Minimal surfaces in AdS 5, Wilson loops and Amplitudes Juan Maldacena Fernando Alday Davide Gaiotto Amit SeverPedro Vieira Collaborators Planar N=4 SYM and Integrability Planar N=4 is integrable and we should be able to compute quantities for all values of the coupling. Great progress in the problem of the operator spectrum. Next: Wilson loops, amplitudes, correlation functions. Solve exactly string theory in a new background with RR fields (beyond flat space, WZW, plane waves and their orbifolds). Minahan Zarembo Beisert Eden Staudacher Gromov Kazakov Vieira Frolov Arutyunov + many others Gromov, Kazakov, Vieira Arutyunov Frolov Bombardelli, Fioravanti, Tateo Why ? Solvable 4d gauge theory Understand better the relation between large N field theories and string theory. Learn useful lessons for connecting large N gauge theories to strings in other cases. Useful for computing amplitudes in QCD (for LHC) Strong coupling and classical strings Classical strings in AdS Structure of integrability is very transparent Useful starting point for learning about the full quantum problem (hopefully) Lots of recent progress from the weak coupling side. Bena, Polchinski, Roiban Arkani-Hamed, Cachazo et al. Strong coupling Weak coupling Amplitudes and Wilson loops Scattering amplitudes in N=4 super Yang Mills at strong coupling. Related via a certain T-duality to a polygonal Wilson loop. At strong coupling they can be computed using a minimal surface that ends on the polygonal Wilson loop. Alday & JM Berkovits & JM n cusps, 3(n-5) cross ratios. cusp k i = x i + 1 x i k 1 ; k 2 ; k n The problem Area(Cross ratios) Solution Consider a family of polygons, depending on Cross ratios Functional equations for Y: Integral equations for Y. - m j appear in the boundary conditions. Y system, similar to equations appearing in the study of the thermodynamic equations (TBA) of integrable theories. Area = Free energy of TBA = Y i ( ; m j ) Y + s Y s = ( 1 + Y s + 1 )( 1 + Y s 1 ) ; Y = Y ( i 2 ) l og Y s = m s cos h + Z d ~ 2 1 cos h ( ~ ) l og ( 1 + Y s 1 )( 1 + Y s + 1 )( ~ ) Yang Yang Zamolodchikov A rea = X s Z 1 1 d 2 m s cos hl og ( 1 + Y s ( )) Method : Use integrability We will not find the surface. Only the area. Analogy: creation and annihilation operators for the harmonic oscillator. a ; a y Integrability and the flat connection G/H, g ~ g h - Trivial equation: - Equations of motion: Combine into: One parameter family of flat connections. SO(2,4)/SO(1,4) or SU(4)/Sp(4) J = g 1 d g = K + H d A + A ^ A = 0 d J + J ^ J = 0 A = K z d ze 2 + H + e 2 K z d z d ( g K g 1 ) = 0 Equations for A are called Hitchin equations. A given Hitchin solution whole one parameter family of flat connections. F romase t o ff ourso l u t i ons ! recons t ruc t g. g 1 ; a = ; a S o l ve: ( d + A ) = 0 ! ( z ; z ; ) = ( ) Connected to wall crossing, N=2 theories, Gaiotto, Moore, Neitzke Description of string states Flat connection conserved charges Infinite number of charges = P e R A ( ) = 1 + Z j 0 + 2 ZZ j 0 j 0 + T r [ ] ( ) Kazakov, Marshakov, Minahan, Zarembo j 0 = j a 0 T a R ; j 1 = j a 1 T a R For Polygonal Wilson loops Worldsheet is the full complex plane We have Stokes sectors in the plane for large z. Each sector is related to a cusp. a = e i a z n = 4 + c : c : 1 2 n In each Stokes sector there is a unique smallest solution: Connection A in SL(4,C) we can contract solutions with epsilon tensors s i s i ^ s j ^ s k ^ s l = h s i s j s k s l i Cross Ratios Y does not depend on rescaling of s i for a fixed J = g -1 dg non trivial gauge invariant information of the flat connection. ( Similar to conserved charges of the cylinder problem. ) Y ( ) Y = h s i s i + 1 s j s j + 1 ih s k s k + 1 s l s l + 1 i h s i s i + 1 s k s k + 1 ih s j s j + 1 s l s l + 1 i = x 2 ij x 2 kl x 2 i k x 2 j l F or = 0, t h eseare t h eor i g i na l space t i mecrossra t i os. Functional Equations We can derive some functional equations for the We will need two facts: 1) What is 2) Relations between inner products Y ( ) s j ( + i 2 ) s j ( + i 2 ) = U s j + 1 ( ) s = e z n = 4 e + Stokes sectors rotate as we shift We have the n solutions s i, and there are many inner products we can make. Not all independent. Relations between them. Plucker relations. h 12 ih 34 i = h 13 ih 24 i h 23 ih 14 ih 1 2 ih 3 4 i = h 1 3 ih 2 4 i h 2 3 ih 1 4 i ; = 56 SU(2) SU(4) 0 1 2 m m+1 m+2 We can choose a set of functions T a,s 1) Relations between inner products 2) Express some of them by shifting Relations functional equations. Equations T + as T a ; s = T a 1 ; s T a + 1 ; s + T a ; s 1 T a ; s + 1 Y a ; s = T a ; s + 1 T a ; s 1 T a + 1 ; s T a 1 ; s Hirota equations Y functions a =1,2,3 s =0,1, , n-4 Y - System Not to be confused with the Y system that gives the quantum operator dimensions m=1, , n-5 Simple Y function equations: 3(n-5) Y functions. We can convert these into integral equations. For this we need the analytic structure. Analytic on a strip and large behavior 3 (n-5) parameters m i ; c i l og Y i m i cos h + c i + Computing the area Area is infinite. We can regularize it in a well understood way By some manipulations one can show that the interesting part of the area is Free energy of TBA A rea = R T r [ K z K z ] J = g 1 d g = K + H A = A d i vergen t + A BDS + A i n t eres t i ng A i n t eres t i ng = X i Z 1 1 d 2 m i cos hl og ( 1 + Y i ( )) R d p l og ( 1 + e ( p ) ) Strategy Pick some parameters m i, c i Solve integral equations for Y Compute the area. all have the same area. Y ( = i ' ) Regular Polygons Z n symmetric polygons One parameter family. Y = constant. Correspond to high temperature of TBA (like a CFT limit of the integrable system) Simple answer, Y i = sines and cosines A = 2 Two loop result in the weak coupling expansion Strong coupling n=6 Bern, Dixon, Kosower, Roiban, Spradlin, Vergu, Volovich / Drummond, Henn, Korchemsky, Sokatchev Del Duca, Duhr, Smirnov A = 2 3 u = 1 cos 2 3 Summary Wilson loops (or amplitudes) in AdS 5 Reduced it to a set of integral equations We solved them in some particular cases. Future Extend to the full quantum problem. Make contact with the weak coupling analysis. (Arkani-Hamed, Cachazo, et al) Learn the useful lessons for other gauge theories, apply it for the LHC, etc