Smooth Wilson Loops and Yangian Symmetry in Planar N = 4 SYM ITP, Niklas Beisert Workshop on Hidden symmetries and integrability methods in super Yang Mills theories and their dual string theories Centre de Recherches Mathematiques, Montreal 10 August 2015 work with J. Plefka, D. Müller and C. Vergu (1506.07047, to appear); and with A. Garus (in progress)
I. Motivation CRM 2015, Niklas Beisert 2
Integrability Yangian Symmetry Spectrum of Local Operators: symmetry broken by boundaries (annulus) Scattering Amplitudes / Null Polygonal Wilson Loops: symmetry broken by IR/UV divergences Smooth Maldacena Wilson loops: finite observable with disc topology! CRM 2015, Niklas Beisert 3
II. Finite Wilson Loops CRM 2015, Niklas Beisert 4
Maldacena Wilson Loops Can define finite Wilson loops in N = 4 SYM: Couple scalars [ hep-th/9803002] Maldacena W = P exp (Aµ dx µ + Φ m q m dτ ). Maldacena Wilson loops where dx = q dτ: path is non-null in 4D; path is null in 10D (4 spacetime + 6 internal); locally supersymmetric object; no perimeter divergence (perimeter is null). Yangian symmetry: finite observable: could make meaningful statements; requires superconformal transformations; best done in superspace; Yangian symmetry demonstrated at leading order in θ s; symmetry up to subtleties regarding boundary terms. [ Müller, Münkler Plefka, Pollok, Zarembo] CRM 2015, Niklas Beisert 5
Conformal and Yangian Symmetry Conformal action (level-zero Yangian) by path deformation. Action equivalent to Wilson line with single insertion τ J k W = dτ W [1, τ] J k A(τ) W [τ, 0]. 0 Level-one Yangian action: bi-local insertion follows coproduct Ĵ k W = fmn k dτ 1 dτ 2 W [1, τ 2 ] J m A(τ 2 ) W [τ 2, τ 1 ] J n A(τ 1 ) W [τ 1, 0]. τ 2 >τ 1 Yangian is symmetry if (higher levels follow) 1 2 0 J k Tr W = 0, Ĵk Tr W = 0. Important issue: Yangian normally does not respect cyclicity. CRM 2015, Niklas Beisert 6
Open Questions (Formalism) How about full superspace (all orders in θ)? Difficulties: [ NB, Müller Plefka, Vergu] How to compute perturbative corrections? How to define finite Wilson loop (precisely)? How to define superconformal action (precisely)? How to define Yangian action (precisely)? CRM 2015, Niklas Beisert 7
Open Questions (Integrability) How about boundary terms? Difficulties: How to deal with boundary terms? How about regularisation and local terms? Is the action consistent with the constraints? Does the Yangian algebra close (and how)? [ NB, Müller Plefka, Vergu (to appear) ] CRM 2015, Niklas Beisert 8
III. Wilson Loops in Superspace CRM 2015, Niklas Beisert 9
N = 4 Superspace Extend spacetime (x µ ) to superspace (x β α, θ βa, θ b α ), a = 1,..., N. Gauge theory on superspace: Extend gauge potential (A µ ) to superspace (A αβ, A aβ, A α b ). way too many component fields for super Yang Mills theory. impose constraint: some components of F fixed (Φ scalar field) F aβ, γ d = 0, F aβ,cδ ε βδ ε acef Φ ef, F α b γ d ε α γ Φ bd. Remaining components of F contain fermionic fields Ψ, Ψ: F aβ, γδ ε βδ Ψa γ, F α b, γδ ε α γ Ψ δ b. Quantised field theory: Can derive gauge propagator A 1 A 2 from equations of motion; other propagators A 1 Φ 2, etc., follow from Φ F da. CRM 2015, Niklas Beisert 10
Wilson Line in Superspace Wilson line with scalar coupling in superspace: [ Ooguri, Rahmfeld Robins, Tannenhauser] W P exp (Ax (dx + idθ θ + iθd θ) + A θ dθ + A θ d θ + Φ q dτ ). Wilson loop expectation value at O(g 2 ): Tr W (A + Φ)1 (A + Φ) 2. Only above gauge propagator needed. 2 1 Kappa-symmetry: 8 local fermionic symmetries; extend path reparametrisation to 1 8 superalgebra (fat path); one bosonic constraint dx = q dτ; UV finiteness expected. CRM 2015, Niklas Beisert 11
Finiteness Expand mixed chiral gauge propagator at (τ 1, τ 2 ) = (τ, τ + ɛ) [ ] 2 A + 1 A 1 2 dτ dɛ ɛ 1 2ip ( θ θ) +.... 2 ɛ p 2 1 Result depends on superspace covariant direction p = ẋ + i θ θ iθ θ. Sub-leading terms cancel between opposite chiralities. Chiral and anti-chiral propagators are non-singular. Scalar propagator contributes similar singularity [ q 2 1 (Φ q)1 (Φ q) 2 p 2 ɛ + 1 q q 1 2 ɛ q 2 ɛ Gauge-scalar propagators are non-singular. ] p ṗ +.... p 2 Singularities cancel for p 2 + q 2 = 0 (10D null) or p 2 + q 2 = const.. CRM 2015, Niklas Beisert 12
Conformal Symmetry of Wilson Loops Wilson loop expectation value at order O(g 2 ): Tr W A1 A 2. 2 1 Conformal action on propagator is non-trivial J k A 1 A 2 = J k A 1 A 2 + A1 JA 2 = d1 f12+d k 2 f21. k JA Total derivative terms: represent gauge transformations of gauge potentials; zero for symmetries respecting gauge fixing (e.g. Poincaré); non-zero for others (e.g. special conformal); remaining function f 12 is finite for 1 2; cancel on closed Wilson loop. A CRM 2015, Niklas Beisert 13
IV. Yangian Invariance CRM 2015, Niklas Beisert 14
Yangian Symmetry of Wilson Loops Level-one momentum (dual conformal) P easiest: P P D + P L + Q Q. Action on propagator: Ĵ k A 1 A 2 = f k mn J m A 1 J n A 2 = d1 f k 12 d 2 f k 21. Almost zero up to derivative terms: no gauge fixing that respects all of P, D, L, Q, Q; bulk-boundary terms do not cancel; not clear how to compensate. Fundamental problems: definition not cyclic, definition not gauge invariant. JA JA CRM 2015, Niklas Beisert 15
Improved Symmetry Action JA is not gauge covariant but can rewrite conformal action Two resulting terms: Field strength F is covariant; D(JX A) is a boundary term. JA = JX F + D(JX A). Drop derivative term, obtain gauge-covariant conformal action: J k W = dτ W [1, τ] J k X(τ) F (τ) W [τ, 0]. In fact, standard superconformal action on fields. Closure of conformal symmetry modulo gauge transformations [J m, J n ] = f mn k J k + G[G mn ], G mn = J m X A J n X B F AB. CRM 2015, Niklas Beisert 16
Improved Yangian Action Improved level-one Yangian action: Ĵ k W = fmn k dτ 1 dτ 2 W [1, τ 2 ] J m X 2 F 2 W [τ 2, τ 1 ] J n X 1 F 1 W [τ 1, 0]. τ 2 >τ 1 Resolves both fundamental problems! Curious identity needed for cyclicity: f k mn[j m, J n ] = 0 f k mnf mn l = 0, f k mng mn f k mn J m X A J n X B F AB = 0. Is satisfied for PSU(2, 2 4) and for fields of N = 4 gauge theory! New problem: Yangian algebra does not close right away. Residual terms can be written as conformal transformations. Sufficient for invariance of Wilson loops. CRM 2015, Niklas Beisert 17
Yangian Invariance of Propagator Level-one generators almost annihilate propagator. Proof: Ĵ k da 1 A 2 = f k mn J m F 1 J n A 2 = f k mnj m F 1 J n A 2 f k mn F1 J m J n A 2 = f k mnj m F 1 (J n X F ) 2 1 2 f k mn F1 [J m, J n ]A 2 = 0. Therefore action on gauge propagator yields double total derivative: Ĵ k A 1 A 2 = d1 d 2 R k 12. For level-one momentum P and mixed chiral gauge fields: P αβ A + 1 A 1 2 d1 d 2 (x 12 iθ 12 θ12 ). β α Scalar fields as components of field strength Φ F : Ĵ k Φ 1 A 2 = Ĵ k Φ 1 Φ 2 = 0. CRM 2015, Niklas Beisert 18
Yangian Invariance of Wilson Loop Action on Wilson line leaves a local contribution: d 1 d 2 R12 k = (d 2 R12) k 1=2 d 2 R02 k = (d 2 R12) k 1=2. 1<2 2 Need to adjust local action of Yangian Ĵ k W = Ĵk bi-localw + dτ W [1, τ] Ĵk A(τ) W [τ, 0], Ĵ k A 1 = (d 1 R k 21) 2=1. Wilson loop expectation value at O(g 2 ) is Yangian invariant! Regularisation and renormalisation: Local term divergent (d 2 R k 21) 2=1 ɛ 2 for cut-off ɛ. Need to renormalise local and boundary part of Yangian action. CRM 2015, Niklas Beisert 19 2 2
V. Yangian Symmetry? CRM 2015, Niklas Beisert 20
What now? Still invariant at higher orders? Are there other invariant objects? Can we prove the symmetry in general? CRM 2015, Niklas Beisert 21
Would like to show: Ĵ S = 0 CRM 2015, Niklas Beisert 22
Invariance of the Action Aim: Show Yangian invariance of the (planar) action. NB, Garus [(in progress)] How to apply Ĵ to the action S? distinction of planar and non-planar parts unclear which representation: free, non-linear, quantum? Hints: Propagator is Yangian invariant (up to gauge artefacts) OPE of (invariant) Wilson loops contains Lagrangian L Lagrangian is a sequence of fields Use above (classical, non-linear) representation! Essential features of the action: action is single-trace (disc topology) action is conformal (required for cyclicity) action is not renormalised (no anomalies) S CRM 2015, Niklas Beisert 23
Equations of Motion Application on the action needs extra care. Consider the equations of motion instead: Ĵ(e.o.m.)? e.o.m. Need this for consistency! Quantum formalism usually on-shell... Dirac equation is easiest: D Ψ + [Φ, Ψ] = 0. Bi-local action of Ĵk on the Dirac equation: if k mn{j m A, J n Ψ} + f k mn{j m Φ, J n Ψ}? = 0. Many terms cancel, however, some have no counterparts. CRM 2015, Niklas Beisert 24
Local Terms in Yangian Action How to make equations of motion invariant? Not yet specified action on single fields: P αβ A γδ ε α γ ε βδ {Φ ef, Φ ef }, P αβ Ψ γ d ε βγ {Φ de, Ψ e α }, P αβ Φ cd = 0. All terms cancel properly. Dirac equation Yangian-invariant! shown invariance for all equations of motion: Φ, Ψ and A; other Yangian generators Ĵk follow from algebra. Equations of Motion Yangian-invariant! Notes: anti-commutators in line with odd parity of Yangian generator Ĵk ; almost irrelevant for Wilson loop expectation value at O(g 2 ); CRM 2015, Niklas Beisert 25
Issues of Lagrangian Would like to show invariance of action S = d 4 x L Ĵ L? = µ Kµ. Difficulties: cyclicity: where to cut open trace? conformal symmetry should help, but different terms/lengths. complexity: many terms, spinor algebra, signs, traces,...? should we use equations of motion? Attempt combination of all possible orderings: to no avail... ; ordering issue persists; result does depend on cyclic representative; must not! Wrong approach. really! [Earlier claim at IGST was in fact trivial (J 2 vs. Ĵ). sorry!] CRM 2015, Niklas Beisert 26
Invariance of the Action Equivalent If invariance of the action cannot be expressed, then what? Reconsider conformal symmetry of action, write as variations: δs 0 = JS = dx Jφ a (x) δφ a (x). Notes: must be zero w/o applying equations of motion! (otherwise trivial); JS is a polynomial in the fields (happens to be zero). Vary again by a field Invariance of e.o.m. follows J δs 0 = δ(js) δ 2 S = Jφ a + δ(jφ a) δφ c δφ c δφ a δφ c + δ(jφ a) δφ c δφ c Stronger statement: no use of e.o.m.! δs δφ a = 0. δs δφ a. CRM 2015, Niklas Beisert 27
Exact Invariance of the E.o.M. How to interpret above statement? Invariance of action is generating functional for exact invariance of the equations of motion. exact invariance sufficient to derive invariance of quantum correlators (up to anomalies). not traces, no cyclicity issues! Could apply to Yangian. Take inspiration from formal variation: Ĵ k δs δs ( δ(ĵk φ a ) fmn k (J m φ a ) δ δs δφ c δφ a δφ c δφ a δφ b δ δφ c ) (J n φ b ) = 0. Correctly predicts appearance of e.o.m. in Yangian variation of e.o.m.. Proper definition of integrability! CRM 2015, Niklas Beisert 28
VI. Implications CRM 2015, Niklas Beisert 29
Higher Loops Above proof of Yangian symmetry is: classical, but non-linear in the fields. How about quantum effects? We know: variation of action is exactly invariant: δs; propagators are linearly invariant: 1/L 0 (up to gauge); interaction vertices are not invariant: S i = S S 0. Non-linear composition in Feynman diagrams is invariant! S i S i S i = S i S i S i CRM 2015, Niklas Beisert 30
Non-Linear Cancellations Example: conformal symmetry at O(g 4 ). Three diagrams (II, Y, Φ): Conformal invariance requires all of them: conformal symmetry acts on perimeter (linearly and non-linearly); vertices and propagators are conformal; linear equation of motion contracts a propagator to a point; non-linear action cancels other end of propagator; residual gauge terms shift along perimeter; cancel by themselves. Cancellation at interface terms: V (II Y), collapsed Y (Y Φ) Exact invariance of e.o.m. sufficient, but all e.o.m. terms needed! CRM 2015, Niklas Beisert 31
Non-Linear Invariance Exact classical non-linear symmetry is sufficient. Derivation by λ, e.g. Wilson loop expectation value W S W. λ For N c use insertions into planar faces only. S Ĵ S S Equations of motion hold in quantum theory (Schwinger Dyson). CRM 2015, Niklas Beisert 32
Anomalies? Classical symmetries may suffer from quantum anomalies: Not clear how to deal with anomalies for non-local action (in colour-space not necessarily in spacetime). Violation of (non-local) current? Cohomological origin? However: Not an issue for Wilson loop expectation value at one loop. Anomalies from interplay of symmetry deformation by regularisation and quantum divergences; N = 4 is finite, no anomalies expected. CRM 2015, Niklas Beisert 33
Gauge Fixing Yangian action is gauge invariant: by construction all fields mapped to gauge covariant fields. What impact does gauge fixing have on Yangian symmetry? gauge equations of motion change; introduction of ghosts. should consider BRST symmetry: how to represent symmetry on unphysical fields and ghosts? Consider conformal symmetry first assume all conformal transformations act trivially on ghosts; action conformal modulo BRST exact terms; equations of motion conformal modulo particular terms; follows from variation of action. Can apparently proceed like this for Yangian action. Gauge fixing does not seem to disturb Yangian symmetry. CRM 2015, Niklas Beisert 34
Does the Yangian algebra close? Algebra Algebra comprises Yangian and gauge transformations. Several terms left over due to gauge transformations. Extra terms are gauge and conformal transformations. Serre relations non-linear and very complicated. CRM 2015, Niklas Beisert 35
VII. Conclusions CRM 2015, Niklas Beisert 36
Conclusions Maldacena Wilson Loops: Definition of Maldacena Wilson loops in full superspace. Finiteness of smooth loops at O(g 2 ). Yangian Invariance of Wilson Loops: How to act with Yangian on Wilson loops. Yangian symmetry of Wilson loop expectation value at O(g 2 ). Yangian Symmetry of Planar N = 4 SYM: Equations of motion & action variation invariant. Planar N = 4 SYM integrable. No anomalies to be expected?! CRM 2015, Niklas Beisert 37