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“ Maths of Gauge and String Theory” London, 5/05/2012 Solving the spectral AdS/CFT Y-system Vladimir Kazakov (ENS,Paris) Collaborations with Gromov, Leurent, Tsuboi, Vieira, Volin Quantum Integrability in AdS/CFT • Y-system (for planar AdS5/CFT4 , AdS4/CFT3 ,...) dimensions of all local operators at any coupling (non-BPS, summing genuine 4D Feynman diagrams!) calculates exact anomalous Gromov, V.K., Vieira • Operators • 2-point correlators define dimensions – complicated functions of coupling • Y-system is an infinite set of functional eqs. We can transform Y-system into a finite system of non-linear integral equations (FiNLIE) using its Hirota discrete integrable dynamics and analyticity properties in spectral parameter Gromov, V.K., Leurent, Volin Alternative approach: Balog, Hegedus Konishi operator : numerics from Y-system Beisert, Eden,Staudacher Gubser,Klebanov,Polyakov ABA Gubser Klebanov Polyakov Y-system numerics Gromov,Shenderovich, Serban, Volin Roiban,Tseytlin Masuccato,Valilio Gromov, Valatka Gromov,V.K.,Vieira (recently confirmed by Frolov) millions of 4D Feynman graphs! 5 loops and BFKL Fiamberti,Santambrogio,Sieg,Zanon Velizhanin Bajnok,Janik Gromov,V.K.,Vieira Bajnok,Janik,Lukowski Lukowski,Rej,Velizhanin,Orlova Eden,Heslop,Korchemsky,Smirnov,Sokatchev Our numerics uses the TBA form of Y-system =2! From quasiclassics AdS/CFT Y-system passes all known tests Cavaglia, Fioravanti, Tatteo Gromov, V.K., Vieira Arutyunov, Frolov Classical integrability of superstring on AdS5×S5 String equations of motion and constraints can be recasted into zero curvature condition AdS time Mikhailov,Zakharov Bena,Roiban,Polchinski for Lax connection - double valued w.r.t. spectral parameter Monodromy matrix encodes infinitely many conservation lows Algebraic curve for quasi-momenta: V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo Dimension of YM operator Energy of a string state world sheet Classical symmetry where Unitary eigenvalues define quasimomenta - conserved quantities symmetry, together with unimodularity of induces a monodromy Trace of classical monodromy matrix is a psu(2,2|4) character. We take it in Gromov,V.K.,Tsuboi irreps for rectangular Young tableaux: symmetry: a s (Super-)group theoretical origins of Y- and T-systems A curious property of gl(N|M) representations with rectangular Young tableaux: = a s s + s-1 a-1 a+1 s+1 For characters – simplified Hirota eq.: Boundary conditions for Hirota eq.: ∞ - dim. unitary highest weight representations of u(2,2|4) in “T-hook” ! a Kwon U(2,2|4) Cheng,Lam,Zhang Gromov, V.K., Tsuboi s Hirota equation for characters promoted to the full quantum equation for T-functions (“transfer matrices”). Gromov,V.K.,Tsuboi AdS/CFT: Dispersion relation in physical and crossing channels Gross,Mikhailov,Roiban Santambrogio,Zanon Beisert,Dippel,Staudacher N.Dorey • Exact one particle dispersion relation: • Bound states (fusion) • Changing physical dispersion to cross channel dispersion Al.Zamolodchikov Ambjorn,Janik,Kristjansen • Parametrization for the dispersion relation by Zhukovsky map: • From physical to crossing (“mirror”) kinematics: continuation through the cut Arutyunov,Frolov cuts in complex u -plane AdS/CFT Y-system for the spectrum of N=4 SYM Gromov,V.K.,Vieira • Y-system is directly related to T-system (Hirota): T-hook • Complicated analyticity structure in u (similar to Hubbard model) cuts in complex • Extra equation: • Energy : (anomalous dimension) • obey the exact Bethe eq.: -plane Krichever,Lipan, Wiegmann,Zabrodin Wronskian solutions of Hirota equation • We can solve Hirota equations in terms of differential forms of functions Q(u). Solution combines dynamics of representations Gromov,V.K.,Leurent,Volin and the quantum fusion. Construction for gl(N): • -form encodes all Q-functions with indices: a s • E.g. for gl(2) : • Solution of Hirota equation in a strip: • For gl(N) spin chain (half-strip) we impose: QQ-relations (Plücker identities) Tsuboi V.K.,Sorin,Zabrodin Gromov,Vieira Tsuboi,Bazhanov • Example: gl(2|2) Hasse diagram: hypercub • E.g. - bosonic QQ-rel. - fermionic QQ rel. • All Q’s expressed through a few basic ones by determinant formulas • T-operators obey Hirota equation: solved by Wronskian determinants of Q’s Wronskian solution of u(2,2|4) T-system in T-hook Gromov,V.K.,Tsuboi Gromov,Tsuboi,V.K.,Leurent Tsuboi Plücker relations express all 256 Q-functions through 8 independent ones Solution of AdS/CFT T-system in terms of finite number of non-linear integral equations (FiNLIE) Gromov,V.K.,Leurent,Volin • Main tools: integrable Hirota dynamics + analyticity (inspired by classics and asymptotic Bethe ansatz) • Original T-system is in mirror sheet (long cuts) • Gauge symmetry definitions: • No single analyticity friendly gauge for T’s of right, left and upper bands. We parameterize T’s of 3 bands in different, analyticity friendly gauges, also respecting their reality and some symmetries, like quantum Magic sheet and solution for the right band • T-functions have certain analyticity strips (between two closest to Zhukovsky cuts) • The property suggests that the functions are much simpler on the “magic” sheet – with only short cuts: Only two cuts left on the magic sheet for by a polynomial S(u), a gauge function and a density ! Right band parameterized: with one magic cut on ℝ Quantum symmetry Gromov,V.K. Leurent, Tsuboi Gromov,V.K.Leurent,Volin can be analytically continued on special magic sheet in labels Analytically continued each in its infinite strip. and satisfy the Hirota equations, Magic sheet for the upper band • Analyticity strips (dictated by Y-functions) • Relation again suggests to go to magic sheet (short cuts), but the solution is more complicated. • Irreps (n,2) and (2,n) are in fact the same typical irrep, so it is natural to impose for our physical gauge • From the unimodularity of the quantum monodromy matrix it follows that the function is i-periodic Wronskian solution and parameterization for the upper band Use Wronskian formula for general solution of Hirota in a band of width N From reality, symmetry and asymptotic properties at large L , and considering only left-right symmetric states it gives We parameterize the upper band in terms of a spectral density the “wing exchange” function and gauge function and two polynomials P(u) and (u) encoding Bethe roots The rest of q’s restored from Plucker QQ relations , Parameterization of the upper band: continuation Remarkably, since and all T-functions have the right analyticity strips! symmetry is also respected…. Closing FiNLIE: sawing together 3 bands • We found and check from TBA the following relation between the upper and right/left bands Greatly inspired by: Bombardelli, Fioravanti, Tatteo Balog, Hegedus We have expressed all T (or Y) functions through 6 functions All but can be expressed through From analyticity properties we get two extra equations, on via spectral Cauchy representation TBA also reproduced and and Bethe roots and energy (anomalous dimension) of a state The Bethe roots characterizing a state (operqtor) are encoded into zeros of some q-functions (in particular ). Can be extracted from absence of poles in T-functions in “physical” gauge. Or the old formula from TBA… The energy of a state can be extracted from the large u asymptotics We managed to close the system of FiNLIE ! Gromov,V.K.,Leurent,Volin Its preliminary numerical analysis matches earlier numerics for TBA Conclusions • Integrability (normally 2D) – a window into D>2 physics: non-BPS, sum of 4D Feynman graphs! • AdS/CFT Y-system for exact spectrum of anomalous dimensions has passed many important checks. • Y-system obeys integrable Hirota dynamics – can be reduced to a finite system of non-linear integral eqs (FiNLIE) in terms of Wronskians of Q-functions. Gromov, V.K., Vieira V.K., Leurent • General method of solving quantum ϭ-models (successfully applied to 2D principal chiral field). Correa, Maldacena, Sever, Drukker • Recently this method was used to find the quark-antiquark potential in N=4 SYM Future directions • Better understanding of analyticity of Q-functions. Quantum algebraic curve for AdS5/CFT4 ? • Why is N=4 SYM integrable? • FiNLIE for another integrable AdS/CFT duality: 3D ABJM gauge theory • What lessons for less supersymmetric SYM and QCD? • BFKL limit from Y-system? • 1/N – expansion integrable? • Gluon amlitudes, correlators …integrable? END