Download Y-system

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