research.ipmu.jpinstitute for the physics and mathematics of the universe, 20 october, 2009, tokyo...

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





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


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


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


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


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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





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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





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rix


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sche

r co

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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





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sche

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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


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sche

r co

rrec

tion

Nondiagonal scattering: S-matrix = scalar . Matrix

18




J





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tion


Matrix:

global symmetry PSU(2|2)2

Q = 1 reps

b1

b2

f3

f4

⊗ b1

b2f3f4


(Q)∆

∆ (Q)




J





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rix


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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





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rix


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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

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nsat

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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



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



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 −

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rix


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r co

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tion


2p

n

p1

20


g



J





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rix


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sche

r co

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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


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Bet

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nsat

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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


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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





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rix


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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

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Wra

ppin

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



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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Bet

he A

nsat

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Wra

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g/Lu

sche

r co

rrec

tion


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

p

22




J





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rix


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he A

nsat

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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


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Bet

he A

nsat

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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)


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

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Bet

he A

nsat

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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)



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





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mat

rix


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he A

nsat

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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




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→∞ 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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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





S −

mat

rix


Asy

mpt

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nsat

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Wra

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g/Lu

sche

r co

rrec

tion



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





S −

mat

rix


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Bet

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sche

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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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mpt

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Bet

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nsat

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Wra

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sche

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tion



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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mat

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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


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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r co

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tion


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

p

28




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rix


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he A

nsat

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ppin

g/Lu

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)




J





S −

mat

rix


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mpt

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Bet

he A

nsat

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Wra

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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



J





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mat

rix


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Bet

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g/Lu

sche

r co

rrec

tion

Conclusion

g



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mat

rix


Asy

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Bet

he A

nsat

z

Wra

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g/Lu

sche

r co

rrec

tion


physical sheet, explanation of all the poles

Conclusion

g



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mat

rix


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Bet

he A

nsat

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g/Lu

sche

r co

rrec

tion



Excited states = analiticity . Y-system

Analytical structure of all excited states

Conclusion

g



J





S −

mat

rix


Asy

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Bet

he A

nsat

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Wra

ppin

g/Lu

sche

r co

rrec

tion



Excited states = analiticity . Y-system

Analytical structure of all excited states

lattice?

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