Solving the spectral AdS /CFT Y-system

1. “ 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

2. Quantum Integrability in AdS/CFT Gromov, V.K., Vieira Gromov, V.K., Leurent, Volin Alternative approach: Balog, Hegedus • Y-system (for planar AdS5/CFT4 , AdS4/CFT3 ,...) calculates exact anomalous dimensions of all local operators at anycoupling (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

3. Konishioperator: numericsfromY-system Beisert, Eden,Staudacher ABA Gubser,Klebanov,Polyakov Gubser Klebanov Polyakov =2! From quasiclassics Y-system numerics Gromov,V.K.,Vieira (recently confirmed by Frolov) Gromov,Shenderovich, Serban, Volin Roiban,Tseytlin Masuccato,Valilio Gromov, Valatka 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 millions of 4D Feynman graphs! Cavaglia, Fioravanti, Tatteo Gromov, V.K., Vieira Arutyunov, Frolov • Our numerics uses the TBA form of Y-system • AdS/CFT Y-system passes all known tests

4. Classical integrability of superstring on AdS5×S5 • String equations of motion and constraints • can be recastedinto zero curvature condition Mikhailov,Zakharov Bena,Roiban,Polchinski for Lax connection - double valuedw.r.t. spectral parameter world sheet • Monodromymatrix encodes infinitely many conservation lows AdS time • Algebraic curve for quasi-momenta: Energy of a string state V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo • Dimension of YM operator

5. 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 • irreps for rectangular Young tableaux: Gromov,V.K.,Tsuboi • symmetry: a s

6. (Super-)group theoretical origins of Y- and T-systems • A curious property of gl(N|M)representations with rectangular Young tableaux: a-1 = + a a+1 s s s-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 U(2,2|4) Kwon 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

7. 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

8. 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) • Extra equation: • Energy : • (anomalous dimension) cuts in complex -plane • obey the exact Bethe eq.:

9. 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 • and the quantum fusion. Construction for gl(N): Gromov,V.K.,Leurent,Volin • -form encodes all Q-functions with indices: • E.g. for gl(2) : • Solution of Hirota equation in a strip: a • For gl(N) spin chain (half-strip) we impose: s

10. Tsuboi V.K.,Sorin,Zabrodin Gromov,Vieira Tsuboi,Bazhanov QQ-relations (Plücker identities) • Example: gl(2|2) Hassediagram: 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

11. 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

12. 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 • 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 definitions:

13. 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 ! Right band parameterized: • by a polynomial S(u), a gauge function with one magic cut on ℝ • and a density

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

15. 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

16. 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

17. Parameterization of the upper band: continuation • Remarkably, since and • all T-functions have the right analyticity strips! • symmetry is also respected….

18. 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 expressedall T (or Y) functions through 6 functions • All but can be expressed through • From analyticity properties and • we get two extra equations, on and • via spectral Cauchy representation • TBA also reproduced

19. 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 ! • Its preliminary numerical analysis matches • earlier numerics for TBA Gromov,V.K.,Leurent,Volin

20. 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 AdS5/CFT4 ? • Why is N=4 SYM integrable? • FiNLIE for another integrableAdS/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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