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

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

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

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

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:

= +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

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

• 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

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

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.

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

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)

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:

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

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

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

Parameterization of the upper band: continuation

Remarkably, since and all T-functions have the right analyticity strips!

symmetry is also respected….

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

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

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

END