Integrability of spectrum of N=4 SYM theory

Transcription

1 Todai/Riken joint workshop on Super Yang-Mills, solvable systems and related subjects University of Tokyo, October 23, 2013 Integrability of spectrum of N=4 SYM theory Vladimir Kazakov (ENS, Paris) Collaborations with N. Gromov, S. Leurent, Z. Tsuboi, P. Vieira, D. Volin

2 Integrability in AdS/CFT Strong coupling dynamics of 4-dimensional YM theories is an unsolved theoretical problem. Since mid-90 s, many exact results in topological, BPS sectors of N=1,2 super-yang- Mills (SYM) theories (Seiberg-Witten etc.) Since early Integrable planar superconformal models: 4-dimensional N=4 SYM, 3d N=8 ABJM: Solvable non-bps! Summing genuine 4D Feynman diagrams for most important physical quantities: anomalous dimensions, correlation functions, Wilson loops, gluon scattering amplitudes Integrability is related to AdS/CFT duality to very special 2D superstring ϭ-models on AdS-background. 2D integrability tools applicable: quantum spin chains, S-matrix, TBA for finite volume, etc Most advanced and accomplished part computation of anomalous dimensions of local operators. A long development has led to the exact (at any YM- thooft coupling) equations for the spectrum of dimensions, with many spectacular analytic and high precision numerical results I will review the set of ideas which allows, for the first time, to hope for the comprehensive solution of a 4-dimensional CFT.

3 CFT: N=4 SYM as a superconformal 4d QFT Gliozzi,Scherk,Olive 77 Zero β-function! Global 4d superconformal symmetry PSU(2,2 4) Operators from local fields 4D Correlators (e.g. for scalars): scaling dimensions structure constants non-trivial functions of thooft coupling λ! They describe the whole conformal theory via operator product expansion

4 Dilatation operator in SYM perturbation theory Anomalous dimensions are encoded into dilatation operator which can be calculated perturbatively, e.g. from the 2-point correlators Point-splitting and renormalization: so that Action of dilatation on a general operator gives a mixture: Conf. dimensions are eigenvalues of ``Hamiltonian'' Perturbative dimensions:

5 Perturbative Konishi: integrability versus Feynman graphs Integrability allows to sum exactly enormous numbers of Feynman diagrams of N=4 SYM Bajnok,Janik Leurent,Serban,Volin Bajnok,Janik,Lukowski Lukowski,Rej, Velizhanin,Orlova Leurent, Volin Leurent, Volin (8 loops from FiNLIE) Volin (9-loops from spectral curve) Confirmed up to 5 loops by direct graph calculus (6 loops promised) Fiamberti,Santambrogio,Sieg,Zanon Velizhanin Eden,Heslop,Korchemsky,Smirnov,Sokatchev

6 SYM is dual to supersting σ-model on AdS5 S 5 2D ϭ-model on a coset fermions Maldacena Gubser, Polyakov, Klebasnov Witten Super-conformal N=4 SYM symmetry PSU(2,2 4) isometry of string target space target space AdS time world sheet Metsaev-Tseytlin action Dimension of YM operator Energy of a string state

7 Classical integrability of superstring on AdS 5 S 5 String equations of motion and constraints can be recast into flat connection condition for Lax connection - double valued w.r.t. spectral parameter Mikhailov,Zakharov Bena,Roiban,Polchinski Monodromy matrix encodes infinitely many conservation lows Algebraic curve for quasi-momenta world sheet Its,Matweev,Dubrovin,Novikov,Krichever V.K.,Marshakov,Minahan,Zarembo Beisert,V.K.,Sakai,Zarembo generates finite gap solutions psu(2,2 4) character of in irreps for rectangular Young tableaux: Gromov,V.K.,Tsuboi a Is a classical analog of quantum T-functions introduced later s

8 AdS 5 quasi-momenta Riemann surface of classical superstring Spectral parameter: bosonic cuts fixed poles fermionic poles S 5 quasi-momenta Quasiclassical quantization: in terms of the spectral parameter

9 Global Charges Conserved charges: angular momenta, spins and energy E: defined from the asymptotics at x=0,. Energy E of state dimension D of operator in SYM.

10 AdS string quasiclassics and numerics in SL(2) sector: twist-l operators of spin S 3 leading strong coupling terms were calculated for any S and L Numerics from Y-system, TBA, FiNLIE, at any coupling: - for Konishi operator - and twist-3 operator They perfectly reproduce the TBA/Y-system or FiNLIE numerics Y-system numerics Gromov,V.K.,Vieira Frolov Gromov,Valatka Gubser, Klebanov, Polyakov Gromov,Shenderovich, Serban, Volin Roiban, Tseytlin Vallilo, Mazzucato Gromov, Valatka Frolov Gromov, Valatka, AdS/CFT Y-system passes all known tests!

11 Asymptotic and thermodynamoic ansatz Beisert, Staudacher Beisert,Eden,Staudacher p j p 1 This periodicity condition is diagonalized by nested Bethe ansatz Dimensions of long YM operators for any coupling: p M finite size corrections, important for short operators! Thermodynamic Bethe Ansatz (TBA) sums up these corrections (exact!) Gromov,V.K.,Vieira Bombardelli,Fioravanti,Tateo Gromov,V.K.,Kozak,Vieira Arutyunov,Frolov New insights into analyticity :: Cavaglia,Fioravanti,Tateo Hegedus,Balog

12 Discrete Hirota eq.: T-system and Y-system Inverting the kernel, TBA can be brought to universal Y-system! Gromov,V.K.,Vieira By change we get Hirota discrete bilinear equation (T-system) Discrete classical integrable dynamics! We will use it to solve Y-system in terms of finite number of Q-functions

13 PSU 2,2 4 T-hook Exact spectral AdS/CFT Y-system Gromov,V.K.,Vieira cuts in complex -plane Analyticity from large L or u asymptotics via one-particle dispersion relation: L cuts in u -plane Zhukovsky map: Extra corner equations:

14 Magic sheet and solution for the right band The property suggests that certain T-functions are much simpler on the physical sheet, with only short cuts: Example: right band in terms of two Q-functions with one magic cut on R Important symmetry is satisfied T-system in T-hook can be solved in terms of 8 Baxter s Q-functions

15 Krichever,Lipan, Wiegmann,Zabrodin Gromov,V.K.,Leurent,Volin Wronskian solutions of Hirota equation Example: solution of Hirota equation in a band of width N in terms of differential forms with 2N coefficient functions Solution combines dynamics of gl(n) representations and quantum fusion: -form encodes all Q-functions with indices: a Multi-index Q-functions: E.g. for gl(2) : s Solution of Hirota equation in a strip (via arbitrary - and -forms): For su(n) spin chain (half-strip) we impose:

16 Hasse diagram and QQ-relations (Plücker id.) Tsuboi V.K.,Sorin,Zabrodin Tsuboi,Bazhanov gl(2 2) example: classification of all Q-functions Hasse diagram: hypercub E.g. - bosonic QQ-rel. - fermionic QQ rel. All Q s expressed through a few basic ones by determinant formulas

17 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 definitions:

18 Basic Q-functions and their asymptotics Consider SL(2) operators (general case in work) We found two 4-vectors of Q-functions with a single cut on defining sheet: with one short, physical cut Gromov, V.K., Leurent, Volin 2013 and asymptotics with one long, mirror cut and asymptotics Asymptotics are found by comparing to the classical and one loop approximations. All other T-functions can be found through these basic Q-functions. But what about other sheets? We propose Riemann-Hilbert eqs to continue through the cuts -- system

19 µ-matrix Another object representing certain Q-functions: 4 4 matrix, i-periodic in mirror (long cuts) Gromov, V.K., Leurent, Volin 2013 with constraint: where we defined a scalar product: From Y-system and (coming) equations: 5 possible asymptotics

20 Gromov, V.K., Leurent, Volin 2013 AdS/CFT spectrum as a Riemann-Hilbert problem Using Hirota integrability, some symmetry and analyticity we brought the spectral AdS/CFT problem to a finite number of non-liner integral equations (FiNLIE), or, equivalently, to a Riemann-Hilbert problem. AdS/CFT spectral problem reduces to Riemann-Hilbert-type relations on 4-vector and 4 4 matrix (physical sheet, short cuts) where is the analytic continuation of through the cut: Eq. on (almost) follows from the previous eq. by linear algebra:

21 Example: SL(2) sector at one loop from P-µ Plugging the asymptotics into the eq. we get for the coefficients of asymptotics of P-functions In weak coupling, since we know that we can put and system of 5 equations on reduces to one 2-nd order difference equation We can argue that From the absence of poles in P s at which brings us to the standard Baxter equation for SL(2) Heisenberg spin chain! where

22 Note that these formulas for follow from P-µ system and not from a particular form of Hamiltonian, as in standard Heisenberg spin chain! One loop anomalous dimensions for SL(2) To find anomalous dimensions demands the solution of P-µ system to the next order in as seen from asymptotics In the regime we split into regular and singular parts Solving Baxter eq. for singular part in this regime we find: Using the asymptotics of Euler s and comparing with the large u asymptotics for we recover standard formula Using mirror periodicity we recover the trace cyclicity property

23 Analytic continuation w.r.t. spin and BFKL from P-µ Qualitative dependence of anomalous dimension of continuous spin The problem is to reproduce it from the P-µ system numerically exactly and study the weak and strong coupling, as well as the BFKL approximation 0-1 S Δ We managed to find the one loop solution of SL(2) Baxter equation: Janik Gromov, V.K Anom. dim: BFKL is a double scaling limit: which leads to

24 Conclusions We proposed a system of matrix Riemann-Hilbert equations for the exact spectrum of anomalous dimensions of planar N=4 SYM theory in 4D. Very efficient for numerics and for calculations in various approximations This P-µ system defines the full quantum spectral curve of AdS5 S 5 duality Works for Wilson loops and quark-antiquark potential in N=4 SYM Future directions Correa, Maldacena, Sever Drucker Gromov, Sever Gromov, Kazakov, Leurent, Volin Why is N=4 SYM integrable? What lessons for less supersymmetric SYM and QCD? 1/N expansion integrable? Gluon amlitudes, correlators, Wilson loops integrable? BFKL (Regge limit) from P-µ -system? We hope to reproduce not only the leading but next-to-leading BFKL corrections

25 END