QCD POMERON FROM ADS/CFT QUANTUM SPECTRAL CURVE

Nikolay Gromov

Based onN. G., V. Kazakov, S. Leurent, D. Volin 1305.1939 , 1405.4857

N. G., F. Levkovich-Maslyuk, G. Sizov, S. Valatka 1402.0871A. Cavaglia , D. Fioravanti, N. G., R. Tateo 1403.1859

N. G., G. Sizov 1403.1894M. Alfimov,N. G., V. Kazakov appeared last week

Strings, Matrices, Integrability, 201418/08/2014 Paris

Perturbative integrability in N=4 SYM

2002

Minahan,Zarembo,Beisert,Kristijanssen,Staudacher

2003-2008

Bena,Polchinski,RoibanKazakov,Marshakov, Minahan,

Zarembo, Frolov, TseytlinSchafer-Nameki

Beisert,Kazakov,Sakai,ZaremboNG,Vieira

Classical integrability of string ϭ-model on AdS5×S5,

quasiclassics

Arutyunov,Frolov,StaudacherStaudacher, Beisert

JanikHernandez,LopezRoiban, Tseytlin

Beisert,Eden,Staudacher

Origins of YM integrability:

Lipatov’s BFKL

Hamiltonian

LipatovFaddeev,Korchemsky

1993

Plan:Review of the QSC

construction

Examples: 1) near BPS 2) BFKL limit

Applications 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]

[Kazakov, Marshakov, Minahan, Zarembo]

[Beisert, Kazakov, Sakai, Zarembo]

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]

Beisert, Staudacher;Beisert, Hernandez, Lopez;Beisert,Eden,Staudacher

I.e. from the asymptotical spectrum (infinite R) we can compute the Ground state energy for ANY finite volume!

…,Matsubara, Zamolodchikov,…

Bombardelli, Fioravanti, TateoN.G., Kazakov, Vieira

Arutynov, Frolov

N.G., Kazakov, Vieira ‘09

Numerics

Is it the final answer to the spectral problem???

N.G., Kazakov, Vieira

+ discontinuity relations Cavaglià, Fioravanti, Tateo

1) We start exploring all DOS of the string

2) Poles open into cuts

Heisenberg, SYM Classical stringQuantum Spectral Curve

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:

-systemis a closed system of

equtions!

- system

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

1) Find ‘‘rotation” matrix (from a 4th order FDE):

[Alfimov, N.G., Kazakov]

-system ↔ -system

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

2) Find and

Short-cut: Solve 4th order FDE:

where:

Examples: near-BPS expansion

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

Important class of single trace operators:

Twist-2 operators

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]

Next order

Result: [NG. Sizov, Valatka, Levkovich-Maslyuk]

Similar to the dressing phase

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

[Kotikov, Lipatov, Rej, Staudacher, Velizhanin]

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:

It is not hard to get more orders

NLO Q-function

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

where

This equation can still be solved explicitly

Resuming BFKL limit

For twist L states we expect

Generalization to more general states

[N.G., Sizov in preparation]

All-loop BES Bethe ansatz

Similarly in the BFKL regimeWe can resum L/2-loops and get BFKL-ABA with its own dressing phase.

All-loop BES ansatz resums L-loops in the perturbation theory.

Features: Zhukovsky variable, BES dressing phase

Another All-loop Bethe ansatz

ABJM Theory

Spectral curve for ABJMSYM ABJM

PSU(2,2|4) OSP(2,2|6)

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:

SYM: ABJM:

i-(anti)periodicABJM:

i-periodicSYM:

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 TseytlinAbbott, Aniceto, Bombardelli

Lopez-Arcos, Nastase

Bergman, HiranoL.Bianchia, M. Bianchia, Bresa,

Forinia,Vescovia

Reproduces ~4 nontrivial coefficients!

Conclusions• QSC unifies all integrable structures: BFKL/ local

operators, classical strings/spin chains. Beisert-Eden-Staudascher equations can be derived from it in complete generality.

• 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?

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