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Solving the AdS/CFT Y-system
Dmytro Volin Nordita, Stockholm
IHP, 2011
1110.0562 N.Gromov, V.Kazakov, S.Leurent.
D.V.
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Setup:
= IIB, AdS5xS5 0 ∞
This is a talk about the spectral problem:
Conformal dimension of local operators Energy of string states = = ?
Can treat this problem at any coupling, because of integrability
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Nature of integrability:
From the point of view of string we are dealing with the conventional integrability in 2d QFT.
𝑇 = 𝑇𝑟 𝑃𝑒 𝐴(𝑢)
(Free) String is a sigma-model on PSU(2,2|4) has ℤ4 grading, coset respects it This allows us to construct the Lax connection -> classical integrability
[Zakharov, Mikhailov] [Bena, Polchinski, Roiban `03]
x AdS5 S5
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SU(2) PCM
• sigma model on:
• Excitations are over the vacuum
• SU(2) x SU(2) x Poincare • (unbroken)
• Massive particles, • Factorized scattering • Mass is dynamically generated
AdS5xS5 String
x AdS S
J
string sigma model on: Excitations over BMN “vacuum”. • Massive particles, • Factorized scattering • Mass is present classically (cf. centrifugal force)
Symmetry
Large volume description:
(PSU(2,2|4) is broken by choice of BMN vacuum)
Asymptotic Bethe Ansatz solution:
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SU(2) PCM
SU(2) x SU(2) x Poincare CFT N/A
AdS5xS5 String
N/A PSU(2,2|4) version of Heisenberg spin chain. • It can be solved by the algebraic Bethe Ansatz which
explicitly preserves the whole PSU(2,2|4) symmetry. • Weak coupling means in particular large volume
Symmetry
Small volume description:
Weak coupling description:
. . .
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SU(2) PCM
SU(2) x SU(2) x Poincare Y-system
AdS5xS5 String
Y-system Remarkably, PSU(2,2|4) symmetry reappears.
Symmetry
Finite volume description:
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Scientific output: about 1000-2000 papers (1306 on the plot):
Integrability 1 loop SYM
[MZ]
Integrability Classical string
[BPR]
Asymptotic Bethe Ansatz (∞ volume)
[BS] First dimension at any coupling (𝑐𝑢𝑠𝑝 𝑎𝑛𝑜𝑚𝑎𝑙𝑦) [BES]….
𝐴𝑑𝑆4/𝐶𝑃3
[ABJM]…
Y-system (finite volume)
[GKV]
First short operator at any coupling (Konishi) [GKV]
Solution of Y-system In finite set of equations [GKLV]
Integrability 1 loop QCD [BDM]
1998
AdS/CFT [M]
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Cusp anomalous dimension
[Klebanov et al, 06] [Kotikov,Lipatov, 06] [Alday et al, 07] [Kostov, Serban, D.V., 07] [Beccaria, Angelis, Forini, 07]
[Casteill, Kristjansen, 07] [Belitsky, 07]
[Basso, Korchemsky, Kotanski, 07] [Kostov, Serban, D.V., 08]
[Gubser, Klebanov, Polyakov, 02]
[Frolov, Tseytlin, 02]
[Roiban, Tseytlin, 07]
[Moch, Vermaseren, Vogt, 04] [Lipatov et al., 04]
[Bern et al., 06] [Cachazo et al., 06]
[Benna, Benvenuti, Klebanov, Scardicchio, 06]
[Beisert, Eden, Staudacher, 06]
𝑻𝒓 𝒁 𝑫𝑺𝒁 Weak coupling:
Numerics:
Strong coupling:
Nonperturbative corrections: [Basso, Korchemsky, 09]
State of art explicit results for infinite volume case.
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Can we do the same for the finite volume case?
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Simple example: SU(2) chiral Gross-Neveu model at finite volume
• Asymptotically free theory
• Dynamically generated mass scale m
• At infinite volume: described by factorized scattering theory
• Particle content: fundamental representation of SU(2)
• Asymptotic Bethe Ansatz solution
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For finite volume one can use TBA & exchange time and space coordinates
[Zamolodchikov]
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Minimization of free energy lead to TBA equations:
a
s
• This is an infinite set of nonlinear integral equations. • From these equations we can derive Y-system:
• Reverse derivation requires additional input
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a
s
• Y-system is equivalent to Hirota system:
• T-functions satisfy exactly the same system as the transfer matrices of SU(2) XXX spin chain!
• Indeed, chiral GN is a special limit of inhomogeneous SU(2) XXX.
Gauge freedom:
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Character identities vs Hirota equation:
SU(n)
0
n
s
a
(a,s)
Hirota equation:
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SU(n)
0
n
s
a
+ = + + +
Hirota equation:
Character identities vs Hirota equation:
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0
n
s
a Wronskian solution:
Generically need 2n+2 functions, 4 can be fixed by gauge freedom
Define fused products:
The most general solution Of Hirota equation is:
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Back to SU(2) XXX a
s
• Those are identity transfer matrices:
• ,
•
• Transfer matrix in trivial representation:
• Transfer matrix in fundamental representation (enters Baxter equation):
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Baxter equation as Plucker identity a
s
Plucker identity: Baxter equation:
• From regularity of T we derive Bethe equations -> Fixes Q -> Solve spectrum of XXX
• Analyticity input: It was important in our considerations that all functions are polynomials
and that we know
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Single integral equation solving GN
Suppose that • Hirota is satisfied
•
• At large volume Bethe Ansatz is reproduced
• Consider large volume first, black node drops out, we get XXX spin chain again, but:
• For simplicity consider only vacuum, then M=0, 𝑇1,1 = 1, 𝑄2 = 𝑄 = 1, 𝑄1 = −𝑖𝑢.
• Finite volume is considered as deformation. We can choose a gauge such that 𝑄2 = 1 always.
a
s
Presented derivation is Close to [GKV’08]
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Single integral equation solving GN
• We can parameterize 𝑄1 as
• 𝑇1,0 = 𝜌
a
s
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In the chiral GN case it is enough to know: • Hirota equation, • Analytical structure of T’s, • Asymptotical behavior at large volume.
Conclusion
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Transfer matrices
Symmetry(Hirota),
Analyticity,
poles/zeroes
/asymptotic
TBA (infinite)
FiNLIE (DdV) (finite)
…
Objects Constraints Equations Explicit solution
Bootstrap for finite volume, relativistic case:
From mirror TBA: [Zamolodchikov]
From counting function&lattice: [Destri, de Vega]
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From mirror TBA: [Bombardelli,Fioravanti,Tateo, 09] [Gromov, Kazakov, Kozak, Vieira, 09] [Arutyunov, Frolov, 09]
Observation: in AdS/CFT case, by applying approach of Zamolodchikov, Hirota equation was derived!
We believe therefore that there is a kind of bootstrap program which will solve spectral problem, more directly, elegantly and rigorously than mirror TBA approach.
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Transfer matrices
(only classically)
Symmetry(Hirota),
Analyticity,
poles/zeroes
/asymptotic
+discontinuity conditions (???)
TBA (infinite)
FiNLIE (finite)
Konishi
(2009:mismatch
At strong coupling)
Objects Constraints Equations Explicit solution
Finite volume bootstrap programm, for AdS/CFT (status prior 2011):
[Gromov,Kazakov, Tsuboi 10]
From mirror TBA: [Bombardelli,Fioravanti,Tateo, 09] [Gromov, Kazakov, Kozak, Vieira, 09] [Arutyunov, Frolov, 09]
[Gromov,Kazakov, Vieira,09] [Roiban, Tseytlin,09 [Cavaglia,Fioravanti,Tateo, 10]
[Balog,Hegedus 11]
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Strong coupling of the sl(2) sector (Konishi et al):
[Gromov, Kazakov, Vieira, 09] [Frolov, 10]
02/2011: Analytical derivations (using yet to be proved
assumptions): [Gromov,Shenderovich,Serban, D.V.]
[Roiban,Tseytlin]
[Masuccato,Valilio]
J=3, S=2, n=1 J=2, S=2, n=1
Konishi
(2009:mismatch
At strong coupling)
Konishi
(2011:agreement)
[Gromov, Kazakov, Vieira, 09] [Roiban, Tseytlin, 09]
2009:
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0
N
a
s
SU(N)
Implementation of symmetry:
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r1 r2
r1
r2
𝑆𝑈(2,2|4) ⊃ 𝑆𝑈 2 ⊕ 𝑆𝑈 4 ⊕ 𝑆𝑈 2 ⊕𝑈 1 ⊕𝑈(1)
[Gromov, Kazakov, Tsuboi,’10] : mapping Young tableaux inside T-hook to highest weight irreps.
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r1 r2
r1
r2
𝑆𝑈(2,2|4) ⊃ 𝑆𝑈 2 ⊕ 𝑆𝑈 4 ⊕ 𝑆𝑈 2 ⊕𝑈 1 ⊕𝑈(1)
[Gromov, Kazakov, Tsuboi,’10] : mapping Young tableaux inside T-hook to highest weight irreps.
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r1 r2
r1
r2
𝑆𝑈(2,2|4) ⊃ 𝑆𝑈 2 ⊕ 𝑆𝑈 4 ⊕ 𝑆𝑈 2 ⊕𝑈 1 ⊕𝑈(1)
[Gromov, Kazakov, Tsuboi,’10] : mapping Young tableaux inside T-hook to highest weight irreps.
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Conjecture [D.V.’10] : T-hook classifies all unitary highest weight representations of SU(n,m|k)
• Proved for all subcases (m=0 or k=0) • Agreees with Dobrev-Petkova for SU(2,2|4)
T-hook and classification of Unitary highest weight representations:
r1
r2
[Gromov, Kazakov, Tsuboi,’10] : mapping T-hook Young tableaux to unitary highest weight irreps.
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• Transfer matrix interpretation of T-functions exist only at strong coupling - classically (T=character): - first nontrivial quantum correction (new!)
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Assuming that T’s are transfer matrices, i.e. physical objects – quantum version of PSU(2,2|4) characters,
what constraints apart Hirota can we put on them?
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a
s
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Unimodularity:
T is a physical gauge
[Gromov, Kazakov, Leurent, Tsuboi, ‘10]
Want to impose sdet=1
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Hirota and branch points
Magic:
Mirror:
Symmetry:
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Symmetry:
Right band:
Upper band:
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Complete set of properties of T and gauges:
Symmetry Analyticity
(Unimodularity)
( ) No poles Minimal # of zeroes
Two cuts for No poles
( )
1110.0562 N.Gromov, V.Kazakov, S.Leurent.
D.V.
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• Listed above properties (symmetry+analyticity) + correct large volume (asymptotic Bethe Ansatz) behavior uniquely fix solution of the Y-system = Hirota system
• In particular, these properties are equivalent to the TBA equations
• We also get a new way to extract energy from the T’s….
Symmetry
(Hirota)
+Analyticity
+Poles/zeres/asymptotics
+ Discontinuity
conditions
Symmetry
(Hirota, det=1, Z4)
+Analyticity
+Poles/zeroes/asymptotics
1110.0562 N.Gromov, V.Kazakov, S.Leurent.
D.V.
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Exact Bethe equations
x x
x x
x x
x x
x x
x x
This is a condition for absence of singularities
In the physical T-gauge
New formula for the energy
x x
x x
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TBA
(infinite set)
• Instead of infinite set of TBA equations we propose a FiNLIE
FiNLIE !
1110.0562 N.Gromov, V.Kazakov, S.Leurent.
D.V.
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Upper band, Wronskian paremeterization
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• Due to reality of T-functions p-s can be expressed through complex conjugation of q-s
• Need to define q-s • Initial basis:
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• Due to reality of T-functions p-s can be expressed through complex conjugation of q-s
• Need to define q-s • Alternative basis:
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1
• Due to reality of T-functions p-s can be expressed through complex conjugation of q-s
• Need to define q-s • Alternative basis, use gauge freedom and LR symmetry to
simplify:
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Closing system of equations
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Complete system of equations: Right band:
Upper band:
Gluing equaitons:
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• We explicitly checked equation numerically for the Konishi state and got a complete agreement with the TBA approach.
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Discussion
• Infinite volume case was solved by infinite volume bootstrap, based on old idea about factorization of the transfer matrix.
• Cusp anomalous dimension can be efficiently computed at any coupling • First computations available for finite volume (e.g. Konishi), though not most efficient and not systematic.
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Discussion
• An important advance in bootstrap program for finite volume. Based on Hirota dynamics.
U(2,2|4) + Det=1 + ℤ𝟒
+ analyticity + large volume explicitly = solution of the spectral problem
1110.0562 N.Gromov, V.Kazakov, S.Leurent.
D.V.
Mirror Magic
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Discussion
Transfer matrices
(classical level only)
Symmetry
(Hirota+det=1+Z4)
+Analyticity
+Poles/Zeroes
/Asymptotics
FiNLIE
Konishi – ok
Some systematics
now?
Approaching now to the systematic study: • Weak coupling (e.g transcendentality structure) • Strong copuling (asymptotic? Borel summable?) • BFKL?
Need to define transfer matrices and Q-operators at weak coupling! Need to quantize transfer matrices, need to define Q-operators at strong coupling Z4 symmetry is not used to its full power, can simplify more FiNLIE (reduce to only finite support densities)
• An important advance in bootstrap program for finite volume. Based on Hirota dynamics.