Introduction to Integrability in AdS/CFT: Lecture 4 Rafael Nepomechie University of Miami
Introduction
Recall: 1-loop dilatation operator for all single-trace operators in N=4 SYM is integrable 1-loop anomalous dimensions are given by a set of Bethe equations Higher loops?
Plan Today: dilatation operator at higher loops all-loop asymptotic S-matrix all-loop asymptotic Bethe equations Final: bound states & their S-matrices
dilatation operator at higher loops
Dilatation operator has been computed perturbatively SU(2) sector: D (1) =2 D (2) =2... in various sectors at small ( 2,3) loop order L k=1 L k=1 (I P k,k+1 ) D = n=0 ( λ 16π 2 ) n D (n) [Beisert, Kristjansen & Staudacher 03,...] n: loop order # coupled nearest neighbors ( 4I +6P k,k+1 P k,k+1 P k+1,k+2 P k+1,k+2 P k,k+1 ) 2 3 D (n) n+1 Interaction range grows with loop order!
Seem to have perturbative integrability : higher charges also constructed perturbatively Q k = n=1 Q 1 = D [Q k,q l ] = 0 λ n Q (n) k i.e., 0= [( λq (1) k + λ 2 Q (2) k + ), ( λq (1) l + λ 2 Q (2) l + [ ] ([ ] [ ]) = λ 2 Q (1) k,q(1) l +λ 3 Q (1) k,q(2) l + Q (2) k,q(1) l + }{{}}{{} 0 0 )] Does not seem to come from an R-matrix New kind of integrability!? Finding D to all loops seems (at least for now) hopeless...
all-loop asymptotic S-matrix
R-matrix does not seem to work... Try S-matrix approach! long-range interaction S-matrix is asymptotic : valid only for widely-separated particles Ψ (12) (x 1,x 2 ) e i(p 1x 1 +p 2 x 2 ) + S(p 2,p 1 )e i(p 2x 1 +p 1 x 2 ) x 1 x 2 Can compute S-matrix perturbatively, from known D, by introducing fudge functions: [Staudacher 04] Ψ (12) (x 1,x 2 )=e i(p 1x 1 +p 2 x 2 ) f(x 2 x 1,p 1,p 2 ) + S(p 2,p 1 )e i(p 2x 1 +p 1 x 2 ) g(x 2 x 1,p 1,p 2 ) S(p 1,p 2 )= f(x, p 1,p 2 ) = 1 + n=0 λ n S (n) (p 1,p 2 ) λ n+x f (n) (p 1,p 2 ) n=0 But will not take us far, since we hardly know... D
Audacious idea: guess the exact S-matrix! [Beisert 05] (analogy: sine-gordon [Zamolodchikov 2 79] ) Guiding principle: symmetry Global symmetry algebra is psu(2,2 4) But it is partially broken by the vacuum : so(6) so(4) = su(2) su(2) psu(2, 2 4) psu(2 2) psu(2 2) R 0 = Z L common central charge H D R 56 R 56 0 = L 0 H 0 =0!
Elementary excitations: L x=1 e ipx 1 Z x χ L Z χ {φ aȧ,d α α Z ; ψ αȧ, ψ a α } a =1, 2, α =3, 4, ȧ = 1, 2 α = 3, 4 (2 2) (2 2) = (4 + 4 4 + 4) = (8 8) Fundamental reps ZF operators: su(2 2) su(2 2) A k k (p) =A k (p) A k(p) k =1,..., 4, k = 1,..., 4
Focus on single copy of su(2 2) Generators: La b, Rα β bosonic a, b = 1, 2, α, β = 3, 4 SUSY central charges Qα a, Q α a H, C, C Algebra: [ L b a, J c ] = δ b c J a 1 2 δb aj c, [ R β α, J γ ] = δ β γ J α 1 2 δβ αj γ [ L b a, J c] = δ c aj b + 1 2 δb aj c, { } Qα a, Qβ b = ɛ αβ ɛ ab C, [ R β α, J γ] = δ γ αj β + 1 2 δβ αj γ { } Q α a, Q β b = ɛ αβ ɛ ab C { } Qα a, Q β b = δ a b R β α + δ β αl a b + 1 2 δa b δ β αh
Action of symmetry generators on ZF operators: bosonic: [ L b a,a c(p) ] =(δcδ b a d 1 2 δb aδc d )A d (p), [ R β α,a γ(p) ] =(δγ β δα δ 1 2 δβ αδγ)a δ δ (p), [ L b a,a γ(p) ] =0 [ R β α,a c(p) ] =0 SUSY: [ ] Qα a A b (p) =e ip/2 a(p)δb a A α(p)+a b (p) Q α a [ ] Qα a A β (p) =e ip/2 b(p)ɛ αβ ɛ ab A b (p) A β (p) Q α a Q α a Q α a [ A b (p) =eip/2 c(p)ɛ ab ɛ αβ A β (p)+a b (p) Q α a A β (p) =eip/2 [ d(p)δ α β A a(p) A β (p) Q α a ] ]
central: C A i (p) =e ip [ a(p)b(p)a i (p)+a i (p) C ] C A i (p) =eip [ c(p)d(p)a i (p)+a i (p) C ] H A i (p) =[a(p)d(p)+b(p)c(p)] A i (p)+a i (p) H Determination of a, b, c, d: A i (p) 0 form a rep of algebra ad bc = 1 rep unitary d = a,c= b C on 2-particle states ab = ig(e ip 1) g constant
Consistent with a = gη, b = g i η ( ) x + x 1, c = g η x +, d = g x+ iη (1 x x + ) where x + + 1 x + x 1 x = i g, x + x = eip, η = e i p 4 i(x x + ) For 1-particle states: H = ig ( x 1 x x+ + 1 x + ) = 1 + 16g 2 sin 2 p 2 Compare with 1 loop to determine g: J =( 0 + γ) J = 1 + γ = 1 + λ 2π 2 sin2 p 2 + g = λ 4π
Consistent with a = gη, b = g i η ( ) x + x 1, c = g η x +, d = g x+ iη (1 x x + ) where x + + 1 x + x 1 x = i g, x + x = eip, η = e i p 4 i(x x + ) For 1-particle states: H = ig J =( 0 + γ) J = 1 + γ ( x 1 x x+ + 1 x + Compare with 1 loop to determine g: = 1 + λ 2π 2 sin2 p 2 + ) = 1 + 16g 2 sin 2 p 2 g = λ 4π exact! C, C essential
Useful parametrization in terms of Jacobi elliptic functions: p = 2 am z, x ± = 1 4g ( cn z sn z ± i ) (1 + dn z) elliptic modulus k = 16g 2 periods 2! 1 (real), 2! 2 (imaginary) [Arutyunov & Frolov 07]
Finally, can determine exact S-matrix: A i (p 1) A j (p 2)=S i j ij (p 1,p 2 ) A j (p 2 ) A i (p 1 ) Demand that the symmetry generators commute with 2-particle scattering J i.e, consider J A i (p 1) A j (p 2) 0 Can first exchange s, then move J to right J 0 = 0 A Or first move J to right, then exchange s A Consistency linear equations for S-matrix elements
bosonic generators S aa aa = A, S αα αα = D, Sab ab = 1 (A B), 2 Sba S αβ αβ = 1 (D E), 2 Sβα ab = 1 (A + B), 2 αβ = 1 (D + E), 2 S αβ ab = 1 2 ɛ abɛ αβ C, S ab αβ = 1 2 ɛab ɛ αβ F, S a α a α = G, S α a a α = H, S a α α a = K, S α a α a = L, a, b {1, 2}, a b α, β {3, 4}, α β
SUSY generators A = S 0 x 2 x+ 1 x + 2 x 1 B = S 0 [ x 2 x + 1 x + 2 x 1 η 1 η 2 η 1 η 2, +2 (x 1 x+ 1 )(x 2 x+ 2 )(x 2 + x+ 1 ) (x 1 x+ 2 )(x 1 x 2 x+ 1 x+ 2 ) C = S 0 2ix 1 x 2 (x+ 1 x+ 2 )η 1η 2 x + 1 x+ 2 (x 1 x+ 2 )(1 x 1 x 2 ), D = S 0, E = S 0 [ 1 2 (x 1 x+ 1 )(x 2 x+ 2 )(x 1 + x+ 2 ) (x 1 x+ 2 )(x 1 x 2 x+ 1 x+ 2 ) F = S 0 2i(x 1 x+ 1 )(x 2 x+ 2 )(x+ 1 x+ 2 ) (x 1 x+ 2 )(1 x 1 x 2 ) η 1 η 2, G = S 0 (x 2 x 1 ) (x + 2 x 1 ) η 1 η 1, H = S 0 (x + 2 x 2 ) (x 1 x+ 2 ) η 1 η 2, K = S 0 (x + 1 x 1 ) (x 1 x+ 2 ) η 2 η 1, L = S 0 (x + 1 x+ 2 ) (x 1 x+ 2 ) η 2 η 2 ] η1 η 2 η 1 η 2, η 1 = η(p 1 )e ip 2/2, η 2 = η(p 2 ), η 1 = η(p 1 ), η 2 = η(p 2 )e ip 1/2 ],
Satisfies YBE Does not have difference property; can be mapped to R-matrix for Hubbard model [Shastry 86] Has Yangian symmetry [more next lecture] [Beisert 07] su(2 2) symmetry determines S-matrix only up to overall scalar factor S 0 Full S-matrix: two copies of su(2 2) S-matrix S = S Ṡ
Determination of scalar factor: Assume unitarity S 12 (p 1,p 2 ) S 21 (p 2,p 1 )=I S 0 (p 1,p 2 ) S 0 (p 2,p 1 )=1 Assume crossing symmetry [Janik 06,...] C 1 1 S t 1 12 (z 1,z 2 ) C 1 S 12 (z 1 + ω 2,z 2 )=I S 0 (z 1,z 2 ) S 0 (z 1,z 2 ω 2 )= ( ( 1 x 1 1 x + 1 x 2 x 2 ) ) (x 1 x+ 2 ) (x + 1 x+ 2 )
Can solve under some additional physical assumptions S 0 (p 1,p 2 ) 2 = x 1 x+ 2 x + 1 x 2 dressing phase : 1 1 x + 1 x 2 1 1 x 1 x+ 2 σ(p 1,p 2 ) 2 σ(x ± 1,x± 2 )=R(x+ 1,x+ 2 ) R(x 1,x 2 ) R(x + 1,x 2 ) R(x 1,x+ 2 ), R(x 1,x 2 )=e i[χ(x 1,x 2 ) χ(x 2,x 1 )] χ(x 1,x 2 )= i z 1 =1 dz 1 2π z 2 =1 dz 2 2π ln Γ ( ) 1+ig(z 1 + 1 z 1 z 2 1 z 2 ) (x 1 z 1 )(x 2 z 2 ) [BES, BHL, DHM; review Vieira & Volin 10] Changrim Ahn
all-loop asymptotic Bethe equations
Recall: Bethe-Yang equations e ip kl = Λ(p k ; p 1,..., p M ), k =1,..., M Λ(p; p 1,..., p M ) eigenvalues of (inhomogeneous) transfer matrix t(p; p 1,..., p M ) = tr 0 S 01 (p, p 1 ) S 0M (p, p M ) Can determine using (nested) algebraic Bethe ansatz, etc. U j (x j,k ) 7 K j j =1 k =1 (j,k ) (j,k) u j,k u j,k + i 2 A j,j u j,k u j,k i 2 A j,j =1, j =1,..., 7 u j,k = g ( x j,k + 1 x j,k ), uj,k ± i 2 = g( x ± j,k + 1 x ± j,k )
Cartan matrix + A = U 2 = U 6 =1, ( x U 4 (x) =U s (x) 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 U 1 (x) =U3 1 1 (x) =U5 (x) =U 7(x) = x + ) L K1 k=1 S 1 aux(x, x 1,k ) K 3 k=1 S aux (x, x 3,k ) K 4 k=1 K 5 k=1 S aux (x 4,k,x) S aux (x, x 5,k ) K 7 k=1 S 1 aux(x, x 7,k ) S aux (x 1,x 2 )= 1 1/x+ 1 x 2 1 1/x 1 x 2 γ =2ig K 4 k=1 ( 1 x + 4,k 1 x 4,k, U s (x) = ) K 4 k=1 σ(x, x 4,k ) 2 First conjectured! Then derived [Beisert & Staudacher 05] [Beisert 05, Martins & Mello 07]
In weak-coupling limit λ 0,! reduce to 1-loop Bethe equations In thermodynamic limit L, λ, reduce to eqs from algebraic curve Valid only for long operators The problem with short operators: for length L, wrapping corrections (to the anomalous dimensions) appear at loop-order L, due to interactions of range L+1
Epilogue
We don t know the all-loop dilatation operator for single-trace operators in N=4 SYM Nevertheless, we know that the (all-loop) anomalous dimensions of long operators are given by a set of BEs! Key: all-loop S-matrix Based on su(2 2) symmetry To compute finite-size corrections for short operators, need also all-loop S-matrices for bound states su(2 2) symmetry is not enough; need also Yangian symmetry - final lecture!