Post on 23-Jan-2016
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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 recently
Ahrenshoop conference, 201429/08/2014
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
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]
Asymptotics – global charges
R-charges
Lorentz, Conformal charges
Similarly to the classical curve
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]
Small P’s imply small discontinuity of mu:
Next order
Small P’s imply small discontinuity of mu:
Result: [NG. Sizov, Valatka, Levkovich-Maslyuk in prog.]
Dressing phase!
Spectrum for different spins: [Brower, Polchinski, Strassler, -Itan `06]
Twist-2 operators
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
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]