research.ipmu.jpInstitute for the Physics and Mathematics of the Universe, 20 October, 2009, Tokyo Finite size e ects in integrable models: planar AdS/CFT Zolt an Bajnok, Theoretical

Institute for the Physics and Mathematics of the Universe, 20 October, 2009, Tokyo

Finite size effects in integrable models: planar AdS/CFTZoltan Bajnok,

Theoretical Physics Research Group of the Hungarian Academy of Sciences,

Eotvos University, Budapest

1





IIB superstring on AdS5 × S5∑51 Y

2i = R2 −+ + +− = −R2

R2

α′∫dτdσ4π

(∂aXM∂aXM + ∂aY M∂aYM

)+ . . .

≡

N = 4 D=4 SU(N) SYMAµ,Ψi,Ψi,Φa

2gYM

∫d4xTr

[−1

4F 2 − 1

2(DΦ)2 + iΨD/Ψ + V

]V (Φ,Ψ) = 1

4[Φ,Φ]2 + Ψ[Φ,Ψ]

gaugeinvariant ope: O = Tr(Φ2)

Couplings:√λ = R2

α′ , gs = λ

N → 0

QM for string: 2D field theoryString spectrum E(λ)

E(λ) = E(∞) + E1√λ

+ E2

λ+ . . .

strong↔weak⇓

λ = g2YMN , N →∞ planar

〈On(x)Om(0)〉 = δnm|x|2∆n(λ)

Anomalous dim ∆(λ)∆(λ) = ∆(0) + λ∆1 + λ2∆2+

AdS←→integrable model←→CFT





g

Classical string theory

Semiclassical string theory

J

1 loop perturbative gauge theory

+ string loop corrections

Nonperturbative gauge theory

Quantum String theory

S −

mat

rix

+ gauge loop corrections

Asy

mpt

otic

Bet

he A

nsat

z

Wra

ppin

g/Lu

sche

r co

rrec

tion

Finite J (volume) integrable models: (Lee-Yang, sinh-Gordon, sine-Gordon)←→AdS/CFT

Motivation: AdS/CFT

2

Motivation: AdS/CFT

g

J

Motivation: AdS/CFT

g


J

Motivation: AdS/CFT

g



J

Motivation: AdS/CFT

g



J



Motivation: AdS/CFT

g



J




Motivation: AdS/CFT

g



J






Motivation: AdS/CFT

g



J





S −

mat

rix


Motivation: AdS/CFT

g



J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Motivation: AdS/CFT

g



J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Wra

ppin

g/Lu

sche

r co

rrec

tion

Need finite J (volume) solution of the spectral problem

Plan of talk I: Finite size effects in integrable models

3


Classical integrable models: sine-Gordon theory0

V

2 πβ φ

AdS5

S5



V

2 πβ φ

AdS5

S5

Bootstrap approach to quantum integrable models: S=scalar.matrix p p

1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2



V

2 πβ φ

AdS5

S5


1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2

Lee-Yang, sinh-Gordon, sine-Gordon+−

Bn

B

B B

B



V

2 πβ φ

AdS5

S5


1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Bn

B

B B

B

Finite volume: Asymptotic Bethe Ansatz:

p

2p

n

p1



V

2 πβ φ

AdS5

S5


1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Bn

B

B B

B


p

2p

n

p1

Luscher correction



V

2 πβ φ

AdS5

S5


1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Bn

B

B B

B


p

2p

n

p1

Luscher correction

Exact groundstate: TBA,

L

Re−H(R) L

L

e−H(L)R

R

Classical integrable models: sin(e/h)-Gordon theory

sine-Gordon0

V

2 πβ φ target 0

2πβ β ↔ ib sinh-Gordon

0

V

φ target οο

οο

L = 12(∂φ)2 − m2

β2 (1− cosβφ) L = 12(∂φ)2 − m2

b2(cosh bφ− 1)

4


sine-Gordon0

V

2 πβ φ target 0


0

V

φ target οο

οο

L = 12(∂φ)2 − m2

β2 (1− cosβφ) L = 12(∂φ)2 − m2

b2(cosh bφ− 1)

Classicalfinite energy

solutions:

x

φ

x

φ

x

φ

sine-Gordon theory soliton anti-soliton breather


sine-Gordon0

V

2 πβ φ target 0


0

V

φ target οο

οο

L = 12(∂φ)2 − m2

β2 (1− cosβφ) L = 12(∂φ)2 − m2

b2(cosh bφ− 1)


solutions:

x

φ

x

φ

x

φ


Classical factorized scattering: time delays sums up ∆T1(v1, v2, . . . , vn) =∑i∆T1(v1, vi)

Τ vvv1 2 n> > ... >

.

v Τ1

v <2< ... <vn

Bootstrap program

Asymptotic states |p1, p2, . . . , pn〉in/outform a representation of global symmetry: p p

21> >...> p

n

5

Bootstrap program


21> >...> p

n

Lorentz: P =∑i pi E =

∑iE(pi)

dispersion relation E(p) =√m2 + p2

Bootstrap program


21> >...> p

n


∑iE(pi)


p p1

pn2

∆(Q)

Bootstrap program


21> >...> p

n


∑iE(pi)


Scattering matrix S: |out〉 → |in〉commutes with symmetry [S,∆(Q)] = 0

Bootstrap program


21> >...> p

n


∑iE(pi)



p p

S−matrix

1

21> >...>

p’p’m

< p’2

<...<

pn

Bootstrap program


21> >...> p

n


∑iE(pi)



p p

S−matrix

1

21> >...>

p’p’m

< p’2

<...<

pn

(Q)∆

(Q)∆

Bootstrap program


21> >...> p

n


∑iE(pi)



p p

S−matrix

1

21> >...>

p’p’m

< p’2

<...<

pn

(Q)∆

(Q)∆

Higher spin concerved charge

factorization + Yang-Baxter equation

S123 = S23S13S12 = S12S13S23

p p

1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2

S-matrix = scalar . Matrix

Bootstrap program

Asymptotic states |p1, p2, . . . , pn〉in/outform a representation of global symmetry:

p p21

> >...> pn


∑iE(pi)



p p

S−matrix

1

21> >...>

p’p’m

< p’2

<...<

pn

(Q)∆

(Q)∆



S123 = S23S13S12 = S12S13S23p p

1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Unitarity S12S21 = Id

Crossing symmetry S12 = S21Maximal analyticity: all poles have physical origin→ boundstates, anomalous thresholds

Bootstrap program: diagonal

Diagonal scattering: S-matrix = scalar S(p1, p2) = S(θ1−θ2) p = m sinh θ

6



Unitarity S(θ)S(−θ) = 1

Crossing symmetry S(θ) = S(iπ − θ)

Maximal analyticity: all poles have physical origin






Minimal solution: S(θ) = 1 Free boson







CDD factor: S(θ) = sinh θ−i sin pπsinh θ+i sin pπ Sinh-Gordon

V ∼ cosh bφ↔ b2

8π+b2= p > 0









8π+b2= p > 0

Maximal analyticity: S(θ) = sinh θ+i sin pπsinh θ−i sin pπ

pole at θ = ipπ →boundstate B2

bootstrap: S12(θ)=S11(θ − ipπ2

)S11(θ + ipπ2

)

new particle if p 6= 23 Lee-Yang



Unitarity S(θ)S(−θ) = 1Crossing symmetry S(θ) = S(iπ − θ)Maximal analyticity: all poles have physical origin




8π+b2= p > 0




)S11(θ + ipπ2

)


Maximal analyticity:

all poles have physical origin

→sine-Gordon solitons

θιπ

S11

ipπ

B2

θιπ

S

ipπ/23ipπ/2

12

B3. . .

θιπ

S1n

π/2ip(n+1)(n−1)ipπ/2

Bn

Bootstrap program: sine-Gordon

Nondiagonal scattering: S-matrix = scalar . Matrix soliton doublet

(ss

)

Matrix:

global symmetry Uq(sl2)

2d evaluation reps

[S,∆(Q)] = 0S−matrix

(Q)∆

∆ (Q)

−1 0 0 0

0 − sinλπsinλ(π+iθ)

sin iλθsinλ(π+iθ) 0

0 sin iλθsinλ(π+iθ)

− sinλπsinλ(π+iθ) 0

0 0 0 −1

7



(ss

)

Matrix:


2d evaluation reps


(Q)∆

∆ (Q)

−1 0 0 0





0 0 0 −1

Unitarity

S(θ)S(−θ) = 1

Crossing symmetry

S(θ) = Sc1(iπ − θ)

∏∞l=1

[Γ(2(l−1)λ+λiθ

π )Γ(2lλ+1+λiθπ )

Γ((2l−1)λ+λiθπ )Γ((2l−1)λ+1+λiθ

π )/(θ → −θ)

]



(ss

)

Matrix:

global symmetry Uq(sl2)2d evaluation reps

[S,∆(Q)] = 0

S−matrix

(Q)∆

∆ (Q)

−1 0 0 0





0 0 0 −1

Unitarity

S(θ)S(−θ) = 1

Crossing symmetry

S(θ) = Sc1(iπ − θ)

∏∞l=1




π )/(θ → −θ)

]



either boundstates or

anomalous thresholds

p = λ−1+−

Bn

B

B B

B



(ss

)

Matrix:


[S,∆(Q)] = 0

S−matrix

(Q)∆

∆ (Q)

−1 0 0 0





0 0 0 −1

Unitarity

S(θ)S(−θ) = 1Crossing symmetry

S(θ) = Sc1(iπ − θ)

∏∞l=1




π )/(θ → −θ)

]





p = λ−1+−

Bn

B

B B

B

θιπ

S+−+−

(1−np)ιπ

...

...

QFTs in finite volume

Finite volume spectrump

2p

n

p1

8



2p

n

p1

L

E

0

2m

m

S−

mat

rix

Bethe−YangLuscherCFT



2p

n

p1

L

E

0

2m

m

S−

mat

rix




2p

n

p1

L

E

0

2m

m

S−

mat

rix


Infinite volume spectrum:

E(p1, . . . , pn) =∑iE(pi) pi ∈ R



2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R

Polynomial volume corrections:

Bethe-Yang; pi quantized. Diagonal



2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R



eipjLS(pj, p1) . . . S(pj, pn) = −1 ; S(0) = −1

pjL+∑k

1i logS(pj, pk) = (2n+ 1)iπ

p2πL



2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R




pjL+∑k


p2πL

Non-diagonal, sine-Gordon

eipjLS(pj, p1) . . .S(pj, pn)Ψ = −Ψ S(0) = −Pn

1 2 j... ...p p p

p



2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R




pjL+∑k


p2πL



1 2 j... ...p p p

p

Inhomogenous XXZ spin-chain spectral problem eiL sinh θjT (θj)S0(θj) = −1



2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R




pjL+∑k


p2πL



1 2 j... ...p p p

p


T (θ)Q(θ) = Q(θ+iπ)T0(θ− iπ2 )+Q(θ−iπ)T0(θ+ iπ2 ) = Q++T−+Q−−T+



2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R




pjL+∑k


p2πL


eipjLS(pj, p1) . . .S(pj, pn)Ψ = −Ψ S(0) = −P

n

1 2 j... ...p p p

p



Q(θ) =∏β sinh(λ(θ − wβ))

T0(θ) =∏j sinh(λ(θ − θj))

Bethe Ansatz:T0(wα−iπ2 )Q(wα+iπ)

T0(wα+iπ2 )Q(wα−iπ)

=T−0 Q

++

T+0 Q−−

|α = −1

Luscher correction of multiparticle states

Finite volume spectrum n

1 2 j... ...p p p

p

9



1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix




1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix




1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix


Luscher originally: O(e−mL) mass correction

m(L) =√

32 m(−iRes

θ=2iπ3S)e−

√3

2 mL

−∫ dθ

2π cosh θ(S(θ + iπ2 )− 1)e−mL cosh θ



1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix



m(L) =√

32 m(−iRes

θ=2iπ3S)e−

√3

2 mL

−∫ dθ


Multiparticle Luscher correction

BY: S(pj, p1) . . .S(pj, pn)Ψ = −e−ipjLΨ

T (p, p1, . . . , pn)Ψ = t(p, p1, . . . , pn,Ψ)Ψ



1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix



m(L) =√

32 m(−iRes

θ=2iπ3S)e−

√3

2 mL

−∫ dθ




T (p, p1, . . . , pn)Ψ = t(p, p1, . . . , pn,Ψ)Ψ

Modified momenta:

pjL− i log t(pj, p1, . . . , pn,Ψ) = (2n+ 1)π + Φj

Φj =∫ dq

2πddpjt(q, p1, . . . , pn,Ψ)e−LE(q)


Finite volume spectrumn

1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix



m(L) =√

32 m(−iRes

θ=2iπ3S)e−

√3

2 mL

−∫ dθ



BY: S(pj, p1) . . .S(pj, pn)Ψ = −e−ipjLΨT (p, p1, . . . , pn)Ψ = t(p, p1, . . . , pn,Ψ)Ψ

Modified momenta:


Φj =∫ dq


Modified energy:

E(p1, . . . , pn) =∑kE(pk + δpk)−

∫ dq2πt(q, p1, . . . , pn,Ψ)e−LE(q)

Thermodynamic Bethe Ansatz: diagonal

Ground-state energy exactly L

10



L

E

0

2m

m

S−

mat

rix




L

E

0

2m

m

S−

mat

rix




L

E

0

2m

m

S−

mat

rix


Euclidien partition function:

Z(L,R) =R→∞ Tr(e−H(L)R)

Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

E

0

2m

m

S−

mat

rix




Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R

Exchange space and Euclidien time

Z(L,R) =R→∞ Tr(e−H(R)L) =R→∞∑n e−En(L)R

L

Re−H(R) L



L

E

0

2m

m

S−

mat

rix




Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

Re−H(R) L

Dominant contribution: finite particle/hole density ρ, ρh:p2π

L



L

E

0

2m

m

S−

mat

rix




Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

Re−H(R) L


L

En(R) =∑

iE(pi)→∫E(p)ρ(p)dp

pjR+∑

k1i

logS(pj, pk) = (2n+ 1)iπ −→ R+∫

(−idp logS(p, p′))ρ(p′)dp

′= 2π(ρ+ ρh)



L

E

0

2m

m

S−

mat

rix




Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

Re−H(R) L


L

En(R) =∑


pjR+∑

k1i

logS(pj, pk) = (2n+ 1)iπ −→ R+∫


′= 2π(ρ+ ρh)

Z(L,R) =∫d[ρ, ρh]e−LE(R)−

∫((ρ+ρh) ln(ρ+ρh)−ρ ln ρ−ρh ln ρh)dp



L

E

0

2m

m

S−

mat

rix



Z(L,R) =R→∞ Tr(e−H(L)R)Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

Re−H(R) L


L

En(R) =∑


pjR+∑

k1i

logS(pj, pk) = (2n+ 1)iπ −→ R+∫


′= 2π(ρ+ ρh)



Saddle point : ε(p) = ln ρh(p)ρ(p) ε(p) = E(p)L+

∫ dp2πidp logS(p′, p) log(1 + e−ε(p

′))

Ground state energy exactly: E0(L) = −∫ dp

2π log(1 + e−ε(p)) Lee-Yang, sinh-Gordon

Plan of talk II: Planar AdS/CFT

11



V

2 πβ φ

AdS5

S5



V

2 πβ φ

AdS5

S5

Quantization of integrable models: sine-Gordon model: PCFT



V

2 πβ φ

AdS5

S5



1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2



V

2 πβ φ

AdS5

S5



1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2

Lee-Yang, sinh-Gordon, sine-Gordon↔ AdS σmodel+−

Bn

B

B B

B



V

2 πβ φ

AdS5

S5



1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Bn

B

B B

B


p

2p

n

p1



V

2 πβ φ

AdS5

S5



1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Bn

B

B B

B


p

2p

n

p1

Luscher correction



V

2 πβ φ

AdS5

S5



1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Bn

B

B B

B


p

2p

n

p1

Luscher correction

Exact: groundstate TBA,

L

Re−H(R) L

L

e−H(L)R

R



V

2 πβ φ

AdS5

S5



1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Bn

B

B B

B


p

2p

n

p1

Luscher correction

Exact: groundstate TBA,

L

Re−H(R) L

L

e−H(L)R

R

Excited TBA, Y-system, NLIE...

p

p+1


sine-Gordon0

V

2 πβ φ target 0


0

V

φ target οο

οο

L = 12(∂φ)2 − m2

β2 (1− cosβφ) L = 12(∂φ)2 − m2

b2(cosh bφ− 1)

12


sine-Gordon0

V

2 πβ φ target 0


0

V

φ target οο

οο

L = 12(∂φ)2 − m2

β2 (1− cosβφ) L = 12(∂φ)2 − m2

b2(cosh bφ− 1)


solutions:

x

φ

x

φ

x

φ



sine-Gordon0

V

2 πβ φ target 0


0

V

φ target οο

οο

L = 12(∂φ)2 − m2

β2 (1− cosβφ) L = 12(∂φ)2 − m2

b2(cosh bφ− 1)


solutions:

x

φ

x

φ

x

φ


Integrability: ∂xAt − ∂tAx + [Ax, At] = 0↔ T (λ) = TrP exp∮A(x)µdxµ

Ax(µ) = i2

(2µ β∂+ϕ

−β∂+ϕ −2µ

)At(µ) = 1

4iµ

(cosβϕ −i sinβϕi sinβϕ − cosβϕ

)T

gTg−1


sine-Gordon0

V

2 πβ φ target 0


0

V

φ target οο

οο

L = 12(∂φ)2 − m2

β2 (1− cosβφ) L = 12(∂φ)2 − m2

b2(cosh bφ− 1)


solutions:

x

φ

x

φ

x

φ



Ax(µ) = i2

(2µ β∂+ϕ

−β∂+ϕ −2µ

)At(µ) = 1

4iµ


)T

gTg−1

conserved Q±1[ϕ] = E[ϕ]± P [ϕ] =∫ {1

2(∂±ϕ)2 + m2

β2 (1− cosβϕ)}dx

charges: Q±3[ϕ] =∫ {1

2(∂2±ϕ)2 − 1

8(∂±ϕ)4 + m2

β2 (∂±ϕ)2(1− cosβϕ)}


sine-Gordon0

V

2 πβ φ target 0


0

V

φ target οο

οο

L = 12(∂φ)2 − m2

β2 (1− cosβφ) L = 12(∂φ)2 − m2

b2(cosh bφ− 1)


solutions:

x

φ

x

φ

x

φ



Ax(µ) = i2

(2µ β∂+ϕ

−β∂+ϕ −2µ

)At(µ) = 1

4iµ


)T

gTg−1

Classical factorized scattering: time delays sums up ∆T1(v1, v2, . . . , vn) =∑i∆T1(v1, vi)

Τ vvv1 2 n> > ... >

.

v Τ1

v <2< ... <vn

Classical integrability: AdSg



J





S −

mat

rix


Asy

mpt

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Bet

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nsat

z

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sche

r co

rrec

tion

AdS σ modelτ

σ target

AdS5

S5

L = R2

α′(∂aXM∂aXM + ∂aYM∂aYM

)+ fermions

13




J





S −

mat

rix


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Bet

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nsat

z

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g/Lu

sche

r co

rrec

tion

AdS σ modelτ

σ target

AdS5

S5

L = R2


)+ fermions

Classical solutions are found, for example magnon:

S5

p

−p

J




J





S −

mat

rix


Asy

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Bet

he A

nsat

z

Wra

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g/Lu

sche

r co

rrec

tion

AdS σ modelτ

σ target

AdS5

S5

L = R2


)+ fermions


S5

p

−p

J

Coset NLσ model: g ∈ PSU(2,2|4)SO(4,1)×SO(5) J = g−1dg = J||+ J⊥

Z4 graded structure:

J⊥ → J0, J1, J2, J3 L ∝ STr(J2 ∧ ∗J2)− STr(J1 ∧ J3)




J





S −

mat

rix


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Bet

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nsat

z

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g/Lu

sche

r co

rrec

tion

AdS σ modelτ

σ target

AdS5

S5

L = R2


)+ fermions


S5

p

−p

J

Coset NLσ model: g ∈ PSU(2,2|4)SO(4,1)×SO(5) J = g−1dg = J||+ J⊥

Z4 graded structure:

J⊥ → J0, J1, J2, J3L ∝ STr(J2 ∧ ∗J2)− STr(J1 ∧ J3)

Integrability from flat connection: dA−A ∧A = 0

A(µ) = J0 + µ−1J1 + (µ2 + µ−2)J2/2 + (µ2 + µ−2)J2/2 + µJ3

Conserved charges from the trace of the monodromy matrix

T (µ) = P exp∮A(x)µdxµ

T

gTg−1

Quantum integrability: sine-Gordon L = 12(∂φ)2 − m2

β2 (1− cosβφ)

Perturbed Conformal Field Theory Lagrangian perturbation theory

LCFT + λLpert = 12(∂φ)2 + λ(Vβ + V−β) L0 + Vpert = 1

2(∂φ)2 − m2

2 φ2 − β2U

hβ = β2 definite scaling Vβ =: eiβφ : semiclassical=free

14

Quantum integrability: sine-Gordon L = 12(∂φ)2 − m2

β2 (1− cosβφ)

Perturbed Conformal Field Theory Lagrangian perturbation theory

LCFT + λLpert = 12(∂φ)2 + λ(Vβ + V−β) L0 + Vpert = 1

2(∂φ)2 − m2

2 φ2 − β2U

hβ = β2 definite scaling Vβ =: eiβφ : semiclassical=free

Quantum conservation laws∂−Λ4 = 0→ ∂−Λ4 = λ∂+Θ2

[λ] = 2− hβ, [Λ4] = 4,

Nonlocal symmetries Uq(sl2)

Correlators=∑loops Feynmandiagrams

Asymptotic states E(p) =√p2 +m2

S-matrix↔correlators LSZunitarity, crossing symmetry, analyticity

Bootstrap scheme

Quantum integrability: AdS no proof !

Bootstrap program


21> >...> p

n

15

Bootstrap program


21> >...> p

n


∑iE(pi)


Bootstrap program


21> >...> p

n


∑iE(pi)


p p1

pn2

∆(Q)

Bootstrap program


21> >...> p

n


∑iE(pi)



Bootstrap program


21> >...> p

n


∑iE(pi)



p p

S−matrix

1

21> >...>

p’p’m

< p’2

<...<

pn

Bootstrap program


21> >...> p

n


∑iE(pi)



p p

S−matrix

1

21> >...>

p’p’m

< p’2

<...<

pn

(Q)∆

(Q)∆

Bootstrap program


21> >...> p

n


∑iE(pi)



p p

S−matrix

1

21> >...>

p’p’m

< p’2

<...<

pn

(Q)∆

(Q)∆



S123 = S23S13S12 = S12S13S23

p p

1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Bootstrap program

Asymptotic states |p1, p2, . . . , pn〉in/outform a representation of global symmetry:

p p21

> >...> pn


∑iE(pi)



p p

S−matrix

1

21> >...>

p’p’m

< p’2

<...<

pn

(Q)∆

(Q)∆



S123 = S23S13S12 = S12S13S23p p

1

21

2

p

3

3

p p p

p

1

1

2

p

3

3

p p p

p2


Unitarity S12S21 = Id

Crossing symmetry S12 = S21Maximal analyticity: all poles have physical origin→ boundstates, anomalous thresholds



16




















8π+b2= p > 0









8π+b2= p > 0




)S11(θ + ipπ2

)




Unitarity S(θ)S(−θ) = 1Crossing symmetry S(θ) = S(iπ − θ)Maximal analyticity: all poles have physical origin




8π+b2= p > 0




)S11(θ + ipπ2

)




→sine-Gordon solitons

θιπ

S11

ipπ

B2

θιπ

S

ipπ/23ipπ/2

12

B3. . .

θιπ

S1n

π/2ip(n+1)(n−1)ipπ/2

Bn



(ss

)

Matrix:


2d evaluation reps


(Q)∆

∆ (Q)

−1 0 0 0





0 0 0 −1

17



(ss

)

Matrix:


2d evaluation reps


(Q)∆

∆ (Q)

−1 0 0 0





0 0 0 −1

Unitarity

S(θ)S(−θ) = 1

Crossing symmetry

S(θ) = Sc1(iπ − θ)

∏∞l=1




π )/(θ → −θ)

]



(ss

)

Matrix:


[S,∆(Q)] = 0

S−matrix

(Q)∆

∆ (Q)

−1 0 0 0





0 0 0 −1

Unitarity

S(θ)S(−θ) = 1

Crossing symmetry

S(θ) = Sc1(iπ − θ)

∏∞l=1




π )/(θ → −θ)

]





p = λ−1+−

Bn

B

B B

B



(ss

)

Matrix:


[S,∆(Q)] = 0

S−matrix

(Q)∆

∆ (Q)

−1 0 0 0





0 0 0 −1

Unitarity

S(θ)S(−θ) = 1Crossing symmetry

S(θ) = Sc1(iπ − θ)

∏∞l=1




π )/(θ → −θ)

]





p = λ−1+−

Bn

B

B B

B

θιπ

S+−+−

(1−np)ιπ

...

...

Bootstrap program: AdS g



J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Wra

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g/Lu

sche

r co

rrec

tion

Nondiagonal scattering: S-matrix = scalar . Matrix

18




J





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mat

rix


Asy

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he A

nsat

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Wra

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g/Lu

sche

r co

rrec

tion


Matrix:

global symmetry PSU(2|2)2

Q = 1 reps

b1

b2

f3

f4

⊗ b1

b2f3f4


(Q)∆

∆ (Q)




J





S −

mat

rix


Asy

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Bet

he A

nsat

z

Wra

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g/Lu

sche

r co

rrec

tion


Matrix:


Q = 1 reps

b1

b2

f3

f4

⊗ b1

b2f3f4


(Q)∆

∆ (Q)

1 . . . . . . . . . . . . . . .. A . . −b . . . . . . −c . . c .. . d . . . . . e . . . . . . .. −b . .d . . . . . . . . e . . .. . . . A . . . . . . c . . −c .. . . . . 1 . . . . . . . . . .. . . . . . d . .e . . . . . . .. . . . . . . d . . . . . e . .. . f . . . . . g . . . . . . .. . . . . . f . . g . . . . . .. . . . . . . . . . a . . . . .. −h . . h . . . . . . B . . −i .. . . f . . . . . . . . g . . .. . . . . . . f . . . . . g . .. h . . −h . . . . . . −i . . B .. . . . . . . . . . . . . . . a




J





S −

mat

rix


Asy

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Bet

he A

nsat

z

Wra

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g/Lu

sche

r co

rrec

tion


Matrix:


Q = 1 reps

b1

b2

f3

f4

⊗ b1

b2f3f4


(Q)∆

∆ (Q)


Unitarity

S(z1, z2)S(z2, z1) = 1

Crossing symmetry

S(z1, z2) = Sc1(z2, z1 + ω2)

S1111 = u1−u2−i

u1−u2+iei2θ(z1,z2)

u = 12

cot p2E(p)

p = 2 am(z)

Bootstrap program: AdSg



J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Wra

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g/Lu

sche

r co

rrec

tion


Matrix:


Q = 1 reps

b1

b2

f3

f4

⊗ b1

b2f3f4


(Q)∆

∆ (Q)


Unitarity

S(z1, z2)S(z2, z1) = 1

Crossing symmetry

S(z1, z2) = Sc1(z2, z1 + ω2)

S1111 = u1−u2−i

u1−u2+iei2θ(z1,z2)

u = 12

cot p2E(p)

p = 2 am(z)


boundstates atyp symrep: Q ∈ N

anomalous thresholds Physical domain?1 Q

Q 1

1Q ω /2

−ω /2 ω /2

ω2

2

1 1



2p

n

p1

19



2p

n

p1

L

E

0

2m

m

S−

mat

rix




2p

n

p1

L

E

0

2m

m

S−

mat

rix




2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R



2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R





2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R




pjL+∑k


p2πL



2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R




pjL+∑k


p2πL



1 2 j... ...p p p

p



2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R




pjL+∑k


p2πL



1 2 j... ...p p p

p




2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R




pjL+∑k


p2πL



1 2 j... ...p p p

p





2p

n

p1

L

E

0

2m

m

S−

mat

rix



E(p1, . . . , pn) =∑iE(pi) pi ∈ R




pjL+∑k


p2πL



n

1 2 j... ...p p p

p



Q(θ) =∏β sinh(λ(θ − wβ))

T0(θ) =∏j sinh(λ(θ − θj))

Bethe Ansatz:T0(wα−iπ2 )Q(wα+iπ)

T0(wα+iπ2 )Q(wα−iπ)

=T−0 Q

++

T+0 Q−−

|α = −1

Bethe-Yang=Asymptotic Bethe Ansatz

g



J





S −

mat

rix


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Bet

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nsat

z

Wra

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g/Lu

sche

r co

rrec

tion


2p

n

p1

20


g



J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

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Wra

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g/Lu

sche

r co

rrec

tion


2p

n

p1


E(p1, . . . , pn) =∑iE(pi) E(p) =

√1 + (4g sin p

2)2 pi ∈ [−π, π]


g



J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Wra

ppin

g/Lu

sche

r co

rrec

tionFinite volume spectrum

p

2p

n

p1


E(p1, . . . , pn) =∑iE(pi) E(p) =

√1 + (4g sin p

2)2 pi ∈ [−π, π]


Asymptotic Bethe Ansatz; pi quantized, .


1 2 j... ...p p p

p


g



J





S −

mat

rix


Asy

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Bet

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nsat

z

Wra

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g/Lu

sche

r co

rrec

tion


2p

n

p1


E(p1, . . . , pn) =∑iE(pi) E(p) =

√1 + (4g sin p

2)2 pi ∈ [−π, π]




1 2 j... ...p p p

p

Inhomogenous Hubbard2 spin-chain: eiLpjS20(uj)

Q++4 (uj)

Q−−4 (uj)T (uj)T (uj) = −1


g



J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Wra

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g/Lu

sche

r co

rrec


p

2p

n

p1


E(p1, . . . , pn) =∑iE(pi) E(p) =

√1 + (4g sin p

2)2 pi ∈ [−π, π]




1 2 j... ...p p p

p


Q++4 (uj)

Q−−4 (uj)T (uj)T (uj) = −1

T (u) =B−1 B

+3 R−(+)4

B+1 B−3 R−(−)4

[Q−−2 Q+

3Q2Q

−3− R

−(−)4 Q+

3

R−(+)4 Q−3

+Q++

2 Q−1Q2Q

+1

− B+(+)4 Q−1

B+(−)4 Q+

1

]

Qj(u) = −Rj(u)Bj(u) and R(±)j =

∏k

x(u)−x∓j,k√x∓j,k

B(±)j =

∏k

1



g



J





S −

mat

rix


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Bet

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nsat

z

Wra

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g/Lu

sche

r co

rrec


p

2p

n

p1


E(p1, . . . , pn) =∑iE(pi) E(p) =

√1 + (4g sin p

2)2 pi ∈ [−π, π]




n

1 2 j... ...p p p

p


Q++4 (uj)

Q−−4 (uj)T (uj)T (uj) = −1

T (u) =B−1 B

+3 R−(+)4

B+1 B−3 R−(−)4

[Q−−2 Q+

3Q2Q

−3− R

−(−)4 Q+

3

R−(+)4 Q−3

+Q++

2 Q−1Q2Q

+1

− B+(+)4 Q−1

B+(−)4 Q+

1

]

Qj(u) = −Rj(u)Bj(u) and R(±)j =

∏k


B(±)j =

∏k

1


Bethe Ansatz:Q+

2 B(−)4

Q−2B(+)4

|1 = 1Q−−2 Q+

1 Q+3

Q++2 Q−1Q

−3

|2 = −1Q+

2 R(−)4

Q−2R(+)4

|3 = 1



1 2 j... ...p p p

p

21



1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix




1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix




1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix



m(L) =√

32 m(−iRes

θ=2iπ3S)e−

√3

2 mL

−∫ dθ




1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix



m(L) =√

32 m(−iRes

θ=2iπ3S)e−

√3

2 mL

−∫ dθ




T (p, p1, . . . , pn)Ψ = t(p, p1, . . . , pn,Ψ)Ψ



1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix



m(L) =√

32 m(−iRes

θ=2iπ3S)e−

√3

2 mL

−∫ dθ




T (p, p1, . . . , pn)Ψ = t(p, p1, . . . , pn,Ψ)Ψ

Modified momenta:


Φj =∫ dq




1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix



m(L) =√

32 m(−iRes

θ=2iπ3S)e−

√3

2 mL

−∫ dθ



BY: S(pj, p1) . . .S(pj, pn)Ψ = −e−ipjLΨT (p, p1, . . . , pn)Ψ = t(p, p1, . . . , pn,Ψ)Ψ

Modified momenta:


Φj =∫ dq


Modified energy:

E(p1, . . . , pn) =∑kE(pk + δpk)−

∫ dq2πt(q, p1, . . . , pn,Ψ)e−LE(q)

Luscher/wrapping correction in AdSg



J





S −

mat

rix


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mpt

otic

Bet

he A

nsat

z

Wra

ppin

g/Lu

sche

r co

rrec

tion


1 2 j... ...p p p

p

22




J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Wra

ppin

g/Lu

sche

r co

rrec

tion


1 2 j... ...p p p

p

One particle correction:

∆(L) =(1− E′(p)E

′(p0)

)(−iResp=p0

∑b S)e−E(p)L

-∑b∫∞−∞

dp2π

(1− E′(p)E

′(p)

)(Sbaba(p, p)− 1)e−E(p)L




J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Wra

ppin

g/Lu

sche

r co

rrec

tion


1 2 j... ...p p p

p


∆(L) =(1− E′(p)E

′(p0)

)(−iResp=p0

∑b S)e−E(p)L

-∑b∫∞−∞

dp2π

(1− E′(p)E

′(p)


Two particle Luscher correction (Konishi)

BY: j = 1,2 S(pj, p1)S(pj, p2)Ψ = −e−ipjLΨ

T (p, p1, p2)Ψ = t(p, p1, p2,Ψ)Ψ




J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Wra

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g/Lu

sche

r co

rrec

tion


1 2 j... ...p p p

p


∆(L) =(1− E′(p)E

′(p0)

)(−iResp=p0

∑b S)e−E(p)L

-∑b∫∞−∞

dp2π

(1− E′(p)E

′(p)



BY: j = 1,2 S(pj, p1)S(pj, p2)Ψ = −e−ipjLΨ

T (p, p1, p2)Ψ = t(p, p1, p2,Ψ)Ψ

Modified momenta:

pjL− i log t(pj, p1, p2,Ψ) = (2n+ 1)π + Φj∫ dp2π

ddp1

t(p, p1, p2,Ψ)e−LE(p) 6= Φ1 = −∫ dp

2π( ddpS(p, p1))S(p, p2)e−LE(p)




J





S −

mat

rix


Asy

mpt

otic

Bet

he A

nsat

z

Wra

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g/Lu

sche

r co

rrec

tion


1 2 j... ...p p p

p


∆(L) =(1− E′(p)E

′(p0)

)(−iResp=p0

∑b S)e−E(p)L

-∑b∫∞−∞

dp2π

(1− E′(p)E

′(p)



BY: j = 1,2 S(pj, p1)S(pj, p2)Ψ = −e−ipjLΨT (p, p1, p2)Ψ = t(p, p1, p2,Ψ)Ψ

Modified momenta:

pjL− i log t(pj, p1, p2,Ψ) = (2n+ 1)π + Φj∫ dp2π

ddp1

t(p, p1, p2,Ψ)e−LE(p) 6= Φ1 = −∫ dp

2π( ddpS(p, p1))S(p, p2)e−LE(p)

Modified energy:

E(p1, p2) =∑kE(pk + δpk)−

∫ dq2πt(q, p1, p2,Ψ)e−LE(q)



23



L

E

0

2m

m

S−

mat

rix




L

E

0

2m

m

S−

mat

rix




L

E

0

2m

m

S−

mat

rix




Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

E

0

2m

m

S−

mat

rix




Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

Re−H(R) L



L

E

0

2m

m

S−

mat

rix




Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

Re−H(R) L


L



L

E

0

2m

m

S−

mat

rix




Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

Re−H(R) L


L

En(R) =∑


pjR+∑

k1i

logS(pj, pk) = (2n+ 1)iπ −→ R+∫


′= 2π(ρ+ ρh)



L

E

0

2m

m

S−

mat

rix




Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

Re−H(R) L


L

En(R) =∑


pjR+∑

k1i

logS(pj, pk) = (2n+ 1)iπ −→ R+∫


′= 2π(ρ+ ρh)





L

E

0

2m

m

S−

mat

rix



Z(L,R) =R→∞ Tr(e−H(L)R)Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)

L

e−H(L)R

R



L

Re−H(R) L


L

En(R) =∑


pjR+∑

k1i

logS(pj, pk) = (2n+ 1)iπ −→ R+∫


′= 2π(ρ+ ρh)



Saddle point : ε(p) = ln ρh(p)ρ(p) ε(p) = E(p)L+

∫ dp2πidp logS(p′, p) log(1 + e−ε(p

′))


2π log(1 + e−ε(p)) Lee-Yang, sinh-Gordon

Thermodynamic Bethe Ansatz: non-diagonal


L

E

0

2m

m

S−

mat

rix



Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L

24



L

E

0

2m

m

S−

mat

rix



Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L

Finite particle/hole + Bethe root density ρ0, ρ0h, ρ

i, ρih:

eiLpT S0|j = −1,T−0 Q

++

T+0 Q−−

|α = −1

p2πL

ιπ2

ιπp

periodicity ιπ /p



L

E

0

2m

m

S−

mat

rix



Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L


i, ρih:


++

T+0 Q−−

|α = −1

p2πL

ιπ2

ιπp

periodicity ιπ /p

En(R) =∑

iE(pi)→∫E(p)ρ0(p)dp

Rδm0 +∫Kmn (p, p′)ρn(p

′)dp

′= 2π(ρm + ρmh )



L

E

0

2m

m

S−

mat

rix



Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L


i, ρih:


++

T+0 Q−−

|α = −1

p2πL

ιπ2

ιπp

periodicity ιπ /p

En(R) =∑



′)dp

′= 2π(ρm + ρmh )

Z(L,R) =∫d[ρi, ρih]e−LE(R)−

∑i

∫((ρi+ρih) ln(ρi+ρih)−ρi ln ρi−ρih ln ρih)dp



L

E

0

2m

m

S−

mat

rix



Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L


i, ρih:


++

T+0 Q−−

|α = −1

p2πL

ιπ2

ιπp

periodicity ιπ /p

En(R) =∑



′)dp

′= 2π(ρm + ρmh )


∑i


Saddle point : εi(θ) = − ln ρi(p)ρih(p) εj(θ) = δ

j0E(p)L−

∫Kji (p′, p) log(1 + e−ε

i(p′))dp′


2π log(1 + e−ε0(θ))dθ

Thermodynamic Bethe Ansatz: AdS g



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rix


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Euclidien E2 + (4g sin p2)2 = 1 partition function:

Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L

25




J





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mat

rix


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nsat

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sche

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rrec

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Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L

Finite particle/hole + Bethe root density ρQ, ρQh , ρ

i, ρih: i

no periodicity




J





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mat

rix


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Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L


i, ρih: i

no periodicity

En(R) =∑

i,Q EQ(pi)→∑

Q

∫EQ(p)ρQ(p)dp

eiLpS20Q++

4

Q−−4

T T |4 = −1 Q+2 B

(−)4

Q−2B(+)4

|1 = 1 Q−−2 Q+1 Q

+3

Q++2 Q−1Q

−3

|2 = −1 Q+2 R

(−)4

Q−2R(+)4

|3 = 1

∫Kmn (p, p′)ρn(p

′)dp

′= 2π(ρm + ρmh )




J





S −

mat

rix


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Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L


i, ρih: i

no periodicity

En(R) =∑

i,Q EQ(pi)→∑

Q

∫EQ(p)ρQ(p)dp

eiLpS20Q++

4

Q−−4

T T |4 = −1 Q+2 B

(−)4

Q−2B(+)4

|1 = 1 Q−−2 Q+1 Q

+3

Q++2 Q−1Q

−3

|2 = −1 Q+2 R

(−)4

Q−2R(+)4

|3 = 1

∫Kmn (p, p′)ρn(p

′)dp

′= 2π(ρm + ρmh )


∑i





J





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Z(L,R) =R→∞ e−E0(L)R(1 + e−∆ER)L

e−H(L)R

R


L

Re−H(R) L


i, ρih: i

no periodicity

En(R) =∑

i,Q EQ(pi)→∑

Q

∫EQ(p)ρQ(p)dp

eiLpS20Q++

4

Q−−4

T T |4 = −1 Q+2 B

(−)4

Q−2B(+)4

|1 = 1 Q−−2 Q+1 Q

+3

Q++2 Q−1Q

−3

|2 = −1 Q+2 R

(−)4

Q−2R(+)4

|3 = 1∫Kmn (p, p′)ρn(p

′)dp

′= 2π(ρm + ρmh )


∑i


Saddle point : εi(p) = − ln ρi(p)ρih(p) εj(θ) = δ

jQEQ(p)L−

∫Kji (p, p′) log(1 + e−ε

i(p′))dp′

Ground state energy exactly: E0(L) = −∑Q∫ dp

2π log(1 + e−εQ(p))dp

Excited states TBA, Y-system: diagonal

Excited states exactlyp

2p

n

p1

26



2p

n

p1

L

E

0

2m

m

S−

mat

rix




2p

n

p1

L

E

0

2m

m

S−

mat

rix


By lattice regularization: sinh-Gordon

i π/2

ε(θ) = mL cosh θ +∫dθ′

2πidθ logS(θ−θ

′) log(1+e−ε(θ

′))

E{nj}(L) = −m∫dθ

2πcosh θ log(1+e−ε(θ))



2p

n

p1

L

E

0

2m

m

S−

mat

rix



i π/2

ε(θ) = mL cosh θ+∑

logS(θ − θj)+∫dθ′



′))

E{nj}(L) = −m∫dθ




2p

n

p1

L

E

0

2m

m

S−

mat

rix



i π/2





′))

E{nj}(L) =m

i

∑sinh θj−m

∫dθ




2p

n

p1

L

E

0

2m

m

S−

mat

rix



i π/2





′))

E{nj}(L) =m

i

∑sinh θj−m

∫dθ

2πcosh θ log(1+e−ε(θ)) ; Y (θj) = eε(θj) = −1



2p

n

p1

L

E

0

2m

m

S−

mat

rix



i π/2





′))

E{nj}(L) =m

i

∑sinh θj−m

∫dθ


By analytical continuation: Lee-Yang

i π/6

π/6−i

ε(θ) = mL cosh θ −∫ ∞−∞

dθ′

2πφ(θ − θ

′) log(1 + e−ε(θ

′))

E{nj}(L) = −m∫ ∞−∞

dθ

2πcosh θ log(1 + e−ε(θ))

Luscher corrections: differ by µ term



2p

n

p1

L

E

0

2m

m

S−

mat

rix



i π/2





′))

E{nj}(L) =m

i

∑sinh θj−m

∫dθ



i π/6

π/6−i

ε(θ) = mL cosh θ+N∑j=1

logS(θ − θj)S(θ − θ∗j)

−∫ ∞−∞

dθ′

2πφ(θ − θ

′) log(1 + e−ε(θ

′))

E{nj}(L) = −m∫ ∞−∞

dθ





2p

n

p1

L

E

0

2m

m

S−

mat

rix



i π/2





′))

E{nj}(L) =m

i

∑sinh θj−m

∫dθ



i π/6

π/6−i

ε(θ) = mL cosh θ+N∑j=1

logS(θ − θj)S(θ − θ∗j)

−∫ ∞−∞

dθ′

2πφ(θ − θ

′) log(1 + e−ε(θ

′))

E{nj}(L) = −im∑

(sinh θj − sinh θ∗j)−m∫ ∞−∞

dθ



S(θ − iπ3 )S(θ + iπ

3 ) = S(θ)→ Y (θ + iπ3 )Y (θ − iπ

3 ) = 1 + Y (θ)

Y-system+analyticity=TBA <-> scalar . Matrix

Excited states TBA, Y-system: Non-diagonal

Excited states exactlyn

1 2 j... ...p p p

p

27



1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix




1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix




1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix


Y-system: sine-Gordon p2πL

ιπ2

ιπp

periodicity ιπ /p

Ys(θ + iπp2 )Ys(θ − iπp

2 ) = (1 + Ys−1)(1 + Ys+1)

(eiπp2 ∂ + e−

iπp2 ∂) logYs =

∑r Isr log(1 + Yr)

...

p

p+1



1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix



ιπ2

ιπp

periodicity ιπ /p


2 ) = (1 + Ys−1)(1 + Ys+1)

(eiπp2 ∂ + e−

iπp2 ∂) logYs =


...

p

p+1

Excited states: analyticity from Luscher

Lattice regularization:Λ

ΛΛ

Λ

ΛΛ

ΛΛ

−Λ−Λ

−Λ−Λ

−Λ−Λ

−Λ−Λ

sG



1 2 j... ...p p p

p

L

E

0

2m

m

S−

mat

rix



ιπ2

ιπp

periodicity ιπ /p


2 ) = (1 + Ys−1)(1 + Ys+1)

(eiπp2 ∂ + e−

iπp2 ∂) logYs =


...

p

p+1


Lattice regularization:Λ

ΛΛ

Λ

ΛΛ

ΛΛ

−Λ−Λ

−Λ−Λ

−Λ−Λ

−Λ−Λ

sG

Z(θ) = ML sinh θ+source(θ|{θk})+2=m∫dxG(θ−x−iε) log

[1− (−1)δeiZ(x+iε)

]source(θ|{θk}) =

∑k sgnk(−i) logS++

++(θ−θk) kernel: G(θ) = −i∂θ logS++++(θ)

Energy: E = M∑k sgnk cosh θk−2M=m

∫dxG(θ+iε) log

[1− (−1)δeiZ(x+iε)

]Bethe-Yang eiZ(θk) = −1

Excited states TBA, Y-system: AdS g



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1 2 j... ...p p p

p

28




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rix


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r co

rrec

tion


1 2 j... ...p p p

p

Y-system: AdS

i

no periodicity

Ya,s(θ+ i2)Ya,s(θ− i

2)Ya+1,sYa−1,s

=(1+Ya,s−1)(1+Ya,s+1)(1+Ya+1,s)(1+Ya−1,s)




J





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rix


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sche

r co

rrec

tion


1 2 j... ...p p p

p

Y-system: AdS

i

no periodicity

Ya,s(θ+ i2)Ya,s(θ− i

2)Ya+1,sYa−1,s

=(1+Ya,s−1)(1+Ya,s+1)(1+Ya+1,s)(1+Ya−1,s)


Lattice regularization: ?

Conclusion

29

Conclusion

g



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r co

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Conclusion

g



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rix


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nsat

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Wra

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sche

r co

rrec

tion

Conclusion

g



J





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rix


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rrec

tion


physical sheet, explanation of all the poles

Conclusion

g



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rix


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tion



Excited states = analiticity . Y-system

Analytical structure of all excited states

Conclusion

g



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rix


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Bet

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sche

r co

rrec

tion



Excited states = analiticity . Y-system

Analytical structure of all excited states

lattice?

Documents

research.ipmu.jpInstitute for the Physics and Mathematics of the Universe, 20 October, 2009, Tokyo Finite size e ects in integrable models: planar AdS/CFT Zolt an Bajnok, Theoretical