Solving the spectral AdS /CFT Y-system
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Solving the spectral AdS/CFT Y-system
Vladimir Kazakov (ENS,Paris)
“ Maths of Gauge and String Theory”London, 5/05/2012
Collaborations with Gromov, Leurent,
Tsuboi, Vieira, Volin
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Quantum Integrability in AdS/CFT
• Y-system (for planar AdS5/CFT4 , AdS4/CFT3 ,...) calculates exact anomalous dimensions of all local operators at any coupling
(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
Gromov, V.K., Vieira
Gromov, V.K., Leurent, Volin
Alternative approach:Balog, Hegedus
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Konishi operator : numerics from Y-system
GubserKlebanovPolyakov
Beisert, Eden,Staudacher ABA
Y-system numerics Gromov,V.K.,Vieira(recently confirmed by Frolov)
Gubser,Klebanov,Polyakov
Our numerics uses the TBA form of Y-system AdS/CFT Y-system passes all known tests
millions of 4D Feynman graphs!
5 loops and BFKL Fiamberti,Santambrogio,Sieg,Zanon
VelizhaninBajnok,Janik
Gromov,V.K.,VieiraBajnok,Janik,LukowskiLukowski,Rej,Velizhanin,OrlovaEden,Heslop,Korchemsky,Smirnov,Sokatchev
=2! From quasiclassicsGromov,Shenderovich,Serban, VolinRoiban,TseytlinMasuccato,ValilioGromov, Valatka
Cavaglia, Fioravanti, TatteoGromov, V.K., VieiraArutyunov, Frolov
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Classical integrability of superstring on AdS5×S5
Monodromy matrix encodes infinitely many conservation lows
String equations of motion and constraints can be recasted into zero curvature condition
Mikhailov,ZakharovBena,Roiban,Polchinski
for Lax connection - double valued w.r.t. spectral parameter
Algebraic curve for quasi-momenta:
V.K.,Marshakov,Minahan,ZaremboBeisert,V.K.,Sakai,Zarembo
Energy of a string state Dimension of YM operator
world sheet
AdS time
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Classical symmetry
symmetry, together with unimodularity of induces a monodromy
Unitary eigenvalues define quasimomenta - conserved quantities
Gromov,V.K.,Tsuboi
Trace of classical monodromy matrix is a psu(2,2|4) character. We take it in irreps for rectangular Young tableaux:
where
a
s
symmetry:
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= +a
s s s-1 s+1
a-1
a+1
(Super-)group theoretical origins of Y- and T-systems A curious property of gl(N|M) representations with rectangular Young tableaux:
For characters – simplified Hirota eq.:
Boundary conditions for Hirota eq.: ∞ - dim. unitary highest weight representations of u(2,2|4) in “T-hook” !
U(2,2|4)a
s
KwonCheng,Lam,Zhang
Gromov, V.K., Tsuboi
Hirota equation for characters promoted to the full quantum equation for T-functions (“transfer matrices”). Gromov,V.K.,Tsuboi
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AdS/CFT: Dispersion relation in physical and crossing channels
• Changing physical dispersion to cross channel dispersion
• From physical to crossing (“mirror”) kinematics: continuation through the cut
Al.Zamolodchikov
Ambjorn,Janik,Kristjansen
• Exact one particle dispersion relation:
• Bound states (fusion)
• Parametrization for the dispersion relation by Zhukovsky map:
cuts in complex u -plane
Gross,Mikhailov,RoibanSantambrogio,ZanonBeisert,Dippel,StaudacherN.Dorey
Arutyunov,Frolov
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• Complicated analyticity structure in u (similar to Hubbard model)
Gromov,V.K.,Vieira
• Extra equation: cuts in complex -plane
• obey the exact Bethe eq.:
• Energy : (anomalous dimension)
AdS/CFT Y-system for the spectrum of N=4 SYM
• Y-system is directly related to T-system (Hirota):
T-hook
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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):
• -form encodes all Q-functions with indices:
• Solution of Hirota equation in a strip:
a
s
• For gl(N) spin chain (half-strip) we impose:
• E.g. for gl(2) :
Krichever,Lipan,Wiegmann,Zabrodin
Gromov,V.K.,Leurent,Volin
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QQ-relations (Plücker identities)
- bosonic QQ-rel.
• Example: gl(2|2)
TsuboiV.K.,Sorin,Zabrodin
Gromov,VieiraTsuboi,Bazhanov
• All Q’s expressed through a few basic ones by determinant formulas • T-operators obey Hirota equation: solved by Wronskian determinants of Q’s
Hasse diagram: hypercub
• E.g.
- fermionic QQ rel.
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Wronskian solution of u(2,2|4) T-system in T-hook Gromov,V.K.,TsuboiGromov,Tsuboi,V.K.,LeurentTsuboi
Plücker relations express all 256 Q-functionsthrough 8 independent ones
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Solution of AdS/CFT T-system in terms offinite number of non-linear integral equations (FiNLIE)
• 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
Gromov,V.K.,Leurent,Volin
• Main tools: integrable Hirota dynamics + analyticity (inspired by classics and asymptotic Bethe ansatz)
• Gauge symmetry definitions:
• Original T-system is in mirror sheet (long cuts)
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Magic sheet and solution for the right band
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
• 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:
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Quantum symmetry
can be analytically continued on special magic sheet in labels
Analytically continued and satisfy the Hirota equations, each in its infinite strip.
Gromov,V.K. Leurent, TsuboiGromov,V.K.Leurent,Volin
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Magic sheet for the upper band
• Relation again suggests to go to magic sheet (short cuts), but the solution is more complicated.
• Analyticity strips (dictated by Y-functions)
• 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
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Wronskian solution and parameterization for the upper band
From reality, symmetry and asymptotic properties at large L , and considering only left-right symmetric states it gives
Use Wronskian formula for general solution of Hirota in a band of width N
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
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Parameterization of the upper band: continuation
Remarkably, since and all T-functions have the right analyticity strips!
symmetry is also respected….
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Closing FiNLIE: sawing together 3 bands
We have expressed all T (or Y) functions through 6 functions
• We found and check from TBA the following relation between the upper and right/left bands
Greatly inspired by:Bombardelli, Fioravanti, TatteoBalog, Hegedus
From analyticity properties and we get two extra equations, on and via spectral Cauchy representation
TBA also reproduced
All but can be expressed through
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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
Gromov,V.K.,Leurent,Volin
We managed to close the system of FiNLIE !
Its preliminary numerical analysis matches earlier numerics for TBA
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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 integrable AdS/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., VieiraV.K., Leurent
Correa, Maldacena, Sever, Drukker
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