Integrability in AdS/CFT: open problems D. Serban, IPhT Saclay - PowerPoint PPT Presentation

Integrability in AdS/CFT: open problems D. Serban, IPhT Saclay Miniworkshop on integrability in string theory, Galileo Galilei Institute Florence, 29-30 October 2008

Summary • the AdS/CFT corespondence • arguments for integrability • conjectured Bethe Ansatz equations • spin chain vs. sigma model features • connection with the Hubbard model • TBA and finite size effects • integrability and the amplitudes

AdS/CFT correspondence N =4 gauge theory: superconformal symmetry PSL(2,2|4) conformal group SO(4,2) ≅ SU(2,2) R-group SO(6) ≅ SU(4) Field content SU(N) matrices: [Maldacena 97] [Witten 98] [Gubser, Klebanov, Polyakov 98] Type IIB string theory on AdS 5 x S 5 : sigma model on PSL(2,2|4)/SO(4,1)xSO(5) [Metsaev, Tseytlin 98]

AdS/CFT correspondence � ‘t Hooft coupling h g 2 = g 2 YM N String tension T = 2 g 16 π 2 g s = g String coupling Number of colors N g N N planar limit free strings strong coupling classical strings String states !"# S Tr ( Φ I 1 Φ I 2 ... Φ I L ) Local operators Energy of the string E Scaling dimension ∆ ( g ) Angular momenta J a R -charges E( g ), S 1 , S 2 , J 1 , J 2 , J 3

Integrability [Lipatov, 98] One loop dilatation operator String sigma model = is integrable spin chain classically integrable [Minahan, Zarembo, 02] [Bena, Polchinski, Roiban, 02] [Bena, Polchinski, Roiban, 02] s Z = Φ 1 + i Φ 2 d W = Φ 3 + i Φ 4 : tr ZZZW W ZZZW W W ZW ZZZZ . . . string solution, e.g. [Frolov, Tseytlin, 02] L ˆ � D 1 = 2 (1 − P l,l +1 ) [Kazakov, Marshakov, l =1 Minahan, Zarembo, 04] solution of the classical sigma model solution in terms of Bethe Ansatz equations in terms of an algebraic curve

Integrability extends to the whole PSL(2,2|4) group [Beisert, Staudacher 03] survives at higher loops [Beisert, Kristjansen, Staudacher 03] [Beisert 03-04] [conjecture] There exists a model which is integrable for any value of the coupling constant g ? spin chain at g → 0 sigma model at g → ∞ perturbative N=4 SYM perturbative string theory on AdS 5 x S 5 survives at higher loops survives at higher loops

� The all-loop Bethe Ansatz equations psu(2,2|4) [Beisert, Staudacher, 05] [Beisert, 05] K 2 K 4 1 − 1 /x 1 ,k x + u 1 ,k − u 2 ,j + i u 1 � � 4 ,j [Arutynov, Frolov, Zamaklar, 06] 2 1 = , u 1 ,k − u 2 ,j − i 1 − 1 /x 1 ,k x − 4 ,j 2 j =1 j =1 K 2 K 3 K 1 u 2 ,k − u 3 ,j + i u 2 ,k − u 1 ,j + i u 2 ,k − u 2 ,j − i u 2 � � � 2 2 1 = , u 2 ,k − u 2 ,j + i u 2 ,k − u 3 ,j − i u 2 ,k − u 1 ,j − i 2 2 j =1 j =1 j � = k K 2 K 4 x 3 ,k − x + u 3 ,k − u 2 ,j + i u 3 � � 4 ,j 2 1 = , x 3 ,k − x − u 3 ,k − u 2 ,j − i 4 ,j 2 j =1 j =1 � � − − − Dressing factor � L � x + K 4 u 4 ,k − u 4 ,j + i 4 ,k � � u 4 ,k − u 4 ,j − i σ 2 ( x 4 ,k , x 4 ,j ) = x − − − u 4 [Janik’06; 4 ,k j � = k Beisert-Hernandez-Lopez’06; K 1 1 − 1 /x − K 3 x − K 5 x − K 7 1 − 1 /x − 4 ,k x 1 ,j 4 ,k − x 3 ,j 4 ,k − x 5 ,j 4 ,k x 7 ,j � � � � Beisert-Eden-Staudacher’06] , × 1 − 1 /x + x + x + 1 − 1 /x + 4 ,k x 1 ,j 4 ,k − x 3 ,j 4 ,k − x 5 ,j 4 ,k x 7 ,j j =1 j =1 j =1 j =1 map � � � � − − − − j =1 j =1 j =1 j =1 x + 1 x = u K 6 K 4 x 5 ,k − x + u 5 ,k − u 6 ,j + i u 5 � � 4 ,j 2 1 = , g x 5 ,k − x − u 5 ,k − u 6 ,j − i 4 ,j 2 j =1 j =1 K 6 K 5 K 7 u 6 ,k − u 5 ,j + i u 6 ,k − u 7 ,j + i u 6 ,k − u 6 ,j − i u 6 � � � 2 2 1 = , u 6 ,k − u 5 ,j − i u 6 ,k − u 7 ,j − i u 6 ,k − u 6 ,j + i 2 2 j � = k j =1 j =1 , x ± + 1 x ± = 1 � u ± i � K 6 K 4 1 − 1 /x 7 ,k x + u 7 ,k − u 6 ,j + i u 7 � � 4 ,j g 2 2 1 = . u 7 ,k − u 6 ,j − i 1 − 1 /x 7 ,k x − 4 ,j 2 j =1 j =1 magnon symmetry: centrally extended [su(2|2)]^2

Particular features of the N=4 SYM integrable model • one-parametrer integrable super-spin chain • long-range interaction ( ∼ Inozemtsev or Hubbard at half filling) ∼ g 2 n ⇒ spin chain n + 1 spins • length-changing interactions g 3 • BAE only asymptotic (L → ∞ ) • crossing-like symmetry (particle/antiparticle) → dressing phase sigma model � 1 + 16 g 2 sin 2 p/ 2 − 1 E ( p ) = ± • not (exactly) relativistically invariant • scattering matrix does not depend on rapidity difference

Connection(s) with the Hubbard model 2 seemingly unrelated connections with the 1d Hubbard model - su(2) sector reproducible from the Hubbard model at half filling (except for the dressing phase) [Rej, Serban, Staudacher, 06] M u k − u j + i � u ( p ) = 1 e ip k L = � 2 cot p 1 + 16 g 2 sin 2 p u k − u j − i , k 2 , 2 j =1 [Beisert, Dippel, j � = k Staudacher, 04] � 1 + 16 g 2 sin 2 p E ( p ) = 2 − 1 . - Beisert’s su(2|2) symmetric S-matrix ∼ Hubbard Shastry’s R-matrix ⇒ hidden supersymmetry in the Hubbard model [Beisert, 06]

The BDS model from Hubbard at half-filling - itinerant fermions with onsite repulsion ( U=1/ g ): L L H = 1 − 1 � � φ = π ( L + 1) e i φ σ c † i, σ c i +1 , σ + e − i φ σ c † � � � c † i, ↑ c i, ↑ c † i +1 , σ c i, σ i, ↓ c i, ↓ 2 g 2 g 2 2 L i =1 σ = ↑ , ↓ i =1 - Hilbert space: 4 states per site: E → 1 /g 2 - at half filling N=L and g → 0 the spin states decouple: | ↑↓↑↑ ... ↑ > singly-occupied states g 2 - fluctuations ~ XXX model spin permutation

The BDS model from Hubbard at half-filling higher orders: itinerant fermion making 2n step walks on the lattice h 2 = (1 − P i,i +1 ) , > > − 2(1 − P i,i +1 ) + 1 g 4 h 4 = 2(1 − P i,i +2 ) , < < 15 2 (1 − P i,i +1 ) − 3(1 − P i,i +2 ) + 1 h 6 = 2(1 − P i,i +3 ) − 1 2(1 − P i,i +3 )(1 − P i +1 ,i +2 ) +1 2(1 − P i,i +2 )(1 − P i +1 ,i +3 ) . - coincides with the dilatation operator up to three loops - corrects a result from [Klein, Seitz, 73]

The BDS model from Hubbard at half-filling Lieb-Wu equations (half filling): [Lieb, Wu, 68] M � u j − 2 g sin( � q n + φ ) − i/ 2 q n L = e i e q n + φ ) + i/ 2 , n = 1 , . . . , L u j − 2 g sin( � L large, M small j =1 � L � M u k − 2 g sin( � q n + φ ) + i/ 2 u k − u j + i q n + φ ) − i/ 2 = u k − u j − i , k = 1 , . . . , M u k − 2 g sin( � n =1 j =1 j � = k � Shiba (particle/hole) transformation: H ( g ; φ , φ ) → − H ( − g ; π − φ , φ ) − M 2 g 2 Dual Lieb-Wu equations: � M u j − 2 g sin( q n − φ ) − i/ 2 e iq n L = u j − 2 g sin( q n − φ ) + i/ 2 , n = 1 , . . ., 2 M j =1 � 2 M � M u k − 2 g sin( q n − φ ) + i/ 2 u k − u j + i u k − 2 g sin( q n − φ ) − i/ 2 = − u k − u j − i , k = 1 , . . . , M n =1 j =1 j � = k

The BDS model from Hubbard at half-filling Bound-state solutions (strings): [Takahashi, 72] L → ∞ 2 + p 2 + p q 1 − φ = π q 2 − φ = π 2 + i β , 2 − i β complex momenta: � 2 g sin( q 2 − φ ) � p � u ± i/ 2 = 2 g cos 2 ∓ i β u � � 2 g sin( q 1 − φ ) � u ( p ) = 1 2 cot p 1 + 16 g 2 sin 2 p 2 2 M u k − u j + i e ip k L = � Lieb-Wu eq. → BDS eq. u k − u j − i , k j =1 j � = k �� � 1 1 + 16 g 2 sin 2 p E ( p ) = 2 − 1 2 g 2

But the Hubbard construction: - does not extends to other sectors than su(2), e.g. su(1|1) - does not take into account the dressing phase - is it possible to define an all-loop hamiltonian? - how to take into account the fine-size effects? Remark: the su(2|2) symmetric S-matrix is also an essential ingredient of the new AdS 4 x CP 3 duality [Aharony, Bergman, Jafferis Maldacena, 08] cf [Gromov, Vieira 08] OSp(2,2|6)

TBA program Use the field-theoretical methods to compute finite-size corrections: [Ambjorn, Janik, Kristjansen 05] - Lüscher terms [Janik, Lukowski 07,...] - put the theory on the cylinder and make a “double Wick rotation” 1/T → R [Arutynov, Frolov 07; Bajnok, Janik,08] - difficulty : the rotated theory is not equivalent to the original one (“mirror theory”) 1/T > > > > R simplest wrapping correction: the four loop L=4 (Konishi operator) [Fiamberti, Santambrogio, Sieg, Zanon ,08]: perturbative computation in N=4 SYM = [Bajnok, Janik,08]: from TBA

The origin of integrability? There is more in N=4 SYM than the dilatation operator... - the multigluon amplitudes have a particular structure at higher loops - > BDS conjecture [Bern, Dixon, Smirnov 05] (fails for n>5) - this structure was checked at strong coupling for 4 (and many) gluons [Alday, Maldacena 07] - dual superconformal symmetry [Drummond, Henn, Korchemsky, Sokatchev, 07-08] (and duality between multigluon amplitudes and the Wilson loops with lightlike cusps) Is there any connection between this structure and the integrability? [Berkovits, Maldacena, 08] [Beisert, Ricci, Tseytlin, Wolf, 08] Integrable open spin chain for gluon amplitudes [Lipatov, 08]

Conclusion • the AdS/CFT correspondence provides an unusual integrable structure • it puts together many known integrable models into a highly symmetric structure • the complete definition still not under control ( what is the hamiltonian?) • what are the consequences of integrability on the overall structure of N=4 SYM? • which are the other (integrable) dualities?