Exploring TBA in the mirror AdS 5 × S 5 Ryo Suzuki School of Mathematics, Trinity College Dublin...

Exploring TBAin the mirror AdS5 × S5

Exploring TBAin the mirror AdS5 × S5

Ryo SuzukiSchool of Mathematics, Trinity College Dublin

Ryo SuzukiSchool of Mathematics, Trinity College Dublin

Based on arXiv:0906.4783, 0911.2224, 1002.1711Based on arXiv:0906.4783, 0911.2224, 1002.1711Collaborators:Collaborators:

Gleb Arutyunov (Utrecht), Sergey Frolov (TCD)Gleb Arutyunov (Utrecht), Sergey Frolov (TCD)Marius de Leeuw and Alessandro Torrilelli (Utrecht)Marius de Leeuw and Alessandro Torrilelli (Utrecht)

MotivationMotivation

AdS/CFT correspondence predictsAdS/CFT correspondence predictsEEstringstring(() = ) = gaugegauge(())

for any string state and for any local gauge theory operatorfor any string state and for any local gauge theory operator

We want to ‘demonstrate’ this predictionWe want to ‘demonstrate’ this predictionby using the integrablity method called by using the integrablity method called Bethe AnsatzBethe Ansatz

The string/gauge theory becomes perturbatively The string/gauge theory becomes perturbatively integrableintegrable

when they have large supersymmetry and global when they have large supersymmetry and global symmetry symmetry

EEstringstring(() E) EBetheBethe(() ) gaugegauge(())

MotivationMotivation

Consider maximally supersymmetric setup at large Consider maximally supersymmetric setup at large NN limit limit

Super Yang-MillsSuper Yang-MillsSuperstring on AdSSuperstring on AdS5 5 × S× S

55

with with NN-units of RR flux-units of RR flux

Both theories are believed as perturbatively integrableBoth theories are believed as perturbatively integrable

Asymptotic Bethe AnsatzAsymptotic Bethe Ansatz can demonstrate the can demonstrate the relationrelation

[Bena Polchiski Roiban (2003)], [Minahan, Zarembo (2002)] [Bena Polchiski Roiban (2003)], [Minahan, Zarembo (2002)] and so onand so on

[Beisert Staudacher [Beisert Staudacher (2005)](2005)]

EEstringstring(() E) EBetheBethe(() ) gaugegauge(())

for general ‘spinning’ strings and ‘long’ SYM for general ‘spinning’ strings and ‘long’ SYM operators,operators,

when wrapping corrections are negligiblewhen wrapping corrections are negligible

MotivationMotivation

The mirror TBA is expected to give the exact form ofThe mirror TBA is expected to give the exact form ofEEBetheBethe(()) for any string/SYM states for any string/SYM states

It is far from trivial to see that the mirror TBA gives It is far from trivial to see that the mirror TBA gives a ‘reasonable’ answer; one has to solve a set of a ‘reasonable’ answer; one has to solve a set of

nonlinear integral equations nonlinear integral equations among infinitely many among infinitely many fieldsfields

[Bombardelli, Fioravanti, Tateo (2009)] [Gromov, Kazakov, Kozak, Vieira (2009)] [Arutyunov, [Bombardelli, Fioravanti, Tateo (2009)] [Gromov, Kazakov, Kozak, Vieira (2009)] [Arutyunov, Frolov (2009)]Frolov (2009)]

I explain our findings on the mirror TBA for two-particle I explain our findings on the mirror TBA for two-particle states,states,

based on numerical study around the asymptotic limitbased on numerical study around the asymptotic limit

All wrapping corrections can be summed up by All wrapping corrections can be summed up by Thermodynamic Bethe Ansatz (TBA) for the mirror Thermodynamic Bethe Ansatz (TBA) for the mirror

AdSAdS5 5 × S× S55

(N.B. no relationship between the mirror TBA and TBA for minimal surface)(N.B. no relationship between the mirror TBA and TBA for minimal surface)

Main ProblemMain ProblemCompute two-point functions in Compute two-point functions in DD=4, =4, NN=4 =4

SU(SU(NN) SYM) SYMConformal symmetry constrains the two-point Conformal symmetry constrains the two-point

functionfunction

[Fiamberti, Santambrogio, Sieg, Zanon (2007)] [Fiamberti, Santambrogio, Sieg, Zanon (2007)] [Velizhanin (2008)][Velizhanin (2008)]

can be computed at weak couplingcan be computed at weak couplingby summng by summng 131,015131,015 diagrams at four loops diagrams at four loops

Konishi stateKonishi state

Plan of TalkPlan of Talk1. Introduction1. Introduction

Overview of AdS/CFT and IntegrabilityOverview of AdS/CFT and Integrability

2. Results from Integrability2. Results from IntegrabilityTransfer-matrix formulaTransfer-matrix formulaT-system and Y-systemT-system and Y-system

3. Excited-State TBA3. Excited-State TBAHow to formulate TBAHow to formulate TBA

Contour deformation trickContour deformation trickKonishi at five loopsKonishi at five loops

1. Introduction1. Introduction

AdS/CFT setupAdS/CFT setupAdS/CFT setupAdS/CFT setup

Super Yang-MillsSuper Yang-Mills

CFT sideCFT sideAdS sideAdS side

Superstring on AdSSuperstring on AdS5 5 × S× S55

NN-units of RR flux-units of RR flux

Global symmetryGlobal symmetry

TheoryTheory

Coupling constantCoupling constant

(( λ is arbitrary, λ is arbitrary, NN is large) is large)

Prediction Prediction

2. Conformal dimension of a local operator in SYM2. Conformal dimension of a local operator in SYM

1. Energy of a string state measured in AdS-1. Energy of a string state measured in AdS-time (t=τ)time (t=τ)

Given global charges (Given global charges ( SS11, S, S22, J, J11, J, J22, J, J33 ), ), comparecompare

DifficultyDifficulty

Anomalous dimension in perturbative Anomalous dimension in perturbative SYMSYM

String energy in semiclassical String energy in semiclassical expansionexpansion

Correspondence as 2D models

Correspondence as 2D models

E(λ) = Worldsheet energy of a E(λ) = Worldsheet energy of a gauge-fixedgauge-fixed Lagrangian Lagrangian

for closed string on AdSfor closed string on AdS5 5 × S× S55

Δ(λ) = Eigenvalue of anomalous Δ(λ) = Eigenvalue of anomalous dimension dimension matrixmatrixHamiltonian on 1-dim. Hamiltonian on 1-dim.

latticelattice(with continuous time (with continuous time direction)direction)

Hamiltonian of 2-dim. field theory on cylinderHamiltonian of 2-dim. field theory on cylinder

Anomalous dimension matrix

Anomalous dimension matrix

[Minahan, Zarembo (2002)][Minahan, Zarembo (2002)]

Hamiltonian of worldsheet theoryHamiltonian of worldsheet theory

Circumference of cylinder = Circumference of cylinder = LL

??

Hamiltonian of Hamiltonian of NN=4 SYM spin chain=4 SYM spin chain

Length of operator = Length of operator = LL

Vacuum of 2D modelsVacuum of 2D models

Residual global symmetry is R × psu(2|2)Residual global symmetry is R × psu(2|2)22

L L = J= J11 fields fields

spin Jspin J11 = = LLE = JE = J11

= Point particle moving= Point particle moving

along the geodesic of Salong the geodesic of S55

1/2 BPS operator1/2 BPS operator

(D, (D, S1S1, , S2S2, J1, , J1, J2J2, , J3J3))

Δ = JΔ = J11

= vacuum of GS action in the light-cone = vacuum of GS action in the light-cone gauge,gauge,

Excitations of 2D modelsExcitations of 2D models

LeftLeftRightRight

singlet (vacuum)singlet (vacuum)

singletsinglet

Fundamental representationFundamental representation of of PSU(2|2)PSU(2|2)22

2D models have a mass gapfor 2D models have a mass gapfor fixed Jfixed J11

Infinite-size limitInfinite-size limit

SYM operator of an infinite lengthSYM operator of an infinite length

Excitations on the decompactified Excitations on the decompactified worldsheetworldsheet

Periodicity conditionPeriodicity condition Extra central chargesExtra central charges

Magic of psuMagic of psu

Closure of commutators constrains the Closure of commutators constrains the dispersion relationdispersion relation

[Beisert (2005)][Beisert (2005)]

Conjecture:Conjecture: for AdSfor AdS5 5 × S× S55//NN=4 SYM=4 SYM

Please remember that this dispersion is non-relativistic.Please remember that this dispersion is non-relativistic.

2D Hamiltonian is a part of2D Hamiltonian is a part of

Multi-particle statesMulti-particle statesParticles scatter (or create a Particles scatter (or create a

bound state)bound state)Excitations on the decompactified Excitations on the decompactified

worldsheetworldsheet

SYM operator of an infinite lengthSYM operator of an infinite length

Multi-particle statesMulti-particle states1. S-matrix between two 1. S-matrix between two fundamentalfundamental representations ofrepresentations of psu(2|2)psu(2|2)22 is determined by symmetry up to a is determined by symmetry up to a scalar factorscalar factor

3. Yang-Baxter implies 3. Yang-Baxter implies integrabilityintegrability;; infinitely many charges are conserved during the infinitely many charges are conserved during the scatteringscattering4. Assuming Yang-Baxter relation, all S-matrices 4. Assuming Yang-Baxter relation, all S-matrices betweenbetween two two boundstatesboundstates are determined are determined

[Arutyunov, de Leeuw, Torrielli [Arutyunov, de Leeuw, Torrielli (2009)](2009)]

[Martins Melo [Martins Melo (2007)](2007)]

2. The psu(2|2) S-matrix is equivalent to Shastry’s 2. The psu(2|2) S-matrix is equivalent to Shastry’s R-matrixR-matrix of 1+1D Hubbard model, and satisfy of 1+1D Hubbard model, and satisfy Yang-Yang-Baxter relationBaxter relation

[Beisert (2005)] [Beisert (2005)]

Evidences from Perturbative Studies

Evidences from Perturbative Studies

Anomalous dimension in perturbative Anomalous dimension in perturbative SYMSYM

String energy in semiclassical String energy in semiclassical expansionexpansion

[Minahan, Zarembo (2002)][Minahan, Zarembo (2002)]and so onand so on

[Bena Polchiski Roiban [Bena Polchiski Roiban (2003)](2003)]

and so onand so on

IntegrableIntegrable

IntegrableIntegrable

Asymptotic SpectrumAsymptotic Spectrum

““Energy”Energy”

(= Bethe Ansatz Equations)(= Bethe Ansatz Equations)

““Momentum”Momentum”

SS00

[Beisert Staudacher [Beisert Staudacher (2005)](2005)]

““dressing phase”dressing phase”[Beisert Hernández López [Beisert Hernández López

(2006)](2006)][Beisert Eden Staudacher [Beisert Eden Staudacher

(2006)](2006)]

Assume AdSAssume AdS5 5 × S× S55 and and NN=4 SYM are =4 SYM are integrableintegrable

OverviewOverviewAdS/CFT correspondence and Integrability methods

Finite-size corrections

?

Asymptotic Bethe Ansatz

Exact Spectrum

Asymptotic Spectrum

?

Wrapping ProblemWrapping ProblemNo symmetry enhancement for the No symmetry enhancement for the finite-lengthfinite-length

operatoroperatorDispersion relation receives Dispersion relation receives wrapping wrapping correctionscorrections!!

Interactions wrap aroundInteractions wrap aroundat high loop ordersat high loop orders

Virtual particlesVirtual particlestravel around the worldtravel around the world

[Lüscher, [Lüscher, Comm Math Phys Comm Math Phys 104 (1986)]104 (1986)][Janik Łukowski (2007)] [Bajnok Janik (2008)][Janik Łukowski (2007)] [Bajnok Janik (2008)]

[Fiamberti, Santambrogio, Sieg, Zanon (2007)][Fiamberti, Santambrogio, Sieg, Zanon (2007)][Velizhanin (2008)][Velizhanin (2008)]

OverviewOverviewAdS/CFT correspondence and Integrability methods

Finite-size corrections

Thermodynamic Bethe Ansatz

Asymptotic Bethe Ansatz

Exact Spectrum

Asymptotic Spectrum

Generalized Lüscher formula

2. Results from Integrability

2. Results from Integrability

Two-particle States in sl(2) sector

Two-particle States in sl(2) sector

AsymptoticAsymptoticBethe AnsatzBethe Ansatz

[Beisert, Staudacher (2005)][Beisert, Staudacher (2005)]

[Beisert Hernández López [Beisert Hernández López (2006)](2006)]

[Beisert Eden Staudacher [Beisert Eden Staudacher (2006)](2006)]

[Dorey Hofman Maldacena [Dorey Hofman Maldacena (2007)](2007)]

Konishi stateKonishi state

[Beisert, Staudacher (2005)][Beisert, Staudacher (2005)]

RapiditRapidity for y for

KonishiKonishi(J=2, (J=2, n=1)n=1)

Asymptotic Bethe Ansatz equation and asymptotic Asymptotic Bethe Ansatz equation and asymptotic energyenergy

This is correct up to three-loop, invalid at four-loopThis is correct up to three-loop, invalid at four-loop

TBA and Y/T-systemsTBA and Y/T-systems

Mirror TBA for two-particle statesMirror TBA for two-particle states

Y-systemY-system

??Discrete LaplacianDiscrete Laplacian

T-systemT-system

Gauge-invariant combination of TGauge-invariant combination of TRedefinitionRedefinition

Transfer matrix for rectangular representaionsTransfer matrix for rectangular representaions

SolutionSolution

Need to know analytic structure of Y’sNeed to know analytic structure of Y’s

Transfer matrix formulaeTransfer matrix formulae

[Bazhanov, Reshetikhin [Bazhanov, Reshetikhin J Phys AJ Phys A23 23 (1990)](1990)]

[Gromov, Kazakov, Vieira (2009)] [Gromov, Kazakov, Vieira (2009)]

Formula for general “rectangular representations” (a,s)Formula for general “rectangular representations” (a,s)

[Beisert (2006)] [Gromov, Kazakov, Vieira (2009)] [Arutyunov, de Leeuw, Torrielli, [Beisert (2006)] [Gromov, Kazakov, Vieira (2009)] [Arutyunov, de Leeuw, Torrielli, R.S. (2009)]R.S. (2009)]

Transfer matrix formulaeTransfer matrix formulae

T-systemT-systemTransfer matrices satisfy T-system equationsTransfer matrices satisfy T-system equations

T=1 along the bottom line of the fat hook of psu(2,2|4)T=1 along the bottom line of the fat hook of psu(2,2|4)T=0 outside the fat hook of psu(2,2|4)T=0 outside the fat hook of psu(2,2|4)

Nonzero gap on the complex Nonzero gap on the complex vv-plane -plane [Arutyunov, Frolov (2009)] [Frolov, R.S. (2009)][Arutyunov, Frolov (2009)] [Frolov, R.S. (2009)]

Gauge Gauge symmetrysymmetry

can be used to fix can be used to fix boundary boundary conditionsconditions

Fat-hook of psu(2,2|4) can split into two fat-hooks of psu(2|2)Fat-hook of psu(2,2|4) can split into two fat-hooks of psu(2|2)

[Gromov, Kazakov, Vieira [Gromov, Kazakov, Vieira (2009)](2009)]

T=1T=1

T=0T=0T=0T=0

Y-systemY-systemAsymptotic Y-functions in terms of transfer Asymptotic Y-functions in terms of transfer

matrixmatrix

auxiliaryauxiliary

momentum-momentum-carryingcarrying

[Gromov, Kazakov, Vieira [Gromov, Kazakov, Vieira (2009)] (2009)]

Solution of the asymptotic Y-Solution of the asymptotic Y-systemsystem

(gauge invariant)(gauge invariant)

(gauge ambiguity)(gauge ambiguity)

Y-systemY-system

However, solution of the asymptotic Y-However, solution of the asymptotic Y-systemsystem

is is non-uniquenon-unique, because the discrete , because the discrete LaplacianLaplacianhas infinitely many solutionshas infinitely many solutions

Non-canonical Y-system follows from the mirror Non-canonical Y-system follows from the mirror TBATBA

(depends on branch choice)(depends on branch choice)

Need to know the analytic structure of Y-Need to know the analytic structure of Y-functions consistent with TBA, to functions consistent with TBA, to

“integrate” the Y-system“integrate” the Y-system

3. Excited-State TBA3. Excited-State TBA

How to formulate TBAHow to formulate TBA1. Compute the partition 1. Compute the partition functionfunction in the in the “mirror” theory“mirror” theory

3. Get 3. Get TBA equationsTBA equations; a set ; a set ofof nonlinear integral nonlinear integral equationsequations

2. Minimum of the partition 2. Minimum of the partition functionfunction is related to is related to ground stateground state energyenergy

4. Analytic continuation gives4. Analytic continuation gives TBA equations for TBA equations for excited statesexcited states

5. Consistency with 5. Consistency with Lüscher formulaLüscher formula

Mirror transformationMirror transformation

Exact ground state energyExact ground state energy Asymptotic free energyAsymptotic free energy

[Zamolodchikov, [Zamolodchikov, Nucl Phys BNucl Phys B342 342 (1990)](1990)]

Wrapping correctionsWrapping corrections No wrappingNo wrapping

Mirror model for AdS AdS5 5 × × SS55

Mirror model for AdS AdS5 5 × × SS55

[Arutyunov, Frolov, Zamaklar (2006)] [Klose, McLoughlin, Roiban, Zarembo (2006)][Arutyunov, Frolov, Zamaklar (2006)] [Klose, McLoughlin, Roiban, Zarembo (2006)]

(Bosonic part of) the gauge-fixed sigma-model on AdS(Bosonic part of) the gauge-fixed sigma-model on AdS5 5 × S× S55

Mirror model is defined by the Mirror model is defined by the rotationrotation

Mirror model for AdS AdS5 5 × × SS55

Mirror model for AdS AdS5 5 × × SS55

After the After the double Wick-rotationdouble Wick-rotation on on worldsheetworldsheet

Different real section of complexified Different real section of complexified psu(2|2)psu(2|2)22

•Residual symmetry is again Residual symmetry is again psu(2|2)psu(2|2)22

•Same S-matrix, except for theSame S-matrix, except for the dressing phase dressing phase

•Particles (giant magnons) live in AdSParticles (giant magnons) live in AdS55, not in S, not in S55

•Dispersion relation are doubly Wick-rotatedDispersion relation are doubly Wick-rotated

•Residual symmetry is again Residual symmetry is again psu(2|2)psu(2|2)22

•Same S-matrix, except for theSame S-matrix, except for the dressing phase dressing phase

•Particles (giant magnons) live in AdSParticles (giant magnons) live in AdS55, not in S, not in S55

•Dispersion relation are doubly Wick-rotatedDispersion relation are doubly Wick-rotated

[Arutyunov Frolov (2009)] [Volin (2009)] [Gromov Kazakov [Arutyunov Frolov (2009)] [Volin (2009)] [Gromov Kazakov Vieira (2009)]Vieira (2009)]

String theory and Mirror theory

String theory and Mirror theory

StringString

MirrorMirror

Asymptotic dispersion for Q-particle boundstate Asymptotic dispersion for Q-particle boundstate (Q=1,2,... ∞) (Q=1,2,... ∞)

Related by Related by analytic continuationanalytic continuation

xx and and uu variablesvariables(rapidity)(rapidity)

Physical RegionsPhysical Regions

Counting Mirror Spectrum

Counting Mirror Spectrum

Assume that the dominant contribution toAssume that the dominant contribution tothe mirror free energy comes from the mirror free energy comes from

thermodynamical statesthermodynamical states

1. Classify mirror boundstates using 1. Classify mirror boundstates using string string hypothesishypothesis2. Take 2. Take thermodynamic limitthermodynamic limit of ABA for mirror of ABA for mirror boundstatesboundstates

⇒ ⇒ the constraint equationsthe constraint equations3. 3. ExtremizeExtremize free energy by varying the density of free energy by varying the density of particlesparticles(or holes)(or holes)

[Takahashi, [Takahashi, Prog Theor Phys Prog Theor Phys 47 (1972)] 47 (1972)]

[Zamolodchikov, [Zamolodchikov, Nucl Phys BNucl Phys B342 342 (1990)](1990)]

Labelling mirror boundstates by Bethe rootsLabelling mirror boundstates by Bethe roots((momentum-carryingmomentum-carrying + auxiliary) + auxiliary)

Dynkin diagram for Dynkin diagram for su(3|2)su(3|2)

Four types of boundstatesFour types of boundstates

Q-particlesQ-particlesy-particlesy-particlesM|w-stringsM|w-stringsM|vw-stringsM|vw-strings

momentum carrying momentum carrying boundstatesboundstatesfermionic auxiliary particlesfermionic auxiliary particlesbosonic auxiliary bosonic auxiliary boundstatesboundstatesbosonic boundstate of y, y, bosonic boundstate of y, y, M|w M|w

(Q=1,2,... and (Q=1,2,... and M=1,2,...)M=1,2,...)

Q-particleQ-particle

M|vw-stringM|vw-string

M|w-stringM|w-string

y-particley-particle

[Bombardelli, Fioravanti, Tateo (2009)] [Gromov, Kazakov, Kozak, Vieira (2009)] [Arutyunov, [Bombardelli, Fioravanti, Tateo (2009)] [Gromov, Kazakov, Kozak, Vieira (2009)] [Arutyunov, Frolov (2009)]Frolov (2009)]

Ground State TBAGround State TBA

Exact Groundstate Energy

Exact Groundstate Energy

[Frolov, R.S. (2009)][Frolov, R.S. (2009)]

Extremum ofExtremum ofmirror free energymirror free energy

YYQQ is meromorphic is meromorphicon the physical region of z-on the physical region of z-

torustorus

Small Small hh expansion (Witten index at expansion (Witten index at hh=0)=0)

Relation to Y-systemRelation to Y-system

[Gromov, Kazakov, Vieira (2009)] [Arutyunov, [Gromov, Kazakov, Vieira (2009)] [Arutyunov, Frolov (2009)]Frolov (2009)]

⇔⇔

Y-system can be derived by applying discrete Laplacian to the Y-system can be derived by applying discrete Laplacian to the TBA equationsTBA equations

This Y-system is same as the Y-system from transfer matrix This Y-system is same as the Y-system from transfer matrix (up to (up to branch choice)branch choice)

Excited-State TBAExcited-State TBAAnalytic continuation of coupling constantAnalytic continuation of coupling constant

brings TBA from the ground state to excited brings TBA from the ground state to excited statesstates

[Dorey Tateo (1996,1997)][Dorey Tateo (1996,1997)]

Deformed Deformed contourcontour Original contourOriginal contour

Excited-State TBAExcited-State TBAAnalytic continuation of coupling constantAnalytic continuation of coupling constant

brings TBA from the ground state to excited brings TBA from the ground state to excited statesstates

[Dorey Tateo (1997)][Dorey Tateo (1997)]

Stokes PhenomenaStokes Phenomena

Analytic continuationAnalytic continuation Asymptotic limit (large Asymptotic limit (large LL))

Analytic continuationAnalytic continuationAsymptotic limitAsymptotic limit

Contour Deformation Trick

Contour Deformation Trick

• TBA equations and the exact energy are universal. Only the integration contours depend on the state under consideration.

• The large L (= small g) solution must be consistent with the generalized Lüscher formula (= transfer matrix formula).

• The contours are deformed smoothly as g increases.

• TBA equations and the exact energy are universal. Only the integration contours depend on the state under consideration.

• The large L (= small g) solution must be consistent with the generalized Lüscher formula (= transfer matrix formula).

• The contours are deformed smoothly as g increases.

Analytic continuation of Analytic continuation of ggDeformation ofDeformation of

integration contoursintegration contours

Contour in z-Contour in z-torustorus

Contour Deformation Trick

Contour Deformation Trick

Contour in z-Contour in z-torustorus

Our contourOur contour Naive guessNaive guessHow to choose?How to choose?

Contours in u-Contours in u-planeplane

Contour Deformation Trick

Contour Deformation Trick

How to determine the deformed contourHow to determine the deformed contour

All modification is explained by our choice of All modification is explained by our choice of contourcontour

Excited-state TBA equations at large Excited-state TBA equations at large LL must be must be solved by asymptotic Y-functions given by solved by asymptotic Y-functions given by

transfer-matrix formulatransfer-matrix formulaConsider 2-particle states in the Consider 2-particle states in the slsl (2) sector (2) sector with general with general LL,,

What are the integral equations they satisfy?What are the integral equations they satisfy?

Naive choice of contour is Naive choice of contour is inconsistentinconsistent when when L L > 6> 6Extra terms are needed to get consistent TBAExtra terms are needed to get consistent TBA

Singularity of YM|vw, 1+YM|

vw

Singularity of YM|vw, 1+YM|

vwYYM|vwM|vw functions are auxiliary Y-functions coupled to functions are auxiliary Y-functions coupled to

the Ythe YQQ functions which carry the energy functions which carry the energy

Singularity of YM|vw, 1+YM|

vw

Singularity of YM|vw, 1+YM|

vw• • All asymptotic YAll asymptotic YM|vwM|vw functions have four zeroes, functions have four zeroes, either on the real axis or on the imaginary axis of either on the real axis or on the imaginary axis of the mirrorthe mirror• • Zeroes of YZeroes of YM-1|vwM-1|vw are related to those of (1+Y are related to those of (1+YM|M|

vwvw), Y), YM+1|vwM+1|vw,, exhibit the universal structure as in the tableexhibit the universal structure as in the table

Type I Y1|vw 2

Type II Y1|vw,Y2|vw 2+2

Type III Y1|vw,Y2|vw, Y3|vw 4+2+2

Type IV Y1|vw,Y2|vw, Y3|vw, Y4|

vw

4+4+2+2

Type V ... ...

Evolution of Zeroes of YM|

vw

Evolution of Zeroes of YM|

vw

QuickTime˛ Ç∆GIF êLí£ÉvÉçÉOÉâÉÄ

ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

Zeroes of Zeroes of YM|vw move as move as gg increasesincreases

Principal-value prescription is Principal-value prescription is neededneeded

for singularities on the real axisfor singularities on the real axis

Subcritical valuesSubcritical values

Evolution of Zeroes of 1+YM|vw

Evolution of Zeroes of 1+YM|vwZeroes of 1+Zeroes of 1+YM|vw move as move as gg

increasesincreases

Critical valuesCritical valuesQuickTime˛ Ç∆

GIF êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

Two of them pinches the real axis,Two of them pinches the real axis,a new residue appears, an old disappearsa new residue appears, an old disappears

Evolution of Zeroes of 1+YM|vw

Evolution of Zeroes of 1+YM|vwZeroes of 1+Zeroes of 1+YM|vw move as move as gg

increasesincreases

Critical valuesCritical valuesQuickTime˛ Ç∆

GIF êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

Two of them pinches the real axis,Two of them pinches the real axis,a new residue appears, an old disappearsa new residue appears, an old disappears

Zeroes of asymptotic YM|

vw

Zeroes of asymptotic YM|

vw• • All asymptotic YAll asymptotic YM|vwM|vw functions have four zeroes, functions have four zeroes, either on the real axis or on the imaginary axis of either on the real axis or on the imaginary axis of the mirrorthe mirror

Konishi at Konishi at gg=0, k>2=0, k>2

Four extra log Four extra log SS terms terms cancelcancel

• • Zeroes of YZeroes of YM-1|vwM-1|vw are related to those of (1+Y are related to those of (1+YM|M|

vwvw), Y), YM+1|vwM+1|vw

Critical pointsCritical points

Three extra log Three extra log SS terms do not terms do not cancelcancel

• • Zeroes of asymptotic YZeroes of asymptotic Yk|vwk|vw move as move as gg increases. increases. The first critical value is approximately The first critical value is approximately gg=4.429 or λ=774=4.429 or λ=774

Konishi (J=2,n=1) at Konishi (J=2,n=1) at gg>4.429>4.429 or or Konishi-like (J>4, n=1) at Konishi-like (J>4, n=1) at g=0g=0 for for k=2k=2

• • Asymptotic Bethe Asymptotic Bethe Ansatz :Ansatz :

Subcritical PointsSubcritical Points

Three extra log Three extra log SS terms do not terms do not cancelcancel

• • Two pure imaginary zeroes collide, split into two Two pure imaginary zeroes collide, split into two real zeroesreal zeroes• • Extra terms in Canonical TBA equations change.Extra terms in Canonical TBA equations change. The first subcritical value is approximately The first subcritical value is approximately gg=4.495 =4.495 or λ=798or λ=798

[Arutyunov, Frolov, R.S. (2009)]; see also [Dorey, Tateo (1997)][Arutyunov, Frolov, R.S. (2009)]; see also [Dorey, Tateo (1997)]

Exact Konishi Spectrum?Exact Konishi Spectrum?

[Gromov, Kazakov, Vieira (2009)][Gromov, Kazakov, Vieira (2009)]

• • Numerical solution of TBA up to λ=700, Numerical solution of TBA up to λ=700, extrapolate to λ=∞extrapolate to λ=∞

• • Disagree slightly with semiclassical string Disagree slightly with semiclassical string resultsresults

[Roiban, Tseytlin (2009)][Roiban, Tseytlin (2009)]

Existence of infinitely many critical points may explain the mismatchExistence of infinitely many critical points may explain the mismatch

Konishi at weak couplingKonishi at weak couplingThe exact energy of TBA for Konishi state is The exact energy of TBA for Konishi state is

expanded asexpanded as

Four-loop part agrees with Four-loop part agrees with the known resultsthe known results in SYMin SYM

There is only There is only a conjecturea conjecture at five at five loopsloops

Konishi at five loopsKonishi at five loopsNeed to know the Need to know the correctioncorrection to the asymptotic to the asymptotic

Bethe rootsBethe rootsA refined generalized Lüscher formula was A refined generalized Lüscher formula was

conjecturedconjectured[Bajnok, Hegedűs, Janik, Łukowski (2009)][Bajnok, Hegedűs, Janik, Łukowski (2009)]

Five-loop anomalous dimension Five-loop anomalous dimension from from LüscherLüscher

Konishi at five loopsKonishi at five loopsThe correction The correction from TBAfrom TBA looks very looks very

differentdifferent

SolveSolve the TBA, linearized around asymptotic the TBA, linearized around asymptotic solutionsolution

We find We find TBA=LüscherTBA=Lüscher numericallynumerically

[Arutyunov, Frolov, R.S. (2010)][Arutyunov, Frolov, R.S. (2010)]

[Balog, Hegedűs (2010)][Balog, Hegedűs (2010)]there is an analytic proofthere is an analytic proof

Conclusion and OutlookConclusion and Outlook•TBA equations for excited states are dynamical;

depend on coupling constant as well as the state under consideration

•Contour deformation trick explains modification

•Five-loop Konishi agrees with the Lüscher formula

•TBA equations for excited states are dynamical; depend on coupling constant as well as the state under consideration

•Contour deformation trick explains modification

•Five-loop Konishi agrees with the Lüscher formula

The Lüscher formula agrees with BFKL agrees with BFKL equation,equation,for five-loop twist-two operators -- how about for five-loop twist-two operators -- how about

TBA?TBA?[Łukowski, Rej, Velizhanin (2009)][Łukowski, Rej, Velizhanin (2009)]

Deeper understanding on analytic structure Deeper understanding on analytic structure of TBAof TBA

Thank you for attentionThank you for attention