Solving the spectral AdS/CFT Y-system Vladimir Kazakov (ENS,Paris) “ Maths of Gauge and String Theory” London, 5/05/2012 Collaborations with Gromov, Leurent, Tsuboi, Vieira, Volin

Quantum Integrability in AdS/CFT Y-system (for planar AdS 5 /CFT 4, AdS 4 /CFT 3,...) calculates exact anomalous dimensions of all local operators at any coupling (non-BPS, summing genuine 4D Feynman diagrams!) 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., Vieira Gromov, V.K., Leurent, Volin Alternative approach: Balog, Hegedus

Konishi operator : numerics from Y-system Gubser Klebanov Polyakov Beisert, Eden,Staudacher ABA Y-system numerics Gromov,V.K.,Vieira (recently confirmed by Frolov) Gubser,Klebanov,Polyakov  Our numerics uses the TBA form of Y-system  AdS/CFT Y-system passes all known tests 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 =2! From quasiclassics Gromov,Shenderovich, Serban, Volin Roiban,Tseytlin Masuccato,Valilio Gromov, Valatka Cavaglia, Fioravanti, Tatteo Gromov, V.K., Vieira Arutyunov, Frolov

Classical integrability of superstring on AdS 5 ×S 5  Monodromy matrix encodes infinitely many conservation lows  String equations of motion and constraints can be recasted into zero curvature condition Mikhailov,Zakharov Bena,Roiban,Polchinski for Lax connection - double valued w.r.t. spectral parameter  Algebraic curve for quasi-momenta: V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo Energy of a string state  Dimension of YM operator world sheet AdS time

Classical symmetry  symmetry, together with unimodularity of induces a monodromy  Unitary eigenvalues define quasimomenta - conserved quantities Gromov,V.K.,Tsuboi  Trace of classical monodromy matrix is a psu(2,2|4) character. We take it in irreps for rectangular Young tableaux: where a s  symmetry:

=+ a sss-1 s+1 a-1 a+1 (Super-)group theoretical origins of Y- and T-systems  A curious property of gl(N|M) representations with rectangular Young tableaux:  For characters – simplified Hirota eq.:  Boundary conditions for Hirota eq.: ∞ - dim. unitary highest weight representations of u(2,2|4) in “T-hook” ! U(2,2|4) a s Kwon Cheng,Lam,Zhang Gromov, V.K., Tsuboi  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 Changing physical dispersion to cross channel dispersion From physical to crossing (“mirror”) kinematics: continuation through the cut Al.Zamolodchikov Ambjorn,Janik,Kristjansen Exact one particle dispersion relation: Bound states (fusion) Parametrization for the dispersion relation by Zhukovsky map: cuts in complex u -plane Gross,Mikhailov,Roiban Santambrogio,Zanon Beisert,Dippel,Staudacher N.Dorey Arutyunov,Frolov

Complicated analyticity structure in u (similar to Hubbard model) Gromov,V.K.,Vieira Extra equation: cuts in complex -plane obey the exact Bethe eq.: Energy : (anomalous dimension) AdS/CFT Y-system for the spectrum of N=4 SYM Y-system is directly related to T-system (Hirota): T-hook

Wronskian solutions of Hirota equation We can solve Hirota equations in terms of differential forms of functions Q(u). Solution combines dynamics of representations and the quantum fusion. Construction for gl(N): -form encodes all Q-functions with indices: Solution of Hirota equation in a strip: a s For gl(N) spin chain (half-strip) we impose: E.g. for gl(2) : Krichever,Lipan, Wiegmann,Zabrodin Gromov,V.K.,Leurent,Volin

QQ-relations (Plücker identities) - bosonic QQ-rel. Example: gl(2|2) Tsuboi V.K.,Sorin,Zabrodin Gromov,Vieira Tsuboi,Bazhanov All Q’s expressed through a few basic ones by determinant formulas T-operators obey Hirota equation: solved by Wronskian determinants of Q’s Hasse diagram: hypercub E.g. - fermionic QQ rel.

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) 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 Gromov,V.K.,Leurent,Volin Main tools: integrable Hirota dynamics + analyticity (inspired by classics and asymptotic Bethe ansatz) Gauge symmetry definitions: Original T-system is in mirror sheet (long cuts)

Magic sheet and solution for the right band  Only two cuts left on the magic sheet for ! Right band parameterized: by a polynomial S(u), a gauge function with one magic cut on ℝ and a density 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:

Quantum symmetry  can be analytically continued on special magic sheet in labels  Analytically continued and satisfy the Hirota equations, each in its infinite strip. Gromov,V.K. Leurent, Tsuboi Gromov,V.K.Leurent,Volin

Magic sheet for the upper band Relation again suggests to go to magic sheet (short cuts), but the solution is more complicated. Analyticity strips (dictated by Y-functions) 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  From reality, symmetry and asymptotic properties at large L, and considering only left-right symmetric states it gives  Use Wronskian formula for general solution of Hirota in a band of width N  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 have expressed all T (or Y) functions through 6 functions We found and check from TBA the following relation between the upper and right/left bands Greatly inspired by: Bombardelli, Fioravanti, Tatteo Balog, Hegedus  From analyticity properties and we get two extra equations, on and via spectral Cauchy representation  TBA also reproduced  All but can be expressed through

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 Gromov,V.K.,Leurent,Volin  We managed to close the system of FiNLIE !  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. General method of solving quantum ϭ -models (successfully applied to 2D principal chiral field). 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 AdS 5 /CFT 4 ? 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? Gromov, V.K., Vieira V.K., Leurent Correa, Maldacena, Sever, Drukker

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