Fluctuations of BPS Wilson loop and AdS 2 /CFT 1 Arkady Tseytlin S. - - PowerPoint PPT Presentation

Fluctuations of BPS Wilson loop and AdS2/CFT1

Arkady Tseytlin

• S. Giombi, R. Roiban, AT

arXiv:1706.00756

• novel sector of observables in AdS/CFT:

gauge-invariant correlators of operators inserted on Wilson loop

• described by an effective (“defect” ) CFT1

”induced” from N = 4 SYM

• case of 1

2-BPS straight-line WL: example of AdS2/CFT1

quantum theory in AdS2 defined by superstring action

• surprisingly, in BPS WL “vacuum” AdS/CFT map:

elementary SYM fields (⊥ to the line)

↔ string coordinates as fields in AdS2 (in static gauge)

[cf. Tr(Φn...DmFk...) ↔ closed-string vertex operators]

• aim: compute 4-point correlators at strong coupling
• Witten diagrams for AdS/CFT correlators,

CFT methods (OPE, etc.), localization, etc. should also have connections to integrability

Review: gauge theory side

N = 4 SYM: special Maldacena-Wilson loop operator

W = trPe

• dt(i ˙

xµAµ+| ˙ x|θIΦI)

generic xµ(t) closed loop, θI(t) unit 6-vector: “locally” susy

• special choices preserve parts of global superconf symm

[Zarembo:02; Drukker,Giombi,Ricci,Trancanelli:07]

• max 16 susy – 1

2 BPS: infinite straight line (or circle), θI=const

x0 = t ∈ (−∞, ∞), θIΦI = Φ6 W = trPe

• dt(iAt+Φ6)
• local Oi(ti) on WL: gauge inv correlator

[Drukker, Kawamoto:06]

O1(t1)O2(t2) · · · On(tn) ≡ trP

• O1(t1) e
• dt(iAt+Φ6) O2(t2) · · · On(tn) e
• dt(iAt+Φ6)

1 = W = 1 and similar normalization for circle

• operator insertions are equivalent to deformations of WL

[Drukker, Kawamoto:06; Cooke, Dekel, Drukker:17]

complete knowledge of correlators ↔ expectation value

• f general Wilson loop – deformation of line or circle
• cf. “wavy line” WL

[Mikhailov:03; Semenoff, Young:04]

• symmetries preserved by 1

2-BPS WL vacuum:

SO(5) ⊂ SO(6) R-symmetry: rotates 5 scalars Φa, a = 1, . . . , 5 SO(2, 1) × SO(3) ⊂ SO(2, 4): SO(3) rotations around line SO(2, 1) – dilatations, transl and special conf along line d = 1 conformal group + 16 supercharges preserved by line: d = 1, N = 8 superconformal group OSp(4∗|4)

• operator insertions O(t) classified by OSp(4∗|4) reps

labelled by dim ∆ and rep of “internal” SO(3) × SO(5)

• correlators define ”defect” CFT1 on the line

[Drukker et al:06; Sakaguchi, Yoshida:07; Cooke et al:17]

determined by spectrum of dims and OPE coeffs

• ... correlators satisfy all usual properties of CFT:

call O(t) “operators in CFT1” without reference to their (non-local) origin in SYM

• “elementary excitations”: short rep of OSp(4∗|4)

8 bosonic (+ 8 fermionic) operators with protected ∆: 5 scalars: Φa (∆ = 1) that do not couple to WL; 3 “displacement operators”: Fti ≡ iFti + DiΦ6 (i = 1, 2, 3) with protected ∆ = 2 (WI for breaking of ⊥ translations)

• protected dims: exact 2-point functions in planar SYM

Φa(t1)Φb(t2) = δab CΦ(λ)

t2

12

, t12 = t1 − t2

Fti(t1)Ftj(t2) = δij

CF(λ) t4

12

CΦ(λ) = 2B(λ) , CF(λ) = 12B(λ) , B(λ) =

√

λ I2(

√

λ) 4π2 I1(

√

λ) B(λ) – Bremsstrahlung function [Correa, Henn, Maldacena, Sever:12]

• 3-point functions of these elementary bosonic operators

vanish by SO(3) × SO(5) symmetry

• 4-point functions: depend on t1, ..., t2 and λ

constrained by 1d conf symm;

• nly leading O(λ2) term in 4-point Fti known [Cooke et al:17]

String theory side Aim: 4-point functions at strong coupling (N = ∞, λ ≫ 1) from string theory in AdS5 × S5

• WL → open string minimal surfaces in AdS5 ending
• n contour defining WL operator at the boundary
• 1

2-BPS Wilson line (or circle): minimal surface –

AdS2 embedded in AdS5 (at fixed point on S5)

• fundamental open string stretched in AdS5:

preserves same OSp(4∗|4) as 1

2-BPS WL

1d conf group SO(2, 1) realized as isometry of AdS2

• expanding string action around AdS2 surface:

AdS2 multiplet of fluctuations transverse to string – 5 (m2 = 0) scalars ya in S5; 3 (m2 = 2) scalars xi in AdS5; 8 (m2 = 1) fermions

[Drukker, Gross, AT:00]

• identify 8+8 fields in AdS2 with elementary CFT1 insertions

[Sakaguchi, Yoshida:07; Faraggi, Pando Zayas:11; Fiol et al:13]

• m2 = ∆(∆ − d) for AdSd+1 scalar masses and CFTd dims:

massless S5 fields ya should be dual to Φa in CFT1 with ∆ = 1 massive AdS5 fields xi should be dual to Fti with ∆ = 2

• same spectrum as in “non-relat. limit” of AdS5 × S5 string

[Gomis et al:05] and in OSp(4∗|4) invariant

N = 8 superconformal QM [Belucci et al:03]

• AdS/CFT: closed superstring vertex operators →

single-trace gauge inv local operators in SYM; add open-string sector (strings ending at bndry) → gauge-inv operators = WL with insertions of local operators

• here: only to insertions of ops with protected dims

dual to “light” fields on AdS2 string world-sheet

• open problem: description of unprotected insertions

with large anom dim at λ = ∞, e.g. insertion of Φ6

[Alday, Maldacena:07]

expect duals of “heavy” insertions to have m2 ∼ 1

α′ ∼

√

λ corresponding to massive states of open string; CFT1 spectrum of ops on WL will have large ∆gap ∼ λ1/4 as for closed string states

• other gauge-invariant correlators:

(i) WL with single-trace ops e.g. W trZJ

• pen-closed string sector: closed string from worldsheet to

bdry point away from line ( tr2: subleading at large N)

[Berenstein et al:98; Semenoff, Zarembo:01; Pestun, Zarembo:02]

(ii) mixed correlators of ops on line and ops away from line

Strategy: string action → interaction vertices for ”light” AdS2 fields

→ tree-level Witten diagrams in AdS2 → prediction for

4-point functions of protected ops on WL: expansion parameter

1

√

λ (action S =

√

λ d2σ

√

h∂x∂x + ...) (cf.

1 N2 in 4-points in AdS5 sugra: S = N2

d5x√gR + ...)

• AdS2 QFT: superstring action UV finite –

AdS2/CFT1 duality should hold for any T =

√

λ 2π :

• AdS2 Witten diagrams with loops should be well defined

e.g. 1-loop correction to boundary-to-boundary propagator protected 2-point function: subleading term in B(λ) =

√

λ 4π2 − 3 8π2 + O( 1

√

λ)

checked earlier

[Buchbinder, AT:13]

AdS2 AdS5 R4 t O(t1) O(t2) O(t3) O(t4)

• Plan:

(i) compute tree-level 4-point functions (ii) use OPE to extract strong coupling corrections to dims

• f “2-particle” ops built of 2 of protected insertions: Φ∂n

t Φ, etc.

(iii) compare with localization to YM2

• e.g. for singlet “2-particle” operator O = ΦaΦa

O(t1)O(t2) = COO

t

2∆O 12

, ∆O = 2 −

5

√

λ + . . .

• correlators on circle WL: get by large conf transf;
• corr. of class of S5 ops on circle WL captured by localization

[Drukker, Giombi, Ricci, Trancanelli:07, Giombi,Pestun:09,12]

compare result of Witten diagram calculation in AdS2 to prediction of localization to solvable YM2 on S2 [Migdal:75; Witten:91]

• open question: connection to integrability approach?

[cf. TBA of Drukker:12; Correa, Maldacena, Sever:12] relation of AdS2 Witten diagrams to factorization

• f 2d S-matrix in flat space?

AdS5 × S5 string in static gauge as AdS2 bulk theory

bosonic part of superstring action in AdS5 × S5 (T =

√

λ 2π )

SB = 1

2T

• d2σ

√

h hµν 1 z2

• ∂µxr∂νxr + ∂µz∂νz

+ ∂µya∂νya

(1 + 1

4y2)2

• σµ = (t, s), r = (0, i) = (0, 1, 2, 3), a = 1, ..., 5

minimal surface for straight Wilson line at Euclidean boundary z = s , x0 = t , xi = 0 , ya = 0 induced metric is AdS2: gµνdσµdσν = 1

s2(dt2 + ds2).

• Aim: study correlators of small fluctuations of “transverse”

coordinates (xi, ya) near AdS2 minimal surface

• global symmetry of action SO(2, 1) × [SO(3) × SO(6)]

• make SO(2, 1) symmetry manifest: choose AdS2 adapted

coordinates and fix static gauge: z = s and x0 = t ds2

AdS5 = (1 + 1

4x2)2

(1 − 1

4x2)2 ds2

AdS2 +

dxidxi

(1 − 1

4x2)2 ,

ds2

AdS2 = 1

z2(dx2

0 + dz2)

• Nambu action in static gauge

SB = T d2σ

√

h = T d2σ LB hµν = (1+ 1

4 x2)2

(1− 1

4 x2)2 gµν(σ) + ∂µxi∂νxi

(1− 1

4 x2)2 + ∂µya∂νya

(1+ 1

4y2)2 ,

gµν = 1

s2 δµν

• action of straight fundamental string in AdS5 × S5 along z:

2d theory of 3+5 scalars in AdS2 with SO(2, 1) × [SO(3) × SO(6)]

• bulk AdS2 theory ↔ CFT1 at z = s = 0 bndry:

CFT1 defined by operator insertions on straight WL

LB = L2 + L4x + L2x,2y + L4y + ... L2 = 1

2gµν∂µxi∂νxi + xixi + 1 2gµν∂µya∂νya

L4x =

1 8(gµν∂µxi∂νxi)2 − 1 4(gµν∂µxi∂νxj) (gρκ∂ρxi∂κxj)

+ 1

4xixi(gµν∂µxj∂νxj) + 1 2xixi xjxj

L2x,2y = 1

4(gµν∂µxi∂νxi)(gρκ∂ρya∂κya) − 1 2(gµν∂µxi∂νya)(gρκ∂ρxi∂κya)

L4y = − 1

4(ybyb)(gµν∂µya∂νya) + 1 8(gµν∂µya∂νya)2

− 1

4(gµν∂µya∂νyb) (gρκ∂ρya∂κyb)

xi – 3 massive and ya – 5 massless scalars in AdS2

• superstring: also 8 fermions with m2 = 1

[Drukker et al 00]

• resulting 2d theory is UV finite and thus dual to

CFT1 for any value of coupling T =

√

λ 2π

• bndry correlators O(t1)O(t2)...O(tn) reproduced

by AdS2 amplitudes of string sigma model – series in

1

√

λ

• O correspond to particular string coordinates X = (x, y)

O(t1)O(t2)...O(tn)SYM = X(t1)X(t2)....X(tn)AdS2 ...AdS2 defined by string path integral

e.g tree Witten diagrams with bulk-to-bndry props

• X ∼ ya → O ∼ Φa (a = 1, ..., 5) with ∆ = 1

X ∼ xi → O ∼ Fit (i = 1, 2, 3) with ∆ = 2 Why?

• λ ≪ 1:

gauge theory

[Cooke, Dekel, Drukker:17]

to find O(t1)O(t2)...O(tn): (i) compute wavy-line W(C); (ii) take

δ δC(t), set C(t)= line

• λ ≫ 1: W(C) from AdS5 × S5 open str path int. with

Dirichlet b.c. (disc or half-plane w-surface ending at bndry)

• λ → ∞: logW(C) = minimal area = string action on solution

• string action in static gauge as 2d bulk theory in AdS2:

same as AdS/CFT procedure of computing gen. funct. for CFT correlators = X(t1)X(t2)....X(tn)AdS2

• expanding on-shell string action (gen.f. for tree ”S-matrix”)

in powers of fluctuations δC(t) from straight line: same correlators as from bulk Green’s functions connected to bndry points by bulk-to-bndry propagators

• previous check: O(t1)O(t2) = X(t1)X(t2)AdS2 ∼ B(λ)

t4

12

using wavy-line solution → get string tree level

[Mikhailov:03; Chernicoff, Gujosa:09]

and 1-loop [Buchbinder, AT:13] terms in B(λ) =

√

λ 4π2 − 3 8π2 + ...

• LB: no 3-vertices: 4-point tree-level correlators from 4-vertices

→ contact Witten diagrams in AdS2

• including fermions: fix κ-symm in superstring action

expanded near straight line minimal surface → globally susy field theory for OSp(4∗|4) 8+8 multiplet in AdS2 same symmetry on dual gauge theory side: should allow to determine correlators with fermions.

Comments:

• present example of AdS2/CFT1:

critical string – no dynamical 2d gravity: fixed AdS2 background defect CFT with no ”stress tensor” ↔ AdS2 with no gravity SO(2, 1) as isometry of AdS2 metric, no 1d reparam inv (cf. dilaton gravity [Ahlmeiri, Polchinski:14] in SYK context

[Maldacena, Stanford:16])

• original WL has a reparam inv, fixed by identification x0 = t;

remaining symm SO(2, 1) ⊂ SO(2, 4) that preserves the line; before fixing static gauge string (“bulk”) action is reparam inv but gravity non-dynamical in critical superstring (no analog of pseudo-Goldstone mode in bndry theory related to spont. broken reparams [Maldacena et al:16])

4-point functions and conformal blocks in CFT1

• local operators in CFT1 on line R = {t}

covariant under SO(2, 1) – labelled just by dimension ∆

• 4-point function of O∆(t) restricted by SO(2, 1)

O∆(t1)O∆(t2)O∆(t3)O∆(t4) =

1

(t12t34)2∆ G(χ)

χ = t12t34 t13t24

∈ (−∞, ∞)

usual cross-ratios u, v are not independent in d = 1 u ≡ t2

12t2 34

t2

13t2 24

= χ2 ,

v ≡ t2

14t2 23

t2

13t2 24

= (1 − χ)2

• ne χ:

SO(2, 1) allows to fix 3 points on the line

• G(χ) has OPE expansion

G(χ) = ∑

h

c∆,∆;h χh 2F1(h, h, 2h, χ) h= dim of exchanged operator; c∆,∆;h =

C2

O∆O∆Oh

C2

O∆O∆COhOh

χh 2F1(h, h, 2h, χ) – conf block in d = 1 [Dolan, Osborn:11]

• case of pairwise equal dimensions (∆12 ≡ ∆1 − ∆2)

O∆1(t1)O∆2(t2)O∆1(t3)O∆2(t4) =

1

(t12t34)∆1+∆2

• t24

t13

• ∆12 G(χ)

G(χ) = ∑

h

c∆1,∆2;h χh 2F1(h + ∆12, h − ∆12, 2h, χ)

“Generalized free fields” (e.g. g = 0 large N CFT) [Heemskerk et al:09, Fitzpatrick et al:11]

• case of identical operators of dim ∆:

G(u, v) = 1 + u∆ + ( u

v)∆, i.e. in d = 1

O∆(t1)O∆(t2)O∆(t3)O∆(t4) =

1

(t12t34)2∆

• 1+ χ2∆ +

χ2∆

(1 − χ)2∆

• ps exchanged in OPE are only 1 and “2-particle” ops

O =

• O∆O∆
• 2n ∼ O∆∂2n

t O∆ ,

∆O = 2∆ + 2n, n = 0, 1, . . . corresponding OPE coeffs: c∆,∆;2∆+2n = 2

• Γ(2n + 2∆)

2Γ(2n + 4∆ − 1)

• Γ(2∆)

2Γ(2n + 1)Γ(4n + 4∆ − 1)

• case of pairwise identical operators

O∆1(t1)O∆2(t2)O∆1(t3)O∆2(t4) =

1 t2∆1

13 t2∆2 24

=

χ∆1+∆2

(t12t34)∆1+∆2

• t24

t13

• ∆12
• ps in χ → 0 OPE: [O∆1O∆2]n ∼ O∆1∂n

t O∆2,

n = 0, 1, 2, 3, ... c∆1,∆2;∆1+∆2+n = (−1)nΓ (n + 2∆1) Γ (n + 2∆2) Γ (n + 2∆1 + 2∆2 − 1) Γ (2∆1) Γ (2∆2) Γ(n + 1) Γ (2n + 2∆1 + 2∆2 − 1)

4-point function of S5 fluctuations

tree-level 4-point Witten diagram of S5 fluctuations ya = strong-coupling limit of Φa1a2a3a4(t1, t2, t3, t4) ≡ Φa1(t1)Φa2(t2)Φa3(t3)Φa4(t4)

= ya1(t1)ya2(t2)ya3(t3)ya4(t4)AdS2 =

• CΦ(λ)

2

t2

12t2 34

Ga1a2a3a4(χ) Φa – protected dimension ∆ = 1

ya1(t1)ya2(t2)AdS2 = Φa1(t1)Φa2(t2) = δa1a2 CΦ(λ)

t2

12

can absorb CΦ(λ) = 2B(λ) into normalization of ops Ga1a2a3a4(χ) = δa1a2δa3a4 + O(χ) – non-trivial function of λ

y y y y F F F F y y F F F F

• decompose into SO(5) singlet, antisymm and symm traceless

Ga1a2a3a4(χ) = GS(χ)δa1a2δa3a4 + GA(χ) (δa1a3δa2a4 − δa2a3δa1a4)

+ GT(χ)

• δa1a3δa2a4 + δa2a3δa1a4 − 2

5δa1a2δa3a4

GS,T,A(χ) = G(0)

S,T,A(χ) + 1

√

λG(1) S,T,A(χ) + . . .

[cf. at λ ≪ 1: Φa1a2a3a4 ∼

λ2 t2

12t2 34(δa1a2δa3a4 +

χ2

(1−χ)2 δa1a4δa2a3) ]

• leading terms G(0)

S,T,A(χ) from disconnected 4-point function

– given by generalized free field result Φa1a2a3a4

• disconn. =
• CΦ(λ)

2

t2

12t2 34

• δa1a2δa3a4 + χ2δa1a3δa2a4 +

χ2

(1−χ)2 δa1a4δa2a3

• G(0)

S (χ) = 1 + 2

5G(0)

T (χ) ,

G(0)

T,A(χ) = 1

2

• χ2 ±

χ2

(1−χ)2

• connected part: using 4-vertices in string action and

normalized bulk-to-bndry prop. O∆(x1)O∆(x2) = C∆

x2∆

12

K∆(z, x; x′) = C∆

• z

z2+(x−x′)2

∆

≡ C∆ ˜

K∆(z, x; x′) ,

C∆ =

Γ(∆) 2πd/2Γ(∆+1− d

2)

for d = 1, ∆ = 1, t ≡ x0 K∆=1(z, t; t′) = 1

π z z2+(t−t′)2 ,

C∆=1 = 1

π

• 4-point in terms of D-functions: in AdSd+1 [D’Hoker et al 89]

D∆1∆2∆3∆4(x1, x2, x3, x4)

= dzddx

zd+1 ˜

K∆1(z, x; x1) ˜ K∆2(z, x; x2) ˜ K∆3(z, x; x3) ˜ K∆4(z, x; x4)

• “reduced” ¯

D (Σ ≡ 1

2 ∑i ∆i)

D∆1∆2∆3∆4 =

π

d 2 Γ(Σ− d 2)

2 Γ(∆1)Γ(∆2)Γ(∆3)Γ(∆4) x

2(Σ−∆1−∆4) 14

x

2(Σ−∆3−∆4) 34

x

2(Σ−∆4) 13

x2∆2

24

¯ D∆1∆2∆3∆4(u, v) ¯ D = dαdβdγ δ(α + β + γ − 1) α∆1−1β∆2−1γ∆3−1

Γ(Σ−∆4)Γ(∆4)

(αγ+αβ u+βγ v)Σ−∆4

• e.g. for ∆ = 1:

u = z¯ z , v = (1 − z)(1 − ¯ z) ¯ D1111(u, v) =

1 z−¯ z

• log(z¯

z) log(1−z

1−¯ z) + 2Li2(z) − 2Li2(¯

z)

• in d = 1:

u = χ2, v = (1 − χ)2 (z = ¯ z = χ) ¯ D1111(χ) = −

2 1−χ log |χ| − 2 χ log |1 − χ|

• explicitly:

Φa1a2a3a4

conn.

= 2π

√

λ(C∆=1)4 Qa1a2a3a4 4y Qa1a2a3a4

4y

=

• 3D1111 − 2t2

13D2121 − 2t2 14D2112 − 2t2 23D1221 − 2t2 24D1212

+ 4(t2

13t2 24 + t2 14t2 23 − t2 12t2 34)D2222

• δa1a2δa3a4

+

• 3D1111 − 2t2

12D2211 − 2t2 14D2112 − 2t2 23D1221 − 2t2 34D1122

+ 4(t2

12t2 34 + t2 14t2 23 − t2 13t2 24)D2222

• δa1a3δa2a4

+

• 3D1111 − 2t2

12D2211 − 2t2 13D2121 − 2t2 24D1212 − 2t2 34D1122

+ 4(t2

12t2 34 + t2 13t2 24 − t2 14t2 23)D2222

• δa1a4δa2a3

express in terms of ¯ D1111 using ¯ D2211(u, v) = −∂u ¯ D1111(u, v) , ¯ D2222(u, v) = u∂2

u ¯

D1111(u, v) + ∂u ¯ D1111(u, v), etc.

Φa1a2a3a4

conn

= (C∆=1)2

t2

12t2 34 Ga1a2a3a4

(1)

(χ)

G(1)

S (χ) = − 2(χ4−4χ3+9χ2−10χ+5) 5(χ−1)2

+

χ2(2χ4−11χ3+21χ2−20χ+10) 5(χ−1)3

log |χ|

− 2χ4−5χ3−5χ+10

5χ

log |1 − χ| , G(1)

T (χ) = − χ2(2χ2−3χ+3) 2(χ−1)2

+

χ4(χ2−3χ+3)

(χ−1)3

log |χ| − χ3 log |1 − χ| , G(1)

A (χ) = χ(−2χ3+5χ2−3χ+2) 2(χ−1)2

+

χ3(χ3−4χ2+6χ−4)

(χ−1)3

log |χ| − (χ3 − χ2 − 1) log |1 − χ|

• OPE limit χ → 0

G(1)

S (χ) = 1 30χ2 − 60 log |χ| − 43

+ 1

30χ3 − 60 log |χ| − 73

+ . . . G(1)

T (χ) = − 3 2χ2 − 3 2χ3 + 1 12χ4 − 36 log |χ| − 18

+ . . . G(1)

A (χ) = 1 6χ3

24 log |χ| + 7 + 3

4χ4

8 log |χ| + 5 + . . .

• G(1)

T (χ): no χ2 log |χ| – traceless Φ(aΦb) has

no anom dim as expected: ZJ = (Φ1 + iΦ2)J inserted into WL is BPS [Drukker et al:06; Correa et al:12]

• G(1)

S (χ): χ2 log |χ| – singlet ΦaΦa has anom dim

χ2+n log |χ|: same for Φa∂2n

t Φa

• G(1)

A (χ): no operator of ∆ = 2 + O( 1

√

λ)

e.g. [Φa, Φb] in multiplet with Φ6 [Cooke et al:17] – should get ∆ ∼ λ1/4 [Alday, Maldacena:07] decoupled in perturbative OPE calculation

Dimensions of two-particle operators from OPE GS(χ), GT(χ) and GA(χ): OPE expansion G(χ) = ∑

h

ch χh Fh(χ) = G(0)(χ) +

1

√

λG(1)(χ) + . . .

Fh(χ) ≡ 2F1(h, h, 2h, χ)

• disconn part: leading O(1) – gen. free fields –

exchanged “2-particle” ops:

[ΦΦ]S

2n ∼ Φa∂2n t Φa ,

[ΦΦ]T

2n ∼ Φ(a∂2n t Φb) ,

[ΦΦ]A

2n+1 ∼ Φ[a∂2n+1 t

Φb]

• connected part:

1

√

λ corrections to ∆ and OPE coeffs

• complication: ops may mix – degeneracies at leading order

[ΦΦ]S

2n with n ≥ 1 can mix with F∂2n−2 t

F and ψ∂2n−1

t

ψ ;

[ΦΦ]A

2n+1 can mix with ψ∂2n t ψ in (1, 10) of SU(2) × Sp(4)

• [ΦΦ]T

2n – no mixing – no other 2-particle ops in same rep

h ≡ ∆[ΦΦ]T

2n = 2 + 2n +

1

√

λγ(1)

[ΦΦ]T

2n + . . .

ch = c(0)

ΦΦ[ΦΦ]T

2n +

1

√

λc(1) ΦΦ[ΦΦ]T

2n + . . .

• from G(0)

T (χ) and generalized free field result

c(0)

ΦΦ[ΦΦ]T

2n =

• Γ(2n+2)

2

Γ(2n+3) Γ(2n+1) Γ(4n+3)

• from log |χ| terms in G(1)

T (χ) – 1

√

λ anom dims

χh = χ2+2n+ 1

√

λ γ(1)+... = χ2+2n

1 +

1

√

λγ(1) log |χ| + . . .

• ∞

∑

n=0

c(0)

ΦΦ[ΦΦ]T

2nγ(1)

[ΦΦ]T

2nχ2+2nF2+2n(χ) =

• G(1)

T (χ)

• log |χ|

• solve using orthogonality relation

[Heemskerk et al:09]

• dz

2πi 1 z2 z∆+n F∆+n(z) z1−∆−n′F1−∆−n′(z) = δn,n′

γ(1)

[ΦΦ]T

2n =

1 c(0)

ΦΦ[ΦΦ]T 2n

• dχ

2πi χ−3−2nF−1−2n(χ)

• G(1)

T (χ)

• log |χ|

γ(1)

[ΦΦ]T

2n = −2n2 − 3n

• strong-coupling dim of O(t) = [ΦΦ]T

2n ∼ Φ(a∂2n t Φb)

∆[ΦΦ]T

2n = 2 + 2n − 2n2+3n

√

λ

+ O( 1

λ)

n = 0: protected Φ(aΦb); n ≥ 1 – unprotected – long multiplet

• correction to OPE coeffs: n = 0, 1, ...;

Hn = ∑n

k=1 1 k

c(1)

ΦΦ[ΦΦ]T

2n = [Γ(2n+2)]2

Γ(4n+3)

− (3 + 34n + 56n2 + 24n3)

+ 4n(n + 1)(2n + 1)(2n + 3)(H4n+3 − H2n)

cΦΦ[ΦΦ]T

0 = 1 −

3 2

√

λ + . . . ,

cΦΦ[ΦΦ]T

2 = 3

5 − 3 20

√

λ + . . . , ...

• as in [Heemskerk et al:09; Fitzpatrick et al:11; Alday, Bissi, Lukowski:14]

c(1)

ΦΦ[ΦΦ]T

2n = 1

2 ∂ ∂n

• c(0)

ΦΦ[ΦΦ]T

2nγ(1)

[ΦΦ]T

2n

• singlet and antisymmetric channels:

c(0)

ΦΦ[ΦΦ]S

2n =

2

• Γ(2n+2)

2

Γ(2n+3) 5 Γ(2n+1) Γ(4n+3)

, c(0)

ΦΦ[ΦΦ]A

2n+1 = −

• Γ(2n+3)

2

Γ(2n+4) Γ(2n+2) Γ(4n+2+3)

γ(1)

[ΦΦ]S

2n = −2n2 − 3n − 5 ,

γ(1)

[ΦΦ]A

2n+1 = −2n2 − 5n − 4

• interpreted as “averages” of dimensions over all ops

appearing in mixing (weighted by OPE coefs)

• n = 0 exception: ΦaΦa does not mix

∆ΦaΦa = 2 −

5

√

λ + O( 1 λ) ,

cΦΦ[ΦΦ]S

0 = 2

5 − 43 30

√

λ + . . .

• large n limit of all dims – same asymptotic form

∆n≫1 = 2n − 2n2

√

λ + . . .

• in ”semiclassical” limit: n,

√

λ ≫ 1 and ν ≡

n

√

λ fixed

“2-particle” operators have a universal strong-coupling form ∆n =

√

λ f (ν) , f (ν) = 2ν − 2ν2 + O(ν3) captured by semiclassical string calculation? (cf. cases with large R-charge [Drukker:06, Miwa:06, Sakaguchi:07])

4-point functions with AdS5 fluctuations

Yi1i2a1a2 ≡ xi1(t1)xi2(t2)ya1(t3)ya2(t4)AdS2

= Fi1

t (t1) Fi2 t (t2) Φa1(t3) Φa2(t4) = δi1i2δa1a2 G(χ)

t4

12t2 34

Xi1i2i3i4 ≡ xi1(t1)xi2(t2)xi3(t3)xi4(t4)AdS2

= Fi1

t (t1) Fi2 t (t2) Fi3 t (t3) Fi4 t (t4) = Gi1i2i3i4(χ)

t4

12t4 34 1 2 BPS WL: related to ya1(t1)ya2(t2)ya3(t3)ya4(t4)AdS2

• connected part of Yi1i2a1a2

Gconn(χ) = 2π

√

λ(C∆=1C∆=2)2t4 12t2 34 Qxy ≡ 2 3π2 G(1)(χ)

C∆=1 = 1

π,

C∆=2 =

2 3π

Qxy = 4

• D2211 + 2t2

12D3311 − 2t2 13D3221 − 2t2 23D2321 − 2t2 14D3212

+ 2t2

34D2222 + 4t2 14t2 23D3322 + 4t2 13t2 24D3322 − 4t2 12t2 34D3322

• G(1)(χ) = − 4

√

λ

• 1 −

1

2 − χ−1

ln |1 − χ|

• disconnected contribution

G(0)(χ) =

1

(t2

12t2 34)3/2

• t24

t13

• χ3 ,

c(0)

ΦF[ΦF]n = Γ(n+2) Γ(n+4) Γ(n+5) 6 Γ(n+1) Γ(2n+5)

• dimensions of 2-particle ops in OPE

[ΦaFit]n ∼ Φa∂n

t Fit ,

∆ = 3 + n + O( 1

√

λ)

• using orthogonality relation Fh,a(z) ≡ 2F1(h + a, h − a, 2h, z)
• dz

2πi 1 z2 z∆+nF∆+n,a(z)z1−∆−n′F1−∆−n′,a(z) = δn,n′

γ(1)

[ΦF]n = − n2

2 − 5n 2 − 2 .

• even n: [ΦF]2n can mix with ψ∂2nψ in same rep;
• dd n: no mixing

∆[ΦF]2n+1 = 4 + 2n − 2n2+7n+5

√

λ

+ O( 1

λ)

same universal form for large n

• note ∆[ΦF]2n+1 = ∆[ΦΦ]T

2n+2: in same long supermultiplet

• 4-x correlator:

Gi1i2i3i4

conn (χ) = (C∆=2)2 Gi1i2i3i4

(1)

(χ)

Gi1i2i3i4

(1)

(χ) = G(1)

S δi1i2δi3i4 + G(1) A (δi1i3δi2i4 − δi1i4δi2i3)

+ G(1)

T (δi1i3δi2i4 + δi1i4δi2i3 − 2 3δi1i2δi3i4)

G(1)

S (χ) = −(24χ8−90χ7+125χ6−76χ5+125χ4−306χ3+438χ2−288χ+72) 9(χ−1)4

−

2(4χ6−χ5−6χ+12) 3χ

log |1 − χ|

+

2χ4(4χ6−21χ5+45χ4−50χ3+30χ2−6χ+2) 3(χ−1)5

log |χ| , etc.

• 2-particle ops in OPE: [FF]S

2n ∼ Fti∂2n t Fit ,

[FF]T

2n ∼ Ft(i∂2n t Fj)t ,

[FF]A

2n+1 ∼ Ft[i∂2n+1 t

Fj]t

• c(0) from disconnected part → γ(1) from connected

G(0)

S (χ) = 1 + 2

3G(0)

T (χ) ,

G(0)

T,A(χ) = 1

2

• χ4 ±

χ4

(1 − χ)4

• c(0)

FF[FF]S

2n =

• Γ(2n+4)

2

Γ(2n+7) 54 Γ(2n+1) Γ(4n+7) ,

c(0)

FF[FF]T

2n =

• Γ(2n+4)

2

Γ(2n+7) 36 Γ(2n+1) Γ(4n+7) ,

c(0)

FF[FF]A

2n+1 = −

• Γ(2n+5)

2

Γ(2n+8) 36 Γ(2n+2) Γ(4n+9)

γ(1)

[FF]S

2n = −2n2 − 7n − 2

γ(1)

[FF]T

2n = −2n2 − 7n − 5 ,

γ(1)

[FF]A

2n+1 = −2n2 − 9n − 7

• singlets [FF]S

2n can mix with Φ∂2n+2Φ and ψ∂2n+1ψ

[FF]A

2n+1 mix with ψ∂2n+2ψ – anom dims are ”averages”

• symmetric traceless [FF]T

2n not expected to mix:

∆[FF]T

2n = 4 + 2n − 2n2+7n+5

√

λ

+ O( 1

λ)

• observe that ∆[FF]T

2n = ∆[ΦF]2n+1 = ∆[ΦΦ]T 2n+2

• ps should belong to same long supermultiplet

Comparison to toy model: ϕ4 scalar in AdS2 S =

• d2σ√g
• 1

2gµν∂µϕ∂νϕ + 1 2m2ϕ2 + g 4! ϕ4

metric ds2 = 1

s2(ds2 + dt2): compute tree level Witten diagrams

→ conf inv correlators of O(t) at bndry with ∆(∆ − 1) = m2 O(t1)O(t2)O(t2)O(t3) = −gC4

∆D∆∆∆∆(t1, t2, t3, t4)

• e.g. massless scalar: ∆ = 1 (C∆ =

Γ(∆) 2√πΓ(∆+ 1

2) → 1

π)

O(t1)O(t2)O(t2)O(t3) = − g

4π3 1 t2

12t2 34 χ2 ¯

D1111(χ)

• extract anom dim of [OO]2n ∼ O∂2n

t O

leading order OPE coeffs c(0)

OO[OO]2n = 2

• Γ(2n+2)

2

Γ(2n+3) Γ(2n+1) Γ(4n+3)

from coeff of log χ in ¯ D1111(χ) γ(1)

[OO]2n =

1 c(0)

OO[OO]2n

• dχ

2πiχ−3−2nF−1−2n(χ) gχ2 2π(1−χ)

∆[OO]2n = 2 + 2n + g

4π 1

(2n+1)(n+1) + O(g2)

• different from string σ-model results:

non-derivative 4-coupling → anom dim (i) positive and (ii) go to 0 at n ≫ 1.

• similar for m2 = 2 scalar, i.e. ∆ = 2

∆[OO]2n = 4 + 2n + g

4π

(n+1)(2n+5) (n+2)(n+3)(2n+1)(2n+3) + O(g2)

Correlators of operators on circular Wilson loop: relation to localization to YM2

• map line to circle by conf. transf: t → tan τ

2, τ ∈ (−π, π)

translate correlators on line to those on circle

O∆(t1)O∆(t2)line = CO

t2∆

12 → O∆(τ1)O∆(τ2)circle =

CO

• 2 sin τ12

2

2∆

• normalize to expect. value of circular 1

2-BPS WL at large N [Erickson, Semenoff, Zarembo:00; Drukker, Gross:00; Pestun:07]

Wcircle =

2

√

λ I1(

√

λ)

• string theory side: change coordinates on Euclidean

AdS2 worldsheet from Poincare to hyperbolic disk ds2

2 = dρ2 + sinh2 ρ dτ2

• map of 4-point functions: replacing

1 t

2∆1 12 t2∆2 34

→

1

(2 sin τ1−τ2

2

)2∆1 (2 sin τ3−τ4

2

)2∆2

and conformally invariant cross ratio χ by χ = t12t34

t13t24

→

sin τ1−τ2

2

sin τ3−τ4

2

sin τ1−τ3

2

sin τ2−τ4

2

• 4-point of S5 fluctuations ya – can compare to localization:

certain WL correlators in N = 4 SYM can be computed via localization in terms of 2d YM

[Drukker, Giombi, Ricci, Trancanelli:07; Giombi, Pestun:09; Pestun:12]

• 1

8-susy WL defined on generic contour C on S2 ⊂ R4

couple to 3 of SYM scalars preserving susy W(C) = trPe

• C(iAj+ǫjklxkΦl)dxj ,

x2

1 + x2 2 + x2 3 = 1

mapped under localization to usual WL in 2d YM on S2

• 1

2-BPS circular WL is special case: C = great circle on S2

x1 = cos τ, x2 = sin τ: couples to Φ3 only (previously Φ6)

• local operators captured by localization:

position-dependent chiral primaries

[Drukker, Plefka:09]

(ǫ · Φ)J = (xjΦj + iΦ4)J,

ǫ2 = 0 , xjxj = 1 , j = 1, 2, 3 mapped by localization to (i ∗ F)J, ∗F = 1

2ǫmnFmn, F =2d YM

• crucial property – exact correlators are x-independent –

from localization to 2d YM: x-indep of correlators of ∗F (∗F =const on 2d YM eqs of motion)

• correlators with (ǫ · Φ)J along the loop: 2d YM theory result

matches

[Bonini, Grigulo, Preti, Seminara:15] result for J-generalized

Bremsstrahlung function [Gromov, Levkovich-Maslyuk, Sizov:13]

• let Ok ≡ ǫ(τk) · Φ(τk) be inserted on great circle in (12)-plane:

ǫ(τk) =

• cos τk, sin τk, 0, i, 0, 0

≡ ǫk , k = 1, . . . , 4 K4 = O1O2O3O4circle

• 2-point function indeed position-independent:

O1O2circle = CΦ(λ) ǫ(τ1)·ǫ(τ2)

• 2 sin τ12

2

2 = − 1

2CΦ(λ) = −B(λ)

• consider Φa1Φa2Φa3Φa4circle and contract with ǫk using

ǫ1 · ǫ2 ǫ3 · ǫ4 =

• 2 sin τ1−τ2

2

2 sin τ3−τ4

2

2

ǫ1·ǫ2 ǫ3·ǫ4 ǫ1·ǫ3 ǫ2·ǫ4 = χ2 , ǫ1·ǫ2 ǫ3·ǫ4 ǫ1·ǫ4 ǫ2·ǫ3 = χ2

(1−χ)2

• from normalized connected part of 4-point function:

τ-dependence indeed cancels out

O1O2O3O4conn

circle

O1O22

circle

=

1

√

λ

• G(1)

S (χ) − 2 5G(1) T (χ)

+ 1

χ2

• G(1)

T (χ) + G(1) A (χ)

+ (1−χ)2

χ2

• G(1)

T (χ) − G(1) A (χ)

• = − 3

√

λ + O( 1 λ)

• compare the value with prediction of localization to YM2:

4-point ˜ F inserted on circular WL

˜

F(τ1) ˜ F(τ2) ˜ F(τ3) ˜ F(τ4)YM2

circle ,

˜ F ≡ i ∗ F

• shortcut: start from general contour C and use that insertions
• f i ∗ F are equivalent to taking derivatives of W(C) over area

W(C + δC) = trP(1 +

dτδxµ ˙ xνiFµν + . . .)e

• iA
• dτδxµ ˙

xνiFµν = dτδxµ ˙ xν√gǫµνi ∗ F, δA = dτδxµ ˙ xν√gǫµν

• general contour C on S2: fixing areas A1,2 with A1 + A2 = 4π

inv of YM2 under area preserving diffeos → exp value same up to rescaling of coupling

WA1 =

2

√

λ′ I1(

√

λ′) , λ′ ≡ A1A2

4π2 λ = A1(4π−A1) 4π2

λ applies to general 1

8-BPS WL

W(C) = trPe

• C(iAj+ǫjklxkΦl)dxj
• case of 1

2-BPS circle corresponds to A1 = 2π, i.e. λ′ = λ

taking derivatives of logWA1 over A and setting A1 = 2π gives connected correlators of i ∗ F inserted on circle, e.g.,

˜

F(τ1) ˜ F(τ2)YM2

circle = ∂2 ∂A2

1 logWA1

• A1=2π = −

√

λI2(

√

λ) 4π2I1(

√

λ)

in agreement with Brehmsstrahlung function [Correa et al:12]

• for connected 4-point function get

˜

F(τ1) ˜ F(τ2) ˜ F(τ3) ˜ F(τ4)YM2

circle, conn

[ ˜

F ˜ FYM2

circle]2

=

∂4 ∂A4 1

logWA1

• A1=2π

( ∂2

∂A2 1

logWA1

• A1=2π)2

=

3(λ+4)

• I1(

√

λ)

2

−3λ

• I0(

√

λ)

2

λ

• I2(

√

λ)

2

= − 3

√

λ + 45 8λ3/2 + . . .

leading term matches the above result found from tree-level connected diagrams in AdS2

Open questions:

• extensions: include fermions; 1-loop corrections in AdS2;

6-point correlators; how to describe unprotected ops like Φ6?

• 1d analog of large spin expansion? semiclassical approxim

to explain universal large n behaviour of ∆ of Φ∂n

t Φ, etc.?

• intrinsic description of “induced” CFT1:

N = 8 superconformal QM?

non-local? (cf. SYK-like models [Gross,Rosenhaus:17])

• relation to integrability? how integrability of AdS5 × S5 string

is encoded in correlators in AdS2 computed in static gauge? connection to factorization of 2d S-matrix in BMN gauge?

• extension to all orders in

1

√

λ? conformal bootstrap in d = 1?

Mellin representation in AdS2 useful?