq stem from bilinears φ∂ℓφ

• f a single 10d field φ(w). Degeneracy occurs when different

4d operators come from the same 10d primary!

Summary and Outlook

◮ Conjectured anomalous dimensions of double-trace operators

ηpq|[a,b,a] = − 2MtMt+ℓ+1

• ℓ + 2p − 2 − a − 1+(−1)a+ℓ

2

• 6

has a residual degeneracy in some SU(4) channels. ◮ This residual degeneracy is explained by an unexpected 10d conformal symmetry of supergravity tree-level amplitudes. Outlook: ◮ Study first example of degeneracy ([0, 2, 0] channel): is it lifted at the next order? i.e. is the 10d tree-level conformal symmetry broken by quantum effects? ◮ Study implications of 10d symmetry for higher-loop correlators

Structure of 4-point functions

Consider the correlator (p ≤ q) ppqq = gp

12gq 34

• i

HRi(u, v), where Ri ∈ [0, p, 0] × [0, p, 0] u = g13g24

g12g34 ≡ x ¯

x and v = g13g24

g14g23 ≡ (1 − x)(1 − ¯

x).

Structure of 4-point functions

Consider the correlator (p ≤ q) ppqq = gp

12gq 34

• i

HRi(u, v), where Ri ∈ [0, p, 0] × [0, p, 0] u = g13g24

g12g34 ≡ x ¯

x and v = g13g24

g14g23 ≡ (1 − x)(1 − ¯

x). Consequences of superconformal symmetry:

• i

HRi(u, v) = (protected) + I · F(u, v), with I = (x − y)(x − ¯ y)(¯ x − y)(¯ x − ¯ y).

Structure of 4-point functions

Consider the correlator (p ≤ q) ppqq = gp

12gq 34

• i

HRi(u, v), where Ri ∈ [0, p, 0] × [0, p, 0] u = g13g24

g12g34 ≡ x ¯

x and v = g13g24

g14g23 ≡ (1 − x)(1 − ¯

x). Focus only on interacting part: F(u, v) = F(0) + 1 N2 · F(1) + . . .

Conformal blocks

Expand contributions from long operators into conformal blocks: F(u, v) =

∞

• t,ℓ≥0

C 2

t,ℓ · Bt,ℓ(u, v)

where the conformal blocks Bt,ℓ

[Dolan-Osborn ’01]

Conformal blocks

Expand contributions from long operators into conformal blocks: F(u, v) =

∞

• t,ℓ≥0

C 2

t,ℓ · Bt,ℓ(u, v)

where the conformal blocks Bt,ℓ

[Dolan-Osborn ’01]

◮ are Eigenfunctions of the quadratic conformal Casimir ◮ capture the contribution of a conformal primary operator and all its descendants ◮ in the OPE-limit u → 0 they behave like Bt,ℓ(u, v) → ut (. . .) , with twist t = ∆ − ℓ