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 R 12 R 23 Yang–Baxter equation: = R 13 R 13 R 23 R 12 ⋆ 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) J (1) bc = , ≃ f a j a j b j c a a x x 1 <x 2 x 1 x 2 x Lorentz boost relates quantum charges (e.g. Gross–Neveu [ Bernard 1991 ]): [B , J (1) ] ≃ J (0) t x 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 ( 1 k − P k ) . k Integrability: Tower of charges [ H ( m ) , H ( n ) ] = 0 , 1) Local Hamiltonians: m, n = 2 , 3 , . . . [J ( n ) 2) Nonlocal Yangian Y [ su (2)] : 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 + 1 u J (0) u 2 J (1) + . . . . a a Boost Operator: [ Tetel’man 1982 ] � d k H (2) Discrete version of 2d d uT ( u ) = [ B, T ( u )] , with B = k Lorentz boost. k 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 ( 1 k − P k ) . k Integrability: Tower of charges [ H ( m ) , H ( n ) ] = 0 , 1) Local Hamiltonians: m, n = 2 , 3 , . . . [J ( n ) 2) Nonlocal Yangian Y [ su (2)] : 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) + . . . , Master Symmetry B B T a ( u ) ≃ 1 + 1 + 1 u J (0) u 2 J (1) + . . . . a a Boost Operator: [ Tetel’man 1982 ] � d k H (2) Discrete version of 2d d uT ( u ) = [ B, T ( u )] , with B = k Lorentz boost. k Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 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: extends to Superconformal symmetry psu (2 , 2 | 4) Yangian Y [ psu (2 , 2 | 4)] planar limit Nonlocal Yangian symmetry has been identified for ◮ String theory: Bena [ Roiban ’03 ][ Hatsuda Yoshida ’05 ] [ Janik ’06 ][ Plefka,Spill Torrielli ’06 ][ Beisert ’06 ][ . . . ] Polchinski • Classical strings on AdS 5 × S 5 • Worldsheet S-matrix Mueller, Muenkler ◮ Gauge theory: [ Dolan, Nappi Plefka Witten ’03 ][ Henn ’09 ] [ Zarembo ’13 ][ Beisert, Garus Rosso ’17 ][ . . . ] Drummond Plefka,Pollok • 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): � x | n i � i � W ( γ ) = 1 x µ +Φ i ( x ) | ˙ γ dσ A µ ( x ) ˙ n 2 = 1 N tr P e , � ◮ Strong-coupling ( λ ≫ 1 ) expectation value determined by area A min of minimal surface (string worldsheet) bounded by γ [ Maldacena 1998 ]: √ λ A min ( γ ) . � W ( γ ) � ≃ e − Ishizeki Strong-coupling observation in [ Ziama ’11 ]: Kruczenski 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? Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 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 d g, d U + U ∧ U = 0 . with U = A + a and projections A = U | h and a = U | m . ◮ Model defined by the action � � d σ 2 √ h h αβ tr( a α a β ) . S = tr ( a ∧ ∗ a ) = 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 ǫ ∈ g . δ ǫ g = ǫg, Associated Noether current j = − 2 gag − 1 is conserved and flat d ∗ j = 0 , d j + 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: u ℓ u = 1 + u 2 ( u j + ∗ j ) , d ℓ u + ℓ u ∧ ℓ u = 0 . ◮ Defines flat deformation of Maurer–Cartan form ( L 0 = U ): L u = U + g − 1 ℓ u g, d L u + L u ∧ L u = 0 . ◮ Tower of nonlocal Yangian charges J ( n ) from expansion of monodromy: � � � u J (0) + u 2 J (1) + . . . T ( u ) = P exp ℓ u ≃ exp . Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 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: d g u = g u L u , g u ( z 0 ) = g ( z 0 ) , with z 0 some reference point. ◮ Solved by g u ( z ) = χ u ( z ) g ( z ) , χ u ( z 0 ) = 1 , if χ u satisfies d χ 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. Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 8 / 22
Master Symmetry Generator � δ of this symmetry from expansion of χ u around u = 0 : � z � δg ( z ) = χ (0) ( z ) g ( z ) , χ (0) ( z ) = ∗ j. (nonlocal) z 0 On components of Maurer–Cartan form U = A + a we have, cf. [ Luecker ’12 ] Beisert � � δA = 0 , δa = − 2 ∗ a. ⋆ Symmetry of the equations of motion since � δ eom: d ∗ a + ∗ a ∧ A + A ∧ ∗ a = 0 − → U flat: d a + 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 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 ∗ (d η + [ j, η ]) g ∈ h . Now consider examples for δ 0 ,u : 1) Yangian 2) Master symmetry Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 10 / 22
1) Yangian from Completion ( δ 0 = δ ǫ ) Completion of Lie algebra symmetry δ 0 = δ ǫ yields Yangian variations: ∞ � u n δ ( n ) δ ǫ,u g = χ − 1 u ǫχ u g, δ ǫ,u = , ǫ n =0 with leading orders δ (0) δ (1) ǫ g = [ ǫ, χ (0) ] g. ǫ g = δ ǫ 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 (1) = 2 δ ∗ j, − → j + [ ∗ j 1 , ∗ j 2 ] . σ 1 <σ 2 Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 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 d u χ u g. u δ ( n ) of “master symmetries” of equations of motion: Tower � ∞ � u n � � δ ( n ) . δ u = n =0 δ ( n> 0) relate to Virasoro algebra [ Schwarz ◮ Generators � 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 d χ u = χ u ℓ u . Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 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 Florian Loebbert: Master Symmetry and Wilson Loops in AdS/CFT 13 / 22
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