Master Symmetry and Wilson Loops in AdS/CFT Florian Loebbert - - PowerPoint PPT Presentation

Master Symmetry and Wilson Loops in AdS/CFT

Florian Loebbert

Humboldt University Berlin Phys.Rev. D94 (2016), arXiv: 1606.04104 Nucl.Phys. B916 (2017), arXiv: 1610.01161

with Thomas Klose and Hagen Münkler

Integrability in Gauge and String Theory Paris, July 2017

.

Motivation

Integrability and Nonlocal Symmetries

Yang–Baxter equation:

R12 R13 R23

=

R23 R13 R12

⋆ Rational Solutions → infinite dimensional Yangian Algebra [Drinfel’d

1985 ]

⋆ Yangian spanned by local level-0 (Lie algebra) and bilocal level-1 generators, e.g. in 2d field theory: J(0)

a

=

• x

x ja

, J(1)

a

≃ fa

bc

• x1<x2

x1 jb x2 jc

Lorentz boost relates quantum charges (e.g. Gross–Neveu [Bernard

1991 ]):

[B, J(1)] ≃ J(0)

x t

Symmetry structure of rational integrable models?

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 1 / 22

Example of a “Master” Symmetry

Heisenberg Spin Chain (rational integrable model) ⋆ Hamiltonian: H(2) =

k H(2) k

=

k(1k −Pk).

Integrability: Tower of charges 1) Local Hamiltonians: [H(m), H(n)] = 0, m, n = 2, 3, . . . 2) Nonlocal Yangian Y [su(2)]: [J(n)

a , H]|bulk = 0,

n = 0, 1, . . . Unified via monodromy matrix T a(u): U −1 tr T(u) ≃ 1 + u H(2) + u2H(3) + . . . , Ta(u) ≃ 1 + 1 uJ(0)

a

+ 1 u2 J(1)

a

+ . . . . Boost Operator: [Tetel’man

1982 ]

d duT(u) = [B, T(u)], with B =

• k

k H(2)

k

Discrete version of 2d Lorentz boost.

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 2 / 22

Example of a “Master” Symmetry

Heisenberg Spin Chain (rational integrable model) ⋆ Hamiltonian: H(2) =

k H(2) k

=

k(1k −Pk).

Integrability: Tower of charges 1) Local Hamiltonians: [H(m), H(n)] = 0, m, n = 2, 3, . . . 2) Nonlocal Yangian Y [su(2)]: [J(n)

a , H]|bulk = 0,

n = 0, 1, . . . Unified via monodromy matrix T a(u):

B B B B

U −1 tr T(u) ≃ 1 + u H(2) + u2H(3) + . . . , Ta(u) ≃ 1 + 1 uJ(0)

a

+ 1 u2 J(1)

a

+ . . . . Master Symmetry Boost Operator: [Tetel’man

1982 ]

d duT(u) = [B, T(u)], with B =

• k

k H(2)

k

Discrete version of 2d Lorentz boost.

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 2 / 22

Planar AdS5/CFT4

Specific integrable model of rational type: Strings on AdS5 × S5

duality

4d N = 4 SYM Theory Two theories with large amount of symmetry: Superconformal symmetry psu(2, 2|4)

extends to planar limit

Yangian Y [psu(2, 2|4)] Nonlocal Yangian symmetry has been identified for

◮ String theory:

[

Bena Polchinski Roiban ’03][ Hatsuda Yoshida ’05] [Janik ’06 ][ Plefka,Spill Torrielli ’06 ][Beisert ’06 ][. . .]

• Classical strings on AdS5 × S5 • Worldsheet S-matrix

◮ Gauge theory: [Dolan, Nappi

Witten ’03 ][ Plefka Drummond Henn ’09 ] [ Mueller, Muenkler Plefka,Pollok Zarembo ’13 ][Beisert, Garus Rosso ’17 ][. . .]

• Dilatation operator • 4d Amplitudes • Wilson loops • The action

Is there an AdS/CFT master symmetry?

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 3 / 22

Holographic Wilson Loops

Consider specific observable:

◮ Maldacena–Wilson loop W(γ) along smooth contour γ in planar

SU(N) N = 4 SYM theory (reduction from 10d WL): W(γ) = 1 N tr Pe

i

γ dσ

• Aµ(x) ˙

xµ+Φi(x)| ˙ x|ni

,

• n2 = 1

◮ Strong-coupling (λ ≫ 1) expectation value determined by area Amin

• f minimal surface (string worldsheet) bounded by γ [Maldacena

1998 ]:

W(γ) ≃ e−

√ λ Amin(γ) .

Strong-coupling observation in [

Ishizeki Kruczenski Ziama ’11]:

One-parameter family of AdS3 Wilson loops such that contour and surface depend on spectral parameter but the area does not. What is the symmetry behind this observation?

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 4 / 22

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The Setup

Symmetric Space Models

AdS5 for instance described by coset SO(2, 4)/SO(1, 4) ⇒ stay general for the moment: Symmetric Z2 coset M = G/H with algebras g = h ⊕ m such that [h , h] ⊂ h , [h , m] ⊂ m , [m , m] ⊂ h .

◮ The dynamical field is group-valued g(z) ∈ G with z = σ + iτ. ◮ Flat g-valued Maurer–Cartan form

U = g−1dg, dU + U ∧ U = 0. with U = A + a and projections A = U|h and a = U|m.

◮ Model defined by the action

S =

• tr (a ∧ ∗a) =
• dσ2√

h hαβ tr(aαaβ). Symmetries?

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 5 / 22

Flat Current and Integrability

Local gauge transformations: g → gR(τ, σ) with R ∈ H. Global G-symmetry: g → Lg with L ∈ G: A → A , a → a , Infinitesimal form: Lie algebra symmetry of the action generated by δǫg = ǫg, ǫ ∈ g. Associated Noether current j = −2gag−1 is conserved and flat d ∗ j = 0, dj + j ∧ j = 0. ⇒ The model is integrable.

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 6 / 22

Spectral Parameter and Lax Connection

Spectral Parameter: Introduce parameter u as auxiliary quantity.

◮ Conservation and flatness of j packaged into flatness of

Lax connection: ℓu = u 1 + u2 (u j + ∗j) , dℓu + ℓu ∧ ℓu = 0 .

◮ Defines flat deformation of Maurer–Cartan form (L0 = U):

Lu = U + g−1ℓug, dLu + Lu ∧ Lu = 0.

◮ Tower of nonlocal Yangian charges J(n) from expansion of

monodromy: T(u) = P exp

• ℓu ≃ exp
• u J(0) + u2 J(1) + . . .
• .

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 7 / 22

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Nonlocal Master Symmetry

Physical Spectral Parameter

Lift spectral parameter to physical field g(z):

◮ Deform g(z) into gu(z) ≡ g(z, u) via (non-)auxiliary linear problem:

dgu = guLu, gu(z0) = g(z0), with z0 some reference point.

◮ Solved by

gu(z) = χu(z)g(z), χu(z0) = 1, if χu satisfies dχu = χuℓu.

◮ Transformation g → gu leaves action and equations of motion

invariant! Observed in [ Eichenherr

Forger 1979]. No deformation of the theory!

Note: This is an on-shell symmetry: Eom ⇒ ℓu flat ⇒ χu well-defined.

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 8 / 22

Master Symmetry

Generator δ of this symmetry from expansion of χu around u = 0:

• δg(z) = χ(0)(z)g(z),

χ(0)(z) = z

z0

∗j. (nonlocal) On components of Maurer–Cartan form U = A + a we have, cf. [

Beisert Luecker ’12]

• δA = 0,
• δa = −2 ∗ a.

⋆ Symmetry of the equations of motion since eom: d ∗ a + ∗ a ∧ A + A ∧ ∗ a = 0

• δ

− → U flat: da + a ∧ A + A ∧ a = 0 “Master” symmetry? Show now: Lie algebra and master symmetry yield all other symmetries.

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 9 / 22

Integrable Completion

If δ0 generates a symmetry, then so does conjugation with χu: δ0,u g = χ−1

u δ0(χug). ◮ Any symmetry δ0 turns into one-parameter family of symmetries. ◮ Refer to δ0,u as the integrable completion of the symmetry δ0.

Show this using following symmetry criterion:

◮ When is variation δg = ηg a symmetry? ◮ Answer: Iff we have

g−1d ∗ (dη + [j, η])g ∈ h. Now consider examples for δ0,u: 1) Yangian 2) Master symmetry

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 10 / 22

1) Yangian from Completion (δ0 = δǫ)

Completion of Lie algebra symmetry δ0 = δǫ yields Yangian variations: δǫ,u g = χ−1

u ǫχu g,

δǫ,u =

∞

• n=0

un δ(n)

ǫ

, with leading orders δ(0)

ǫ g = δǫg,

δ(1)

ǫ g = [ǫ, χ(0)]g.

Action of master symmetry on Lie algebra Noether current j gives

• δj = −2 ∗ j + [χ(0), j].

Yields standard expressions for Yangian level-zero and level-one charge: J(0) =

• ∗j,
• δ

− → J(1) = 2

• j +
• σ1<σ2

[∗j1, ∗j2].

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 11 / 22

2) Completion of Master Symmetry (δ0 = δ)

Consider conjugation of δ0 = δ with χu:

• δu g = χ−1

u

δ(χug) = · · · = χ−1

u d duχu g.

Tower δ(n) of “master symmetries” of equations of motion:

• δu =

∞

• n=0

un δ(n).

◮ Generators

δ(n>0) relate to Virasoro algebra [Schwarz

1995 ].

◮ Associated charges are Casimirs of Lie algebra charges J = J(0):

J(0) := tr(JJ) , J(1) := tr(JJ(1)) , . . . Note: δ on-shell symmetry. Strict Noether procedure would require off- shell continuation. Use eom only via dχu = χuℓu.

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 12 / 22

Master Symmetry and Noether Charges

For some symmetry δ0 with associated charge J0:

δ0 J0 δ0,u J0,u Noether Master Master Noether

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 13 / 22

Raising Operator on Yangian Charges

Introduce angle parametrization θ(u): eiθ = 1 − iu 1 + iu such that deformation of Maurer–Cartan form U = A + a becomes: A → A, a → au(θ) = e−iθaz dz + eiθa¯

z d¯

z. Define tower of Yangian charges J(n) as expansion in angle-coordinate θ: J(θ) =

∞

• n=0

θn n! J(n) . Master symmetry acts as level-raising operator on Yangian algebra:

• δ J(n) = J(n+1),

d dθJ(θ) =

δ J(θ). Note: Same charges from Noether procedure for Yangian variations δ(n).

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 14 / 22

Raising Operator on Yangian Charges

Introduce angle parametrization θ(u): eiθ = 1 − iu 1 + iu such that deformation of Maurer–Cartan form U = A + a becomes:

reminiscent of Lorentz boost?

A → A, a → au(θ) = e−iθaz dz + eiθa¯

z d¯

z. Define tower of Yangian charges J(n) as expansion in angle-coordinate θ: J(θ) =

∞

• n=0

θn n! J(n) . Master symmetry acts as level-raising operator on Yangian algebra:

• δ J(n) = J(n+1),

d dθJ(θ) =

δ J(θ). Note: Same charges from Noether procedure for Yangian variations δ(n).

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 14 / 22

Comparison to Lorentz Boost

σ τ

Remember Heisenberg chain: Discrete boost is level-lowering operator for Yangian charges. Master symmetry vs worldsheet Lorentz boost: Infinitesimal version of the master symmetry δAz = 0, δaz = −iθaz .

• 1. Only the m-valued part of the connection, az, is transformed.
• 2. Worldsheet rotation z → eiϑz gives an extra term:

δaz = −iϑaz − i ϑ(z∂ − ¯ z ¯ ∂)az . ⇒ Master symmetry differs from standard Lorentz boost.

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 15 / 22

Overview of Symmetries

• δ
• δ

Yangian Symmetry Variation Charge Level-0 δǫg = ǫg J(0) ≡ J =

• ∗j

Level-1 δ(1)

ǫ g = [ǫ, χ(0)]g

J(1) = 2

• j +
• [∗j1, ∗j2]

Completion δǫ,ug = χ−1

u ǫχu g

Ju =

• ∗ju

Master Symmetry Variation Charge Level-0

• δg = χ(0)g

J(0) = tr

• J J
• Level-1
• δ(1)g = [χ(1) − (χ(0))2]g

J(1) = tr

• J J(1)

Completion

• δug = χ−1

u d duχu g

Ju = 1

2 tr

• JuJu
• Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT

16 / 22

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Back To Wilson Loops

Wilson Loops and Minimal Surfaces

Strong-coupling expectation value of Maldacena–Wilson loop given by minimal surface area: [Maldacena

1998 ]

W(γ)

λ≫1

= exp

• −

√ λ 2π Aren(γ)

• Take minimal surface in AdS5 ≃ SO(2, 4)/SO(1, 4) with metric

ds2 = y−2 dXµdXµ + dy2 in Poincaré coordinates.

◮ Surface described by boundary conditions

y(τ = 0, σ) = 0 , Xµ(τ = 0, σ) = xµ(σ) .

◮ Area divergent ⇒ use cutoff y = ǫ and subtract divergence ⇒ Aren ◮ Symmetry of area functional is symmetry of renormalized area Aren

Master symmetry deforms minimal surface with contour γ into min- imal surface with contour γu (area preserved) →

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 17 / 22

Master Symmetry Beyond Strong Coupling?

Weak coupling observation in [Dekel

2015]: Wilson loop expectation value

not invariant under strong-coupling contour deformation! No symmetry beyond strong coupling? At strong coupling we now understand deformation as nonlocal master symmetry of the underlying symmetric space model. Strong coupling (derived): JWL =

• σ1<σ2

dσ1dσ2 ξνa(x1) δAren(γ) δxν

1

ξµ

a(x2)

δ δxµ

2

ξ: conformal Killing vector

◮ Representation of symmetry generator may be coupling-dependent. ◮ Contour deformation not the same at any coupling λ.

Generic coupling? E.g.: J(λ)

WL =

• σ1<σ2

dσ1dσ2 ξνa(x1) δ log W(γ) δxν

1

ξµ

a(x2)

δ δxµ

2

⇒ Easy to see that J(λ)

WL annihilates Wilson loop expectation value. Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 18 / 22

Discrete Geometry

⋆ Master symmetry is nonlocal: Transformation at some point of min- imal surface depends on shape of entire surface. ⋆ Explicit application of master symmetry to Wilson loop contour re- quires full minimal surface solutions, which are rare. ⇒ Employ discrete approximation for Euclidean case! Sequence of master transformations applied to discrete minimal surfaces at cutoff y =

1 10 in EAdS3:

Ellipse: Triangle: θ = 0 θ = 3π

16

θ = 3π

4

θ = π

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 19 / 22

Another Numerical Example: The Cat

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 20 / 22

Lightlike Polygonal Wilson Loops?

⋆ Lightlike polygonal Wilson loops

dual

← → scattering amplitudes ⋆ Numerics not good in Lorentzian signature → need analytic solutions Four-cusp minimal surface [

Alday Maldacena ’07]

◮ is the only explicitly known solution, ◮ conformally equivalent to single cusp.

⇒ In the four-cusp/single-cusp case the master symmetry is equivalent to a conformal transformation. How to approach higher point Wilson loops?

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 21 / 22

Summary & Outlook

Summary Symmetric space models allow for nonlocal master symmetry that

◮ generates spectral parameter, ◮ acts as a level-raising operator on the Yangian charges, ◮ parametrizes classes of minimal surfaces.

Generalizes to supersymmetric AdS5 × S5 case [Chandia,Linch,

Vallilo ’16 ] [ Münkler PhD Thesis ’17]

Future Directions

◮ Off-shell formulation of Master and Yangian symmetry?

• see [ Dolan

Roos ’80][Hou,Ge Wu ’81] for off-shell Yangian of principal chiral model

◮ Go beyond strong coupling? Quantization of master symmetry? ◮ Application to generic cusp Wilson loops (amplitudes)? ◮ Extension to other AdS/CFT dualities? ◮ Relation to novel quantum boost symmetry of exact AdS5/CFT4

S-matrix [

Borsato Torrielli 2017]?

→ see Riccardo Borsato’s talk!

Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 22 / 22