Master Symmetry and Wilson Loops in AdS/CFT

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 = R13 R23 R23 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 = ja, J (1) a f a bc jb jc x x x 1<x 2 x1 x2 Lorentz boost relates quantum charges (e.g. Gross Neveu [ Bernard 1991 ]): [B, J (1) ] J (0) t x Symmetry structure of rational integrable models? 1 / 22

Example of a Master Symmetry Heisenberg Spin Chain (rational integrable model) Hamiltonian: H (2) = k H(2) k = k (1 k P k ). 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) + u 2 H (3) +..., T a (u) 1 + 1 u J(0) a + 1 u 2 J(1) a +.... Boost Operator: [ Tetel man 1982 ] d du T (u) = [B, T (u)], with B = k k H (2) k Discrete version of 2d Lorentz boost. 2 / 22

Example of a Master Symmetry Heisenberg Spin Chain (rational integrable model) Hamiltonian: H (2) = k H(2) k = k (1 k P k ). 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 U 1 tr T (u) 1 + u H (2) + u 2 H (3) +..., T a (u) 1 + 1 u J(0) a Boost Operator: [ Tetel man 1982 ] d du T (u) = [B, T (u)], with B = k B B + 1 u 2 J(1) a +.... k H (2) k Master Symmetry Discrete version of 2d Lorentz boost. 2 / 22

Planar AdS 5 /CFT 4 Specific integrable model of rational type: Strings on AdS 5 S 5 4d N = 4 SYM Theory duality Two theories with large amount of symmetry: Superconformal symmetry psu(2, 2 4) extends to planar limit Nonlocal Yangian symmetry has been identified for Bena String theory: [ Polchinski Hatsuda ][ Roiban 03 Yoshida 05] [ Janik 06 ][ Plefka,Spill Torrielli 06 ][ Beisert 06 ][... ] Classical strings on AdS 5 S 5 Worldsheet S-matrix Yangian Y [psu(2, 2 4)] Dolan, Nappi Gauge theory: [ Witten 03 ][ Drummond Plefka Muenkler Beisert, Garus ] [Mueller, Plefka,Pollok ][ Henn 09 Rosso 17 ][... ] Zarembo 13 Dilatation operator 4d Amplitudes Wilson loops The action Is there an AdS/CFT master symmetry? 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 Pei dσ A γ µ(x)ẋ µ +Φ i(x) ẋ n i, n 2 = 1 Strong-coupling (λ 1) expectation value determined by area A min of minimal surface (string worldsheet) bounded by γ [ Maldacena 1998 ]: W (γ) e λ A min(γ). Strong-coupling observation in [ Ishizeki Kruczenski Ziama 11 ]: One-parameter family of AdS 3 Wilson loops such that contour and surface depend on spectral parameter but the area does not. What is the symmetry behind this observation? 4 / 22

. The Setup

Symmetric Space Models AdS 5 for instance described by coset SO(2, 4)/SO(1, 4) stay general for the moment: Symmetric Z 2 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 1 dg, 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) = dσ 2 h h αβ tr(a α a β ). Symmetries? 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 δ ɛ g = ɛg, ɛ g. Associated Noether current j = 2gag 1 is conserved and flat d j = 0, dj + j j = 0. The model is integrable. 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: l u = u 1 + u 2 (u j + j), dl u + l u l u = 0. Defines flat deformation of Maurer Cartan form (L 0 = U): L u = U + g 1 l u g, dl u + L u L u = 0. Tower of nonlocal Yangian charges J (n) from expansion of monodromy: T (u) = P exp l u exp ( u J (0) + u 2 J (1) +... ). 7 / 22

. Nonlocal Master Symmetry

Physical Spectral Parameter Lift spectral parameter to physical field g(z): Deform g(z) into g u (z) g(z, u) via (non-)auxiliary linear problem: dg u = g u L u, g u (z 0 ) = g(z 0 ), with z 0 some reference point. Solved by if χ u satisfies g u (z) = χ u (z)g(z), χ u (z 0 ) = 1, dχ u = χ u l u. Transformation g g u 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 l u flat χ u well-defined. 8 / 22

Master Symmetry Generator δ of this symmetry from expansion of χ u around u = 0: δg(z) = χ (0) (z)g(z), χ (0) (z) = z z 0 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. 9 / 22

Integrable Completion If δ 0 generates a symmetry, then so does conjugation with χ u : δ 0,u g = χ 1 u δ 0 (χ u g). Any symmetry δ 0 turns into one-parameter family of symmetries. Refer to δ 0,u as the integrable completion of the symmetry δ 0. Show this using following symmetry criterion: When is variation δg = ηg a symmetry? Answer: Iff we have g 1 d (dη + [j, η])g h. Now consider examples for δ 0,u : 1) Yangian 2) Master symmetry 10 / 22

1) Yangian from Completion (δ 0 = δ ɛ ) Completion of Lie algebra symmetry δ 0 = δ ɛ yields Yangian variations: with leading orders δ ɛ,u g = χ 1 u ɛχ u g, δ ɛ,u = δ (0) n=0 u n δ (n) ɛ, ɛ g = δ ɛ g, δ ɛ (1) g = [ɛ, 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) δ = j, J (1) = 2 j + [ j 1, j 2 ]. σ 1<σ 2 11 / 22

2) Completion of Master Symmetry (δ 0 = δ) Consider conjugation of δ 0 = δ with χ u : δ u g = χ 1 u δ(χ u g) = = χ 1 d u du χ u g. Tower δ (n) of master symmetries of equations of motion: δ u = u n δ (n). n=0 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 offshell continuation. Use eom only via dχ u = χ u l u. 12 / 22

Master Symmetry and Noether Charges For some symmetry δ 0 with associated charge J 0 : Noether δ 0 J 0 Master Master Noether δ 0,u J 0,u 13 / 22

Raising Operator on Yangian Charges Introduce angle parametrization θ(u): e iθ = 1 iu 1 + iu such that deformation of Maurer Cartan form U = A + a becomes: A A, a a u(θ) = e iθ a z dz + e iθ a z d z. Define tower of Yangian charges J (n) as expansion in angle-coordinate θ: J(θ) = n=0 θ n n! J(n). Master symmetry acts as level-raising operator on Yangian algebra: δ J (n) = J (n+1), d dθ J(θ) = δ J(θ). Note: Same charges from Noether procedure for Yangian variations δ (n). 14 / 22

Raising Operator on Yangian Charges Introduce angle parametrization θ(u): e iθ = 1 iu 1 + iu such that deformation of Maurer Cartan form U = A + a becomes: reminiscent of Lorentz boost? A A, a a u(θ) = e iθ a z dz + e iθ a z d z. Define tower of Yangian charges J (n) as expansion in angle-coordinate θ: J(θ) = n=0 θ n n! J(n). Master symmetry acts as level-raising operator on Yangian algebra: δ J (n) = J (n+1), d dθ J(θ) = δ J(θ). Note: Same charges from Noether procedure for Yangian variations δ (n). 14 / 22

Comparison to Lorentz Boost τ σ Remember Heisenberg chain: Discrete boost is level-lowering operator for Yangian charges. Master symmetry vs worldsheet Lorentz boost: Infinitesimal version of the master symmetry δa z = 0, δa z = iθa z. 1. Only the m-valued part of the connection, a z, is transformed. 2. Worldsheet rotation z e iϑ z gives an extra term: δa z = iϑa z i ϑ(z z )a z. Master symmetry differs from standard Lorentz boost. 15 / 22

Overview of Symmetries Variation Yangian Symmetry Charge δ Level-0 δ ɛ g = ɛg J (0) J = j Level-1 δ (1) ɛ g = [ɛ, χ (0) ]g J (1) = 2 j + [ j 1, j 2 ] Completion δ ɛ,u g = χ 1 u ɛχ u g J u = j u δ 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 δ u g = χ 1 d u du χ u g J u = 1 2 tr ( ) J u J u 16 / 22

. 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 A ren(γ) Take minimal surface in AdS 5 SO(2, 4)/SO(1, 4) with metric ds 2 = y 2 ( dx µ dx µ + dy 2) in Poincaré coordinates. Surface described by boundary conditions y(τ = 0, σ) = 0, X µ (τ = 0, σ) = x µ (σ). Area divergent use cutoff y = ɛ and subtract divergence A ren Symmetry of area functional is symmetry of renormalized area A ren Master symmetry deforms minimal surface with contour γ into minimal surface with contour γ u (area preserved) 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): J WL = dσ 1 dσ 2 ξ νa δa ren (γ) (x 1 ) δx ν ξ µ δ a (x 2 ) 1 δx µ 2 σ 1<σ 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 = dσ 1 dσ 2 ξ νa δ log W (γ) (x 1 ) δx ν ξ µ δ a (x 2 ) 1 σ 1<σ 2 Easy to see that J (λ) WL annihilates Wilson loop expectation value. δx µ 2 18 / 22

Discrete Geometry Master symmetry is nonlocal: Transformation at some point of minimal surface depends on shape of entire surface. Explicit application of master symmetry to Wilson loop contour requires 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 EAdS 3: Ellipse: Triangle: θ = 0 θ = 3π 16 θ = 3π 4 θ = π 19 / 22

Another Numerical Example: The Cat 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? 21 / 22

Summary Summary & Outlook 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 AdS 5 S 5 case [ Chandia,Linch, Vallilo 16 ] [ Münkler PhD Thesis 17] Future Directions Off-shell formulation of Master and Yangian symmetry? see [ Dolan ] for off-shell Yangian of principal chiral model Roos 80][ Hou,Ge Wu 81 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 AdS 5 /CFT 4 S-matrix [ Borsato Torrielli 2017]? see Riccardo Borsato s talk! 22 / 22