Amplitudes & Wilson Loops at weak & strong coupling

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1 Amplitudes & Wilson Loops at weak & strong coupling David Skinner - Perimeter Institute Caltech - 29 th March 2012 N=4 SYM, 35 years a!er

2 Twistor space is CP 3, described by co-ords It carries a natural action of SL(4; C) Z A rz A CP 3 R 3,1 X Y Z x y x x Line in twistor space Two lines intersect Point in space-time Separation is null X AB = Z [A 1 ZB] 2 X X = 0 X r X

3 [Witten; Nair; Boels,Mason,DS] N =4 S = 1 2 SYM is described on twistor space by the action Ω Tr (A A + 23 A3 ) +g 2 d 4 8 x log det ( + A ) X! A(Z, χ) =a(z)+χ a ψ a (Z)+ +(χ) 4 g(z) [Ferber]! In axial gauge, Feynman diagrams are CSW diagrams + + +! Reduces to standard form if A is harmonic on each X

4 The Amplitude / Wilson Loop Duality

5 [Alday,Maldacena] MHV Amplitudes are Null Polygonal Wilson Loops p 2 p 2 x 2 x 3 p 1 p 3 p 1 x 1 p 3 p 4 p 4 x 4 momentum closed massless null conserved polygon particles edges! For n =4, 5 agrees with expectation from BDS ansatz! Duality checked at weak coupling by explicit calculation [Drummond,Henn,Korchemsky,Sokatchev;Brandhuber,Heslop,Travaglini; Bern,Dixon,Kosower,Roiban,Spradlin,Vergu,Volovich]

6 [Mason,DS] Space-time vertices Null edges of polygon Twistor lines Twistor vertices x 1 Z 1 Z 2 x n x 2 Z n X 1 The duality extends to all helicities if one constructs a supersymmetric extension of the Wilson Loop ( + A ) X U(σ 1, σ 2 ) = 0 U(σ 1, σ 1 ) = id N =4 twistor superfield

7 Superloop in sd N =4 Tree superamplitude x i X j X i x j Superloop in full N =4 Planar superamplitude X X j x i x X i x j

8 BCFW Recursion from the Loop Equations [Bullimore,DS] BCFW recursion begins by deforming momenta p i p i (r) subject to the constraints pi (r) = 0 p 2 i (r) = 0 Z n (r) Z n 1 Z n Z 1 For any such deformation, the twistors vary freely! Behaviour of twistor Wilson Loop is governed by (a holomorphic version of) the loop equations [Polyakov; Migdal,Makeenko] δ W[C] = ω ω δ 3 4 (Z, Z ) W[C 1 ] W[C 2 ] C C ds s WL origin of all dlogs

9 Z n r Z 1 Z n 1

10 Z n Z i 1 r Z(r i ) Z i Z 1 Z n 1 Z i

11 Z n Z i 1 r Z(r i ) Z i Z 1 Z n 1 Z i Intersecting twistor lines Null separation Factorization channel p i 1 x i p i p 1 (r) x 1 (r) p n (r)

12 For above simple deformation, the loop equations give W[C n ] = W[C n 1 ] + n 1 i=3 This is tree-level BCFW recursion. [n 1, n, 1,i 1,i] W[C i ] W[C i]! Repeating derivation using the full twistor action (including MHV vertices) gives all-loop generalization W[C n ] = W[C n 1 ] + + n 1 i=3 [n 1, n, 1,i 1,i] W[C i ] W[C i] DZ A DZ B [n 1, n, 1, A, B] W[C AB n+2]! Recursive construction of all-loop planar integrand [Arkani-Hamed,Bourjaily,Cachazo,Caron-Huot,Trnka]! dlog parameters describe locations around WL

13 Towards Strong Coupling

14 Z CP 3 4 defines a totally null super-ray Z i+1 (x i+1, θ i+1 ) Z i 1 Z i (x i, θ i ) x i+1 µ α =ix α α λ α x i χ a = θ αa λ α The corresponding space-time superloop ( ) 1 Tr P exp i A A = A N α α dx α α + Γ αa dθ αa involves a superconnection that is ] =0 λ α λ β { constrained by ferm αa λ α λ β [ bos α α, ferm βb, ferm } βb =0

15 Integrability along super null rays is worldline κ-symmetry! Agrees with restriction of Type IIB worldsheet κ-symmetry to boundary [Ooguri,Rahmfeld,Robins,Tannenhauser]! Chirality from QS gauge [Beisert,Ricci,Tseytlin,Wolf] (or from non-chiral superloop restricted to θ i =0) Niklas talk [Witten]!"#$%&"'&( "'%)*+',- Suggests that strong coupling limit of non-mhv amplitudes should be obtained from the natural generalization of MHV! " & "! "%$ case to full Type IIB [Alday,Bullimore,DS work in progress]! "#$ [c.f. Berkovits,Maldacena]

16 .. [Grigoriev,Tseytlin; Mikhailov,Schafer-Nameki] Pohlmeyer reduction of Type IIB on supercoset! Eom of sigma model on PSU(2,2 4) / SO(1,4) SO(5) equivalent to flatness of Z 4 -graded Lax connection L[ξ] = + A + ξ 1 Q 1 + ξ 2 K L[ξ] = + Ā + ξ Q 3 + ξ 2 K after partial κ fixing For strings, also need to impose Virasoro constraints str(kk) = 0 str( K K) = 0! Scalars appear in superloop only from order O(θ 2 )! On the boundary, Virasoro becomes condition 0 = (ẋ + θ θ θ θ) 2 +ẏ 2 [Ooguri,Rahmfeld,Robins,Tannenhauser] We choose a restricted solution where tr (KK) = 0 and tr AdS5 S 5(KK) = 0

17 [Alday,Gaiotto,Maldacena,Sever,Vieira] As in bosonic case, choose a boost generator q s.t. [ [q, K] =K q, K] = K Decomposing Lax flatness conditions in terms of q-charge forces some current components to vanish [ξ] := ξ q L ξ q [ξ] := ξ q L ξ q is a Z 8 -graded Lax connection, whose right (left) flat sections correspond to (dual) supertwistors! To see that we have boundary conditions corresponding to the superloop, need to impose correct asymptotics on...! Fixing residual κ-symmetry important to find conserved quantities

18 Controlling The Yangian Anomaly

19 twistor space momentum twistor space space-time dual space-time momentum space Original & dual superconformal algebras combine to form Yangian [Drummond,Henn,Plefka] + ks

20 Although much of this symmetry is broken at loop level, it can still be powerful so long as the structure of the breaking is understood W n =Z n F n e.g. [K µ, F n ]= Γ cusp 2 n i=1 x µ i,i+1 log x2 i,i+2 x 2 i 1,i+1! Fixes BDS ansatz, and remainder function must be dual conformally invariant [Drummond,Henn,Korchemsky,Sokatchev] Ideally, we would like to know how to constrain all the Yangian anomalies! cf complementary approach of [Bargheer,Beisert,Galleas,Henn, Loebbert,McLoughlin,Plefka]

21 Which dual supercharges are broken? Amplitudes in planar M = ( N =4 algebraic Yangian invariant can be written as ) ( bosonic loop integral ) Annihilated by Q, S Z / χ, but not by Q, S χ / Z! These (dual) supersymmetries also fail for finite quantities such as the remainder or ratio functions Recall: Self-dual theory Full theory Tree amplitudes Loop amplitudes! The broken supercharges are represented differently on the space of fields for self-dual and full N =4

22 Any susy transformation deforms the superloop δ Q(W n )= i N = i N Tr P (δx F(x, θ) Hol[A]) fermion-boson curvature Tr P ( ɛθ (F + + F + Ψ) Hol[A] )! In the self-dual theory, up to field equations and gauge transformations, the insertion is just [ Qsd, A ]! In the full theory, both the field equations and susy transformations are different, and there is a mismatch δ Q W = ig2 ε abcd 3!N Tr P[ɛ a dx b Φ cb Hol[A] fermion-fermion curvature

23 x i+1 Z i+1 X Z i X W x O X (i 1, i, i+1) Z i x i Z i 1 The anomaly can itself be expressed in terms of a WL δ Q W[C n ] = Γ cusp Z i+1 2 i V Ω = ɛ α a χ a V Ω W(..., Z i, Z, Z i+1,...) µ α D3 4 Z Z Contour takes residue as Z Z i Recursive construction with Z i Z i 1 correct transcendentality [Bullimore,DS; Caron-Huot,He]

24 Under the BCFW deformation Z Z(r) =Z + rz i+1 W(..., i, Z,i+1,...) = W(..., i, i+1,...) + i 2 j=i+2 [i, Z,i+1, j, j+1] W(j+1,..., i, Z j ) W(Z j, Z j,i+1,..., j)! Only inhomogeneous terms have required poles Z i+1 Z j p i x i x i+1 p i 1 p i+1 x Z i Z i 1 Z j+1 p j+1 x j+1 p j! On residue, and subloops share X (iz j ) Z j Z i! Dispersion integral over momentum fraction remains [Beisert,Henn,McLoughlin,Plefka; Sever,Vieira]

25 The anomaly for remainder function R n W n (W Ab n ) Γ cusp/g 2 χi =0 =e Γ cuspm (1) MHV Wn follows from Leibnitz rule & collinear behaviour of BDS Q(R n )=Γ cusp = Γ cusp V Ω [ W n+1 (Z) (W Ab n ) Γ cusp/g 2 χi =0 V Ω [ R n+1 (Z) R n W tree NMHV(Z) ] R n W tree NMHV(Z) ]! Also holds for superconformal transformations! With parity conjugate, fixes all Yangian anomalies

26 Conclusions

27 Unitarity & Analyticity Topological Strings Twistor Theory Grassmannians & Motives AdS/CFT N =4 SYM Yangians & Integrability Vast increase in technical power New structures at the heart of QFT