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Perturbative integrability in N=4 SYM 2002 Minahan,Zarembo, Beisert,Kristijanssen,Staudacher 200 3 -2008 Bena,Polchinski,Roiban Kazakov,Marshakov, Minahan, Zarembo, Frolov, Tseytlin Sch

Nikolay Gromov

Based on

N G., V Kazakov, S Leurent , D Volin 1305.1939 , 1405.4857

N G., F Levkovich-Maslyuk, G Sizov, S Valatka 1402.0871

A Cavaglia , D Fioravanti, N G., R Tateo 1403.1859

N G., G Sizov 1403.1894

M Alfimov,N G., V Kazakov to appear

Strings 2014

Perturbative integrability in N=4 SYM

2002

Minahan,Zarembo, Beisert,Kristijanssen,Staudacher

200 3 -2008

Bena,Polchinski,Roiban Kazakov,Marshakov, Minahan, Zarembo, Frolov, Tseytlin Schafer-Nameki Beisert,Kazakov,Sakai,Zarembo

NG,Vieira

Classical integrability of string ϭ-model

on AdS5×S5, quasiclassics

2004-2006

Arutyunov,Frolov,Staudacher Staudacher, Beisert

Janik Hernandez,Lopez Roiban, Tseytlin Beisert,Eden,Staudacher

S-matrix, asymptotic Bethe Ansatz

Cusp dimension

2005-2008

Ambjorn,Janik,Kristijanssen Arutyunov,Frolov Bajnok,Janik, Lukowski

Finite size corrections and mirror model

2009-2010

NG , Kazakov, Vieira Bombardelli,Fioravanti, Tateo

NG , Kazakov,Vieira Arutyunov, Frolov Covagia,Fioravanti,N.G.,Tateo

Analytic Y-system for exact spectrum,

TBA

2011-2014

NG, Kazakov, Leurent, Volin

Quantum spectral curve for all operators

Origins of YM

integrability:

Lipatov’s BFKL

Hamiltonian

Lipatov

Faddeev,Korchemsky

1993

Origins of YM integrability: Lipatov’s BFKL Hamiltonian

Lipatov Faddeev,Korchemsky Alfimov,N.G.,Kazakov to

appear

1993

Plan:

Review of the QSC

construction

Examples:

1) near BPS 2) BFKL limit Generalization to ABJM

Integrability in gauge theory

EOM equivalent to where

on EOM

current

State-dependent cuts

Eigenvalues of the monodromy matrix:

Analytic properties:

Motivation from classics

[Bena, Polchinski, Roiban]

[Dorey, Vicedo]

Can be mapped to a spin chain state:

The one-loop dilatation operator coincides with Heisenberg spin chain Hamiltonian Sklyanin separation of variables allows to factorize the wave function

where

In the simplest case

Two solutions: polynomial singular solution

From weak coupling

[Beisert, Sctaudacher]

1) We start exploring all DOS of the string

2) Poles open into cuts

Heisenberg, SYM Quantum Spectral Curve Classical string

3) Need to know monodromies, when going under the cuts

Generalization to finite coupling

[N.G., Kazakov, Leuren, Volin]

Charges in S5 are integer Charges in AdS5 contain anom.dimension

“Miraculous” simplification

[N.G., Kazakov, Leuren, Volin]

The system reduced to 4+6 functions:

Analytical continuation to the next sheet:

Quadratic branch cuts:

-system

is a closed system of

equtions!

- system

[N.G., Kazakov, Leuren, Volin]

Examples: near-BPS expansion

In the BPS limit: entire periodic function

For in the small S limit:

Solution:

- simple Riemann-Hilbert problem

Near BPS limit: small S

Result:

Similar to the localization results!

[NG Sizov, Valatka, Levkovich-Maslyuk]

[Basso] [Zarembo; Pestun]

Extrapolating results to finite spin

Gromov, Serban, Shenderovich,

Volin`11;

Roiban, Tseytlin`11;

Vallilo, Mazzucato`11 Plefka, Frolov`13

Gubser, Klebanov,

Polyakov `98

Not hard to iterate the procedure and go further away from BPS

Kotikov, Lipatov`13

Costa, Goncalves,

Penedones` 12

We also extract pomeron intercept:

More orders in small S

Gubser, Klebanov,

Polyakov `98

[Basso][NG Sizov, Valatka, Levkovich-Maslyuk]

BFKL regime

Spectrum for different spins: [Brower, Polchinski, Strassler, -Itan `06]

Important class of single trace operators:

BFKL regime

BFKL regime:

So that: Resumming to all loops terms

In this regime SYM is undistinguishable from the real QCD

Small coupling no branch cuts

The problem is essentially about gluons, i.e it is more natural to pass to AdS

S= -1 is when for the first time this ansatz is consistent for non-integer ∆

Can be solved explicitly

Enters into the Q-function of Lipatov, de Vega; Korchemsky, Faddeev!

BFKL limit of -system

[Alfimov, N.G., Kazakov to appear]

[Kotikov, Lipatov] Plugging it into - system we get:

ABJM Theory

Spectral curve for ABJM

define

Discontinuities Constrains

[ A Cavaglia , D Fioravanti, N G., R Tateo ]

Spectral curve for ABJM Algebraically and interchanged their roles, but not analytically

Another important difference is the position of the branch points:

i-(anti)periodic ABJM:

i-periodic SYM:

enters into many important quantities: cusp dimension, magnon dispertion

Finding Interpolation function h

In the near BPS limit we should be able to match with localization

Integrability:

Elliptic type integral

ABJM Matric model integral in its planar limit:

Localization:

Comparing cross-ratios of the branch points:

[ N.G., Sizov ]

[Pestun][Kapustin, Willett, Yaakov] [Marino, Putrov]

Interpolation function h

Minahan, Ohlsson Sax, Sieg &

Leoni, Mauri, Minahan, Ohlsson Sax, Santambrogio, Sieg, Tartaglino-

Mazzucchelli, Minahan, Zarembo

?

McLoughlin, Roiban Tseytlin Abbott, Aniceto, Bombardelli Lopez-Arcos, Nastase

Bergman, Hirano

Reproduces ~4

nontrivial coefficients!

Conclusions

•  QSC unifies all integrable structures: BFKL/ local

operators, classical strings/spin chains

•  Mysterious relation between ABJM and N=4 SYM

integrable structures Sign for an unifying theory? What

is QSC for AdS3?

•  Q-functions should give a way to the exact wave

function in separated variables Can we use it to

compute general 3-point correlation functions to all

loops?

•  Established links between exact results in integrability and localization Does there exist a unified structure

which works for both non-BPS and non-planar?

Discretization of Zhukovsky cut?