AdS/CFT correspondence from an integrable point of view Z. Bajnok

Transcription

1 Matrix Worksho: Integrability in ow-dimensional Quantum systems Creswick 17 June-July AdS/CFT corresondence from an integrable oint of view Z. Bajnok MTA Wigner Research Center for Physics, Holograhic QFT Grou, Budaest 1

2 Matrix Worksho: Integrability in ow-dimensional Quantum systems Creswick 17 June-July AdS/CFT corresondence from an integrable oint of view Z. Bajnok MTA Wigner Research Center for Physics, Holograhic QFT Grou, Budaest Prologue: QFT as the continuum limit of XXZ Sine-Gordon / massive Thirring duality Anti-de Sitter / Conformal Field Theory duality

3 Prologue: QFT as a continuum limit Consider the inhomogenous XXZ sin chain ξ = { ξ, ξ +,..., ξ + }, R(λ) = T(λ ξ) = R 1 (λ ξ 1 )R (λ ξ )......R N (λ ξ N ) = 1 sinh(λ) sinh( iγ) sinh(λ iγ) sinh(λ iγ) sinh( iγ) sinh(λ) sinh(λ iγ) sinh(λ iγ) 1 ( ) A(λ) B(λ) C(λ) D(λ)

4 Prologue: QFT as a continuum limit Consider the inhomogenous XXZ sin chain ξ = { ξ, ξ +,..., ξ + }, R(λ) = T(λ ξ) = R 1 (λ ξ 1 )R (λ ξ )......R N (λ ξ N ) = 1 sinh(λ) sinh( iγ) sinh(λ iγ) sinh(λ iγ) sinh( iγ) sinh(λ) sinh(λ iγ) sinh(λ iγ) 1 ( ) A(λ) B(λ) C(λ) D(λ) Integrability: T (λ ξ) = Tr T(λ ξ) commute [ T (λ ξ),t (λ ξ) ] = conserved charges U ± = T (ξ ± ξ) = e i a (H±P) H P

5 Prologue: QFT as a continuum limit Consider the inhomogenous XXZ sin chain ξ = { ξ, ξ +,..., ξ + }, R(λ) = T(λ ξ) = R 1 (λ ξ 1 )R (λ ξ )......R N (λ ξ N ) = 1 sinh(λ) sinh( iγ) sinh(λ iγ) sinh(λ iγ) sinh( iγ) sinh(λ) sinh(λ iγ) sinh(λ iγ) 1 ( ) A(λ) B(λ) C(λ) D(λ) Integrability: T (λ ξ) = Tr T(λ ξ) commute [ T (λ ξ),t (λ ξ) ] = conserved charges U ± = T (ξ ± ξ) = e i a (H±P) H P Eigenvectors: B(λ 1 )B(λ )...B(λ m ), Bethe Ansatz: N i=1 sinh(λ a ξ i iγ) sinh(λ a ξ i ) m b=1 Alternating inhomogeneities sinh(λ a λ b +iγ) sinh(λ a λ b iγ) = 1 ξ ± = ± γ π Λ iγ Grd st. N roots λ Exc. st. +holes: λ

6 Prologue: QFT as a continuum limit Counting function ( 1) δ e iz λ(λ) = N i=1 sinh(λ ξ i iγ) sinh(λ ξ i ) m b=1 sinh(λ λ b +iγ) sinh(λ λ b iγ) Bethe Ansatz: e iz λ(λ ) a = 1 take δ = and redefine Z N (θ) = Z λ ( γ θ), which satisfies π Z N (θ) = N { arctan[sinh(θ Λ)]+ arctan[sinh(θ + Λ)] } + m H k=1 χ(θ θ k )+Im [Klümer, Pearce, Destri, de Vega,...] G(θ) = i θ logs(θ) = dθ πi G(θ θ iη)ln ( 1+e ) iz N(θ +iη) dωe iωθ sinh( ( 1)πω ) cosh( πω ) sinh(πω ) 3

7 Prologue: QFT as a continuum limit Counting function ( 1) δ e iz λ(λ) = N i=1 sinh(λ ξ i iγ) sinh(λ ξ i ) m b=1 sinh(λ λ b +iγ) sinh(λ λ b iγ) Bethe Ansatz: e iz λ(λ ) a = 1 take δ = and redefine Z N (θ) = Z λ ( γ θ), which satisfies π Z N (θ) = N { arctan[sinh(θ Λ)]+ arctan[sinh(θ + Λ)] } + m H k=1 χ(θ θ k )+Im [Klümer, Pearce, Destri, de Vega,...] G(θ) = i θ logs(θ) = dθ πi G(θ θ iη)ln ( 1+e ) iz N(θ +iη) dωe iωθ sinh( ( 1)πω ) cosh( πω ) sinh(πω ) QFT = Scaled continuum limit N : Λ = ln 4 Ma = ln N M Z(θ) = Msinhθ i m H k=1 logs(θ θ k )+ dθ πi G(θ θ iη) ln ( 1+e iz(θ +iη) ) E ±P = M m H k=1 e±θ k MRe dθ πi e±θ+iη ln ( 1+e iz(θ+iη)) λ

8 Prologue: QFT as a continuum limit Counting function ( 1) δ e iz λ(λ) = N i=1 sinh(λ ξ i iγ) sinh(λ ξ i ) m b=1 sinh(λ λ b +iγ) sinh(λ λ b iγ) Bethe Ansatz: e iz λ(λ a ) = 1 take δ = and redefine Z N (θ) = Z λ ( γ π Z N (θ) = N { arctan[sinh(θ Λ)]+ arctan[sinh(θ + Λ)] } + m H k=1 χ(θ θ k )+Im [Klümer, Pearce, Destri, de Vega,...] G(θ) = i θ logs(θ) = QFT = Scaled continuum limit N : Λ = ln 4 Ma Z(θ) = Msinhθ i m H k=1 logs(θ θ k )+ E ±P = M m H k=1 e±θ k MRe θ), which satisfies dθ πi G(θ θ iη)ln ( 1+e ) iz N(θ +iη) = ln N M dωe iωθ sinh( ( 1)πω ) cosh( πω ) sinh(πω ) dθ πi G(θ θ iη) ln ( 1+e iz(θ +iη) ) dθ πi e±θ+iη ln ( 1+e iz(θ+iη)) λ large volume equation: e iz(θ k) = e imsinhθ k m H j=1 S(θ k θ j ) = 1 relativistic energy sectrum: E = m H j=1 Mcoshθ k

9 Sine-Gordon/massive Thirring duality SG = 1 νφ ν Φ+ m β : cos(βφ) : < β < 8π, strong-weak duality: 1+ g 4π = 4π β = +1 MT = Ψ(iγ ν ν +m )Ψ g Ψγ ν Ψ Ψγ ν Ψ 4

10 Sine-Gordon/massive Thirring duality SG = 1 νφ ν Φ+ m β : cos(βφ) : < β < 8π, g Classical sine Gordon theory Semiclassical sine Gordon theory + loo corrections Quantum sine Gordon theory Massive Thirring model + erturbative corrections strong-weak duality: 1+ g 4π = 4π β = +1 erturbative Thirring model MT = Ψ(iγ ν ν +m )Ψ g Ψγ ν Ψ Ψγ ν Ψ

11 Sine-Gordon/massive Thirring duality SG = 1 νφ ν Φ+ m β : cos(βφ) : < β < 8π, g Classical sine Gordon theory Semiclassical sine Gordon theory + loo corrections Quantum sine Gordon theory Finite size corrections Massive Thirring model uscher correction Bethe Yang equations S matrix, masses strong-weak duality: + erturbative corrections 1+ g 4π = 4π β = +1 erturbative Thirring model MT = Ψ(iγ ν ν +m )Ψ g Ψγ ν Ψ Ψγ ν Ψ

12 Classical integrable models: sin(e/h)-gordon theory V V οο sine-gordon π β φ target π β = 1 β ( βϕ) m β (1 cosβϕ) β ib sinh-gordon φ target οο = 1 ( ϕ) m b (coshbϕ 1) 5

13 Classical integrable models: sin(e/h)-gordon theory V V οο sine-gordon π β φ target π β = 1 β ( βϕ) m β (1 cosβϕ) β ib sinh-gordon φ target οο = 1 ( ϕ) m b (coshbϕ 1) φ Classical finite energy solutions: x x sine-gordon theory soliton anti-soliton breather φ φ x

14 Classical integrable models: sin(e/h)-gordon theory V V οο sine-gordon π β φ target π β = 1 β ( βϕ) m β (1 cosβϕ) β ib sinh-gordon φ target οο = 1 ( ϕ) m b (coshbϕ 1) φ Classical finite energy solutions: x x sine-gordon theory soliton anti-soliton breather φ φ x Integrability: x A t t A x +[A x,a t ] = T(λ) = P ex A(x) ν dx ν A x (λ) = i ( λ β + ϕ β + ϕ λ ) ( A t (λ) = 1 cosβϕ isinβϕ 4iλ isinβϕ cosβϕ ) gtg 1 T

15 Classical integrable models: sin(e/h)-gordon theory V V οο sine-gordon π β φ target π β = 1 β ( βϕ) m β (1 cosβϕ) β ib sinh-gordon φ target οο = 1 ( ϕ) m b (coshbϕ 1) φ Classical finite energy solutions: x x sine-gordon theory soliton anti-soliton breather φ φ x Integrability: x A t t A x +[A x,a t ] = T(λ) = P ex A(x) ν dx ν A x (λ) = i ( λ β + ϕ β + ϕ λ ) ( A t (λ) = 1 cosβϕ isinβϕ 4iλ isinβϕ cosβϕ conserved Q ±1 [ϕ] = E[ϕ]±P[ϕ] = { 1 ( ± ϕ) + m β (1 cosβϕ) } charges: Q ±3 [ϕ] = { 1 β ( ± ϕ) 1 8 ( ±ϕ) 4 + m β ( ±ϕ) (1 cosβϕ) ) dx } dx gtg 1 T

16 Classical integrable models: sin(e/h)-gordon theory V V οο sine-gordon π β φ target π β = 1 β ( βϕ) m β (1 cosβϕ) β ib sinh-gordon φ target οο = 1 ( ϕ) m b (coshbϕ 1) φ Classical finite energy solutions: x x sine-gordon theory soliton anti-soliton breather φ φ x Integrability: x A t t A x +[A x,a t ] = T(λ) = P ex A(x) ν dx ν A x (λ) = i ( λ β + ϕ β + ϕ λ ) A t (λ) = 1 4iλ ( cosβϕ isinβϕ isinβϕ cosβϕ ) gtg 1 T Classical factorized scattering: T 1 (v 1,v,...,v n ) = i T 1 (v 1,v i ) Τ v 1 > v >... > v n v n <... < v < v 1 Τ.

17 Quantum integrability: sine-gordon = 1 ( φ) m β (1 cosβφ) Perturbed Conformal Field Theory agrangian erturbation theory CFT +µ ert = 1 ( φ) +µ(v β +V β ) +V ert = 1 ( φ) m φ β U h β = β 4π definite scaling V β =: e iβφ : free massive boson+ert. 6

18 Quantum integrability: sine-gordon = 1 ( φ) m β (1 cosβφ) Perturbed Conformal Field Theory agrangian erturbation theory CFT +µ ert = 1 ( φ) +µ(v β +V β ) +V ert = 1 ( φ) m φ β U h β = β 4π definite scaling V β =: e iβφ : free massive boson+ert. Quantum conservation laws Λ 4 = Λ 4 = λ + Θ [λ] = h β, [Λ 4 ] = 4, counting argument Nonlocal symmetry: U q ( sl ˆ ) [h,j ± ] = ±J ± [J +,J ] = qh q h q q 1 (J ± ) = q h J ± +J ± q h [S, (J ± )] = arameter relation: q = e iπ Correlators= loosfeynmandiagrams Asymtotic states E() = +m S-matrix correlators SZ 1, O 1, = D 1 D D 1D T(Oϕ(1)ϕ()ϕ(3)ϕ(4)) D j = i d x j e i jx ie j t { t x +m} unitarity, crossing symmetry, analyticity raidity i = mcoshθ i S(θ) = S( θ) 1 = S(iπ θ) = = ib cschθ b4 (cschθ(πcschθ i)) 3π +... Bootstra scheme

19 Bootstra rogram Asymtotic states 1,,..., n in/out form a reresentation of global symmetry: 1 > >...> n 7

20 Bootstra rogram Asymtotic states 1,,..., n in/out form a reresentation of global symmetry: 1 > >...> n orentz: P = i i E = ie( i ) disersion relation E() = m +

21 Bootstra rogram Asymtotic states 1,,..., n in/out form a reresentation of global symmetry: 1 > >...> n orentz: P = i i E = ie( i ) disersion relation E() = (Q) m + 1 n

22 Bootstra rogram Asymtotic states 1,,..., n in/out form a reresentation of global symmetry: 1 > >...> n orentz: P = i i E = ie( i ) disersion relation E() = m + Scattering matrix S: out in commutes with symmetry [S, (Q)] =

23 Bootstra rogram Asymtotic states 1,,..., n in/out form a reresentation of global symmetry: 1 > >...> n orentz: P = i i E = ie( i ) disersion relation E() = m + Scattering matrix S: out in commutes with symmetry [S, (Q)] = m < <...< S matrix 1 1 > >...> n

24 Bootstra rogram Asymtotic states 1,,..., n in/out form a reresentation of global symmetry: 1 > >...> n orentz: P = i i E = ie( i ) disersion relation E() = m + (Q) m < <...< 1 Scattering matrix S: out in commutes with symmetry [S, (Q)] = S matrix (Q) 1 > >...> n

25 Bootstra rogram Asymtotic states 1,,..., n in/out form a reresentation of global symmetry: 1 > >...> n orentz: P = i i E = ie( i ) disersion relation E() = m + (Q) m < <...< 1 Scattering matrix S: out in commutes with symmetry [S, (Q)] = S matrix (Q) > >...> n 3 1 Higher sin concerved charge factorization + Yang-Baxter equation S 13 = S 3 S 13 S 1 = S 1 S 13 S S-matrix = scalar. Matrix

26 Bootstra rogram Asymtotic states 1,,..., n in/out form a reresentation of global symmetry: > >...> 1 n orentz: P = i i E = ie( i ) disersion relation E() = m + (Q) m < <...< 1 Scattering matrix S: out in commutes with symmetry [S, (Q)] = S matrix 3 1 (Q) 3 1 Higher sin concerved charge factorization + Yang-Baxter equation 1 > >...> n S 13 = S 3 S 13 S 1 = S 1 S 13 S 3 S-matrix = scalar. Matrix Unitarity S 1 S 1 = Id Crossing symmetry S 1 = S 1 Maximal analyticity: all oles have hysical origin boundstates, anomalous thresholds

27 Bootstra rogram: diagonal Diagonal: S-matrix = scalar S( 1, ) = S(θ 1 θ ) = msinhθ 8

28 Bootstra rogram: diagonal Diagonal: S-matrix = scalar S( 1, ) = S(θ 1 θ ) = msinhθ Unitarity S(θ)S( θ) = 1 Crossing symmetry S(θ) = S(iπ θ) Maximal analyticity: all oles have hysical origin

29 Bootstra rogram: diagonal Diagonal: S-matrix = scalar S( 1, ) = S(θ 1 θ ) = msinhθ Unitarity S(θ)S( θ) = 1 Crossing symmetry S(θ) = S(iπ θ) Maximal analyticity: all oles have hysical origin Minimal solution: S(θ) = 1 Free boson

30 Bootstra rogram: diagonal Diagonal: S-matrix = scalar S( 1, ) = S(θ 1 θ ) = msinhθ Unitarity S(θ)S( θ) = 1 Crossing symmetry S(θ) = S(iπ θ) Maximal analyticity: all oles have hysical origin Minimal solution: S(θ) = 1 Free boson CDD factor: S(θ) = sinhθ isinπ sinhθ+isinπ

31 Bootstra rogram: diagonal Diagonal: S-matrix = scalar S( 1, ) = S(θ 1 θ ) = msinhθ Unitarity S(θ)S( θ) = 1 Crossing symmetry S(θ) = S(iπ θ) Maximal analyticity: all oles have hysical origin Minimal solution: S(θ) = 1 Free boson V coshbφ 1 = 1+ CDD factor: S(θ) = sinhθ isinπ 8π b > sinhθ+isinπ Sinh-Gordon

32 Bootstra rogram: diagonal Diagonal: S-matrix = scalar S( 1, ) = S(θ 1 θ ) = msinhθ Unitarity S(θ)S( θ) = 1 Crossing symmetry S(θ) = S(iπ θ) Maximal analyticity: all oles have hysical origin Minimal solution: S(θ) = 1 Free boson V coshbφ 1 = 1+ CDD factor: S(θ) = sinhθ isinπ 8π b > sinhθ+isinπ Sinh-Gordon Maximal analyticity: S(θ) = sinhθ+isinπ sinhθ isinπ ole at θ = iπ boundstate B bootstra: S 1 (θ)=s 11 (θ iπ )S 11(θ + iπ ) new article if 3 otherwise ee-yang

33 Bootstra rogram: diagonal Diagonal: S-matrix = scalar S( 1, ) = S(θ 1 θ ) = msinhθ Unitarity S(θ)S( θ) = 1 Crossing symmetry S(θ) = S(iπ θ) Maximal analyticity: all oles have hysical origin Minimal solution: S(θ) = 1 Free boson CDD factor: S(θ) = sinhθ isinπ sinhθ+isinπ Sinh-Gordon Maximal analyticity: S(θ) = sinhθ+isinπ sinhθ isinπ ole at θ = iπ boundstate B bootstra: S 1 (θ)=s 11 (θ iπ )S 11(θ + iπ ) new article if 3 otherwise ee-yang Maximal analyticity: all oles have hysical origin sine-gordon solitons ιπ S 11 θ V coshbφ 1 = 1+ b 8π > ιπ S 1 θ ιπ S 1n (n+1) iπ/ (n 1)iπ/ θ iπ B 3iπ/ iπ/ B 3... B n

34 Bootstra rogram: sine-gordon Nondiagonal scattering: S-matrix = scalar. Matrix soliton doublet ( s s ) Matrix: global symmetry U q ( ˆ sl ) d evaluation res [S, (Q)] = (Q) S matrix 1 sinαπ sin iθα sin α(π+iθ) sin iαθ sin α(π+iθ) sin α(π+iθ) sinαπ sin α(π+iθ) 1 (Q) 9

35 Bootstra rogram: sine-gordon Nondiagonal scattering: S-matrix = scalar. Matrix soliton doublet ( s s ) Matrix: global symmetry U q ( ˆ sl ) d evaluation res [S, (Q)] = (Q) S matrix 1 sinαπ sin iθα sin α(π+iθ) sin iαθ sin α(π+iθ) sin α(π+iθ) sinαπ sin α(π+iθ) 1 (Q) Unitarity S(θ)S( θ) = 1 Crossing symmetry S(θ) = S c 1(iπ θ) [ l=1 Γ((l 1)α+ αiθ π )Γ(lα+1+αiθ π ) Γ((l 1)α+ αiθ π )Γ((l 1)α+1+αiθ π )/(θ θ) ]

36 Bootstra rogram: sine-gordon Nondiagonal scattering: S-matrix = scalar. Matrix soliton doublet Matrix: global symmetry U q ( sl ˆ ) d evaluation res [S, (Q)] = (Q) S matrix 1 sinαπ sin iθα sin α(π+iθ) sin iαθ sin α(π+iθ) sin α(π+iθ) sinαπ sin α(π+iθ) 1 ( s s ) Unitarity S(θ)S( θ) = 1 Crossing symmetry S(θ) = S c 1(iπ θ) (Q) [ l=1 Γ((l 1)α+ αiθ π )Γ(lα+1+αiθ π ) Γ((l 1)α+ αiθ π )Γ((l 1)α+1+αiθ π )/(θ θ) ] Maximal analyticity: all oles have hysical origin either boundstates or anomalous thresholds = α 1 [Zamolodchikov ] B n + B B B B

37 Bootstra rogram: sine-gordon Nondiagonal scattering: S-matrix = scalar. Matrix soliton doublet Matrix: global symmetry U q ( sl ˆ ) d evaluation res [S, (Q)] = Unitarity S(θ)S( θ) = 1 Crossing symmetry S(θ) = S c 1(iπ θ) (Q) S matrix 1 sinαπ sin iθα sin α(π+iθ) sin iαθ sin α(π+iθ) sin α(π+iθ) sinαπ sin α(π+iθ) 1 (Q) [ l=1 Γ((l 1)α+ αiθ π )Γ(lα+1+αiθ π ) Γ((l 1)α+ αiθ π )Γ((l 1)α+1+αiθ π ιπ )/(θ θ) θ ( s ] s ) Maximal analyticity: all oles have hysical origin either boundstates or anomalous thresholds = α 1 [Zamolodchikov ] B n + B B B B + S +... (1 n)ιπ...

38 QFTs in finite volume Finite volume sectrum 1 n 1

39 QFTs in finite volume Finite volume sectrum 1 n CFT uscher Bethe Yang E m m S matrix

40 QFTs in finite volume CFT uscher Bethe Yang E Finite volume sectrum 1 n m m S matrix

41 QFTs in finite volume CFT uscher Bethe Yang E Finite volume sectrum Infinite volume sectrum: E( 1,..., n ) = ie( i ) 1 i R n m m S matrix

42 QFTs in finite volume CFT uscher Bethe Yang E Finite volume sectrum Infinite volume sectrum: E( 1,..., n ) = ie( i ) 1 i R Polynomial volume corrections: Bethe-Yang; i quantized. Diagonal n m m S matrix

43 QFTs in finite volume CFT uscher Bethe Yang E Finite volume sectrum Infinite volume sectrum: E( 1,..., n ) = ie( i ) 1 i R Polynomial volume corrections: Bethe-Yang; i quantized. Diagonal n m m S matrix e i j S( j, 1 )...S( j, n ) = 1 ; S() = 1 j + k 1 i logs( j, k ) = (n+1)iπ π

44 QFTs in finite volume CFT uscher Bethe Yang E Finite volume sectrum Infinite volume sectrum: E( 1,..., n ) = ie( i ) 1 i R Polynomial volume corrections: Bethe-Yang; i quantized. Diagonal n m m S matrix e i j S(j, 1 )...S( j, n ) = 1 ; S() = 1 j + k 1 i logs( j, k ) = (n+1)iπ π Non-diagonal, sine-gordon e i j S(j, 1 )...S( j, n )Ψ = Ψ S() = P... 1 j... n

45 QFTs in finite volume CFT uscher Bethe Yang E Finite volume sectrum Infinite volume sectrum: E( 1,..., n ) = ie( i ) 1 i R Polynomial volume corrections: Bethe-Yang; i quantized. Diagonal n m m S matrix e i j S(j, 1 )...S( j, n ) = 1 ; S() = 1 j + k 1 i logs( j, k ) = (n+1)iπ π Non-diagonal, sine-gordon e i j S(j, 1 )...S( j, n )Ψ = Ψ S() = P... 1 j... n Inhomogenous XXZ sin-chain sectral roblem e isinhθ jt(θ j )S (θ j ) = 1

46 QFTs in finite volume E CFT uscher Bethe Yang Finite volume sectrum Infinite volume sectrum: E( 1,..., n ) = ie( i ) 1 i R Polynomial volume corrections: Bethe-Yang; i quantized. Diagonal n m m S matrix e i j S(j, 1 )...S( j, n ) = 1 ; S() = 1 j + k 1 i logs( j, k ) = (n+1)iπ π Non-diagonal, sine-gordon e i j S(j, 1 )...S( j, n )Ψ = Ψ S() = P... 1 j... n Inhomogenous XXZ sin-chain sectral roblem e isinhθ jt(θ j )S (θ j ) = 1 T(θ)Q(θ) = Q(θ +iπ)t (θ iπ )+Q(θ iπ)t (θ + iπ ) = Q++ T +Q T +

47 QFTs in finite volume E CFT uscher Bethe Yang Finite volume sectrum Infinite volume sectrum: E( 1,..., n ) = ie( i ) 1 i R n m m S matrix Polynomial volume corrections: Bethe-Yang; i quantized. Diagonal e i j S(j, 1 )...S( j, n ) = 1 ; S() = 1 j + k 1 i logs( j, k ) = (n+1)iπ π Non-diagonal, sine-gordon e i j S( j, 1 )...S( j, n )Ψ = Ψ S() = P... 1 j... n Inhomogenous XXZ sin-chain sectral roblem e isinhθ jt(θ j )S (θ j ) = 1 T(θ)Q(θ) = Q(θ +iπ)t (θ iπ )+Q(θ iπ)t (θ + iπ ) = Q++ T +Q T + Q(θ) = β sinh(λ(θ w β)) T (θ) = j sinh(λ(θ θ j)) Bethe Ansatz: T (w α iπ )Q(w α+iπ) T (w α + iπ )Q(w α iπ) = T Q++ T + Q α = 1

48 Thermodynamic Bethe Ansatz: diagonal Ground-state energy exactly [Zamolodchikov] 11

49 Thermodynamic Bethe Ansatz: diagonal Ground-state energy exactly [Zamolodchikov] CFT uscher Bethe Yang E m m S matrix

50 Thermodynamic Bethe Ansatz: diagonal CFT uscher Bethe Yang Ground-state energy exactly [Zamolodchikov] E m m S matrix

51 Thermodynamic Bethe Ansatz: diagonal CFT uscher Bethe Yang Ground-state energy exactly E m [Zamolodchikov] R m S matrix Euclidian artition function: Z(,R) = R Tr(e H()R ) Z(,R) = R e E ()R (1+e ER ) e H()R

52 Thermodynamic Bethe Ansatz: diagonal CFT uscher Bethe Yang Ground-state energy exactly E m [Zamolodchikov] R m S matrix Euclidian artition function: Z(,R) = R Tr(e H()R ) Z(,R) = R e E ()R (1+e ER ) e H()R Exchange sace and Euclidian time Z(,R) = R Tr(e H(R) ) = R ne E n()r e H(R) R

53 Thermodynamic Bethe Ansatz: diagonal CFT uscher Bethe Yang Ground-state energy exactly E m [Zamolodchikov] R m S matrix Euclidian artition function: Z(,R) = R Tr(e H()R ) Z(,R) = R e E ()R (1+e ER ) e H()R Exchange sace and Euclidian time Z(,R) = R Tr(e H(R) ) = R ne E n()r e H(R) R Dominant contribution: finite article/hole density ρ,ρ h : π

54 Thermodynamic Bethe Ansatz: diagonal CFT uscher Bethe Yang Ground-state energy exactly E m [Zamolodchikov] R m S matrix Euclidian artition function: Z(,R) = R Tr(e H()R ) Z(,R) = R e E ()R (1+e ER ) e H()R Exchange sace and Euclidian time Z(,R) = R Tr(e H(R) ) = R ne E n()r e H(R) R Dominant contribution: finite article/hole density ρ,ρ h : π E n (R) = i E( i) E()ρ()d j R+ k 1 i logs( j, k ) = (n+1)iπ R+ ( id logs(, ))ρ( )d = π(ρ+ρ h )

55 Thermodynamic Bethe Ansatz: diagonal CFT uscher Bethe Yang Ground-state energy exactly E m [Zamolodchikov] R m S matrix Euclidian artition function: Z(,R) = R Tr(e H()R ) Z(,R) = R e E ()R (1+e ER ) e H()R Exchange sace and Euclidian time Z(,R) = R Tr(e H(R) ) = R ne E n()r e H(R) R Dominant contribution: finite article/hole density ρ,ρ h : π E n (R) = i E( i) E()ρ()d j R+ k 1 i logs( j, k ) = (n+1)iπ R+ ( id logs(, ))ρ( )d = π(ρ+ρ h ) Z(,R) = d[ρ,ρ h ]e E(R) ((ρ+ρ h )ln(ρ+ρ h ) ρlnρ ρ h lnρ h )d

56 Thermodynamic Bethe Ansatz: diagonal CFT uscher Bethe Yang E Ground-state energy exactly [Zamolodchikov] R m m S matrix Euclidian artition function: Z(,R) = R Tr(e H()R ) Z(,R) = R e E ()R (1+e ER ) Exchange sace and Euclidian time Z(,R) = R Tr(e H(R) ) = R e H()R ne E n()r e H(R) R Dominant contribution: finite article/hole density ρ,ρ h : π E n (R) = i E( i) E()ρ()d j R+ k 1 i logs( j, k ) = (n+1)iπ R+ ( id logs(, ))ρ( )d = π(ρ+ρ h ) Z(,R) = d[ρ,ρ h ]e E(R) ((ρ+ρ h )ln(ρ+ρ h ) ρlnρ ρ h lnρ h )d Saddle oint : ǫ() = ln ρ h() ρ() ǫ() = E()+ d π id logs(,)log(1+e ǫ( ) ) Ground state energy exactly: E () = d π log(1+e ǫ() ) ee-yang, sh-g

57 Thermodynamic Bethe Ansatz: non-diagonal CFT uscher Bethe Yang Ground-state energy exactly E m [Tateo] R m S matrix Euclidian artition function: Z(,R) = R e E ()R (1+e ER ) Z(,R) = R Tr(e H(R) ) = R ne E n()r e H()R e H(R) R 1

58 Thermodynamic Bethe Ansatz: non-diagonal CFT uscher Bethe Yang Ground-state energy exactly E m [Tateo] R m S matrix Euclidian artition function: Z(,R) = R e E ()R (1+e ER ) Z(,R) = R Tr(e H(R) ) = R ne E n()r e H()R e H(R) R Finite article/hole + Bethe root density ρ,ρ h,ρi,ρ i h : e i T S j = 1, T Q++ T + Q α = 1 ιπ ιπ eriodicity ιπ / π

59 Thermodynamic Bethe Ansatz: non-diagonal CFT uscher Bethe Yang Ground-state energy exactly E m [Tateo] R m S matrix Euclidian artition function: Z(,R) = R e E ()R (1+e ER ) Z(,R) = R Tr(e H(R) ) = R ne E n()r e H()R e H(R) R Finite article/hole + Bethe root density ρ,ρ h,ρi,ρ i h : e i T S j = 1, T Q++ T + Q α = 1 ιπ ιπ eriodicity ιπ / E n (R) = i E( i) E()ρ ()d Rδ m + K m n (, )ρ n ( )d = π(ρ m +ρ m h ) π

60 Thermodynamic Bethe Ansatz: non-diagonal CFT uscher Bethe Yang Ground-state energy exactly [Tateo] R E m m S matrix Euclidian artition function: Z(,R) = R e E ()R (1+e ER ) Z(,R) = R Tr(e H(R) ) = R ne E n()r e H()R e H(R) R Finite article/hole + Bethe root density ρ,ρ h,ρi,ρ i h : e i T S j = 1, T Q++ T + Q α = 1 ιπ ιπ eriodicity ιπ / E n (R) = i E( i) E()ρ ()d Rδ m + K m n (, )ρ n ( )d = π(ρ m +ρ m h ) π Z(,R) = d[ρ i,ρ i h ]e E(R) i ((ρ i +ρ i h )ln(ρi +ρ i h ) ρi lnρ i ρ i h lnρi h )d

61 Thermodynamic Bethe Ansatz: non-diagonal CFT uscher Ground-state energy exactly [Tateo] E R m m Bethe Yang S matrix Euclidian artition function: Z(,R) = R e E()R (1+e ER ) Z(,R) = R Tr(e H(R) ) = R ne E n()r e H()R Finite article/hole + Bethe root density ρ,ρ h,ρi,ρ i h : e i T S j = 1, T Q++ T + Q α = 1 ιπ ιπ e H(R) R eriodicity ιπ / E n (R) = i E( i) E()ρ ()d Rδ m + K m n (, )ρ n ( )d = π(ρ m +ρ m h ) π Z(,R) = d[ρ i,ρ i h ]e E(R) i ((ρ i +ρ i h )ln(ρi +ρ i h ) ρi lnρ i ρ i h lnρi h )d Saddle oint : ǫ i (θ) = ln ρi () ρ i h () ǫ j (θ) = δ j E() K j i (,)log(1+e ǫi ( ) )d Ground state energy exactly: E () = d π log(1+e ǫ (θ) )dθ

62 Thermodynamic Bethe Ansatz: excited diagonal Excited state energy exactly [an idea] 1 n 13

63 Thermodynamic Bethe Ansatz: excited diagonal Excited state energy exactly [an idea] 1 n CFT uscher Bethe Yang E m m S matrix

64 Thermodynamic Bethe Ansatz: excited diagonal CFT uscher Bethe Yang Excited state energy exactly [an idea] 1 n E m m S matrix

65 Thermodynamic Bethe Ansatz: excited diagonal CFT uscher Bethe Yang Excited state energy exactly [an idea] 1 n R E m m S matrix Particles are like defects: Z d (,R) = R Tr(e H d()r ) Z d (,R) = R e E d()r (1+e ER ) e H()R

66 Thermodynamic Bethe Ansatz: excited diagonal CFT uscher Bethe Yang Excited state energy exactly [an idea] 1 n R E m m S matrix Particles are like defects: Z d (,R) = R Tr(e H d()r ) Z d (,R) = R e E d()r (1+e ER ) e H()R Particles are like defect oerators Z d (,R) = R Tr(e H(R) D) = R n n D n e E n()r

67 Thermodynamic Bethe Ansatz: excited diagonal CFT uscher Bethe Yang Excited state energy exactly [an idea] 1 n R E m m S matrix Particles are like defects: Z d (,R) = R Tr(e H d()r ) Z d (,R) = R e E d()r (1+e ER ) e H()R Particles are like defect oerators Z d (,R) = R Tr(e H(R) D) = R n n D n e E n()r e H(R) R

68 Thermodynamic Bethe Ansatz: excited diagonal CFT uscher Bethe Yang Excited state energy exactly [an idea] 1 n R E m m S matrix Particles are like defects: Z d (,R) = R Tr(e H d()r ) Z d (,R) = R e E d()r (1+e ER ) e H()R Particles are like defect oerators Z d (,R) = R Tr(e H(R) D) = R n n D n e E n()r e H(R) R

69 Thermodynamic Bethe Ansatz: excited diagonal CFT uscher Bethe Yang Excited state energy exactly [an idea] 1 n R E m m S matrix Particles are like defects: Z d (,R) = R Tr(e H d()r ) Z d (,R) = R e E d()r (1+e ER ) e H()R Particles are like defect oerators Z d (,R) = R Tr(e H(R) D) = R n n D n e E n()r e H(R) R Dominant contribution: finite article/hole density ρ,ρ h : π

70 Thermodynamic Bethe Ansatz: excited diagonal CFT uscher Bethe Yang Excited state energy exactly [an idea] 1 n R E m m S matrix Particles are like defects: Z d (,R) = R Tr(e H d()r ) Z d (,R) = R e E d()r (1+e ER ) e H()R Particles are like defect oerators Z d (,R) = R Tr(e H(R) D) = R n n D n e E n()r e H(R) R Dominant contribution: finite article/hole density ρ,ρ h : π E() E()+logS(, ) j R+ k 1 i logs( j, k ) = (n+1)iπ R+ ( id logs(, ))ρ( )d = π(ρ+ρ h )

71 Thermodynamic Bethe Ansatz: excited diagonal CFT uscher Bethe Yang Excited state energy exactly [an idea] 1 n R E m m S matrix Particles are like defects: Z d (,R) = R Tr(e H d()r ) Z d (,R) = R e E d()r (1+e ER ) e H()R Particles are like defect oerators Z d (,R) = R Tr(e H(R) D) = R n n D n e E n()r e H(R) R Dominant contribution: finite article/hole density ρ,ρ h : π E() E()+logS(, ) j R+ k 1 i logs( j, k ) = (n+1)iπ R+ ( id logs(, ))ρ( )d = π(ρ+ρ h ) forbidden quantizations ρ( )+ρ h ( ) = e ǫ( ) = 1

72 Thermodynamic Bethe Ansatz: excited diagonal CFT uscher Bethe Yang Excited state energy exactly [an idea] 1 n R E m m S matrix Particles are like defects: Z d (,R) = R Tr(e H d()r ) Z d (,R) = R e E d()r (1+e ER ) e H()R Particles are like defect oerators Z d (,R) = R Tr(e H(R) D) = R n n D n e E n()r Dominant contribution: finite article/hole density ρ,ρ h : π e H(R) R E() E()+logS(, ) j R+ k 1 i logs( j, k ) = (n+1)iπ R+ ( id logs(, ))ρ( )d = π(ρ+ρ h ) forbidden quantizations ρ( )+ρ h ( ) = e ǫ( ) = 1 Saddle oint : ǫ() = E()+logS(, )+ d π id logs(,)log(1+e ǫ( ) ) Excited state energy exactly: E d () = E( ) d π log(1+e ǫ() ) sh-g

73 Excited states TBA, Y-system: diagonal Excited states exactly 1 n 14

74 Excited states TBA, Y-system: diagonal Excited states exactly 1 n CFT uscher Bethe Yang E m m S matrix

75 Excited states TBA, Y-system: diagonal CFT uscher Bethe Yang E Excited states exactly 1 n i π/ m m S matrix By lattice regularization: sinh-gordon [Teschner] dθ ǫ(θ) = mcoshθ + π id θlogs(θ θ )log(1+e ǫ(θ E {nj } () = m dθ π coshθ log(1+e ǫ(θ) ) ) )

76 Excited states TBA, Y-system: diagonal CFT uscher Bethe Yang E Excited states exactly 1 n i π/ m m S matrix By lattice regularization: sinh-gordon [Teschner] ǫ(θ) = mcoshθ+ dθ logs(θ θ j )+ π id θlogs(θ θ )log(1+e ǫ(θ E {nj } () = m dθ π coshθ log(1+e ǫ(θ) ) ) )

77 Excited states TBA, Y-system: diagonal CFT uscher Bethe Yang E Excited states exactly 1 n i π/ m m S matrix By lattice regularization: sinh-gordon [Teschner] ǫ(θ) = mcoshθ+ dθ logs(θ θ j )+ π id θlogs(θ θ )log(1+e ǫ(θ E {nj } () = m i sinhθj m dθ π coshθ log(1+e ǫ(θ) ) ) )

78 Excited states TBA, Y-system: diagonal CFT uscher Bethe Yang E Excited states exactly 1 n i π/ m m S matrix By lattice regularization: sinh-gordon [Teschner] ǫ(θ) = mcoshθ+ dθ logs(θ θ j )+ π id θlogs(θ θ )log(1+e ǫ(θ E {nj } () = m i sinhθj m dθ π coshθ log(1+e ǫ(θ) ); Y(θ j ) = e ǫ(θ j) = 1 ) )

79 Excited states TBA, Y-system: diagonal CFT uscher Bethe Yang E Excited states exactly 1 n i π/ m m S matrix By lattice regularization: sinh-gordon [Teschner] ǫ(θ) = mcoshθ+ dθ logs(θ θ j )+ π id θlogs(θ θ )log(1+e ǫ(θ E {nj } () = m i sinhθj m dθ π coshθ log(1+e ǫ(θ) ); Y(θ j ) = e ǫ(θ j) = 1 By analytical continuation: ee-yang [P.Dorey,Tateo] iπ/6 dθ ǫ(θ) = mcoshθ + π id θlogs(θ θ )log(1+e ǫ(θ E {nj } () = üscher corrections: differ by µ term m i π/6 ) ) dθ π coshθ log(1+e ǫ(θ) ) ) )

80 Excited states TBA, Y-system: diagonal CFT uscher Bethe Yang E Excited states exactly 1 n i π/ m m S matrix By lattice regularization: sinh-gordon [Teschner] ǫ(θ) = mcoshθ+ dθ logs(θ θ j )+ π id θlogs(θ θ )log(1+e ǫ(θ E {nj } () = m i sinhθj m dθ π coshθ log(1+e ǫ(θ) ); Y(θ j ) = e ǫ(θ j) = 1 By analytical continuation: ee-yang [P.Dorey,Tateo] iπ/6 N ǫ(θ) = mcoshθ+ log S(θ θ j) S(θ θj ) + dθ π id θlogs(θ θ )log(1+e ǫ(θ E {nj } () = j=1 üscher corrections: differ by µ term m i π/6 ) ) dθ π coshθ log(1+e ǫ(θ) ) ) )

81 Excited states TBA, Y-system: diagonal CFT uscher Bethe Yang E Excited states exactly 1 n i π/ m m S matrix By lattice regularization: sinh-gordon [Teschner] ǫ(θ) = mcoshθ+ dθ logs(θ θ j )+ π id θlogs(θ θ )log(1+e ǫ(θ E {nj } () = m i sinhθj m dθ π coshθ log(1+e ǫ(θ) ); Y(θ j ) = e ǫ(θ j) = 1 By analytical continuation: ee-yang [P.Dorey,Tateo] iπ/6 N ǫ(θ) = mcoshθ+ log S(θ θ j) S(θ θj ) + dθ π id θlogs(θ θ )log(1+e ǫ(θ j=1 E {nj } () = im (sinhθ j sinhθ j ) m üscher corrections: differ by µ term i π/6 ) ) dθ π coshθ log(1+e ǫ(θ) ) ) )

82 CFT Excited states TBA, Y-system: diagonal E uscher Bethe Yang Excited states exactly 1 n i π/ m m S matrix By lattice regularization: sinh-gordon [Teschner] ǫ(θ) = mcoshθ+ dθ logs(θ θ j )+ π id θlogs(θ θ )log(1+e ǫ(θ )) E {nj } () = m dθ sinhθj m i π coshθ log(1+e ǫ(θ) ); Y(θ j ) = e ǫ(θ j) = 1 i π/6 By analytical continuation: ee-yang [P.Dorey,Tateo] iπ/6 N ǫ(θ) = mcoshθ+ log S(θ θ j) j=1 S(θ θj ) + dθ π id θlogs(θ θ )log(1+e ǫ(θ )) E {nj } () = im (sinhθ j sinhθj ) m dθ π coshθ log(1+e ǫ(θ) ) üscher corrections: differ by µ term S(θ iπ 3 )S(θ + iπ 3 ) = S(θ) Y(θ + iπ 3 )Y(θ iπ 3 ) = 1+Y(θ) Y-system+analyticity=TBA scalar. Matrix [Bazhanov,ukyanov,Zamolodchikov]

83 Excited states TBA, Y-system: Non-diagonal Excited states exactly 1... j n... 15

84 Excited states TBA, Y-system: Non-diagonal Excited states exactly 1... j n... CFT uscher Bethe Yang E m m S matrix

85 Excited states TBA, Y-system: Non-diagonal E CFT uscher Bethe Yang Excited states exactly 1... j n... m m S matrix

86 Excited states TBA, Y-system: Non-diagonal E CFT uscher Bethe Yang Excited states exactly 1... j n... ιπ eriodicity ιπ / m m S matrix ιπ Y-system: sine-gordon π Y s (θ + iπ )Y s(θ iπ ) = (1+Y s 1)(1+Y s+1 ) (e iπ +e iπ )logy s = ri sr log(1+y r )... +1

87 Excited states TBA, Y-system: Non-diagonal CFT uscher E Bethe Yang Excited states exactly 1... j n... ιπ eriodicity ιπ / m m S matrix ιπ Y-system: sine-gordon π Y s (θ + iπ )Y s(θ iπ ) = (1+Y s 1)(1+Y s+1 ) (e iπ +e iπ )logy s = ri sr log(1+y r ) Excited states: analyticity from üscher [Balog, Hegedus]

88 Excited states TBA, Y-system: Non-diagonal E CFT uscher Bethe Yang Excited states exactly 1... j n... ιπ eriodicity ιπ / m m S matrix ιπ Y-system: sine-gordon π Y s (θ + iπ )Y s(θ iπ ) = (1+Y s 1)(1+Y s+1 ) (e iπ +e iπ )logy s = ri sr log(1+y r ) Excited states: analyticity from üscher [Balog, Hegedus] attice regularization: [Destri,de Vega,Ravanini,...] sg Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ

89 Excited states TBA, Y-system: Non-diagonal CFT uscher E Bethe Yang Excited states exactly 1... j n... ιπ eriodicity ιπ / m m S matrix ιπ Y-system: sine-gordon Y s (θ + iπ )Y s(θ iπ ) = (1+Y s 1)(1+Y s+1 ) (e iπ +e iπ )logy s = ri sr log(1+y r ) π Excited states: analyticity from üscher [Balog, Hegedus] attice regularization: [Destri,de Vega,Ravanini,...] sg Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Z(θ) = Msinhθ +source(θ {θ k })+Im dxg(θ x iǫ)log [ 1 e iz(x+iǫ)] source(θ {θ k }) = i klogs (θ θ k) kernel: G(θ) = i θ logs (θ) Energy: E = M kcoshθ k MIm dxg(θ +iǫ)log [ 1 e iz(x+iǫ)] Bethe-Yang e iz(θ k) = 1

90 Sine-Gordon/massive Thirring duality SG = 1 νφ ν Φ+ m β : cos(βφ) : < β < 8π, g Classical sine Gordon theory Semiclassical sine Gordon theory + loo corrections Quantum sine Gordon theory Finite size corrections Massive Thirring model uscher correction Bethe Yang equations S matrix, masses strong-weak duality: + erturbative corrections 1+ g 4π = 4π β = +1 erturbative Thirring model MT = Ψ(iγ ν ν +m )Ψ g Ψγ ν Ψ Ψγ ν Ψ 16

91 AdS/CFT duality: t Hooft Integrability in QCD: Feynman Richard Feynman ( ) What I cannot create I do not understand QED: Feynman grahs To learn: Bethe Ansatz Probs., Kondo, D Hall, accel tem, Non linear classical Hydro Strong interaction? 17

92 CFT: maximally suersymmetric gauge theory Feynman: If you want to learn about nature, to areciate nature, it is necessary to understand the language she seaks in. Fundamental interactions: gauge theory Couling non erturbative no analytical tool interaction articles gauge theory electromagnetic hoton+electron U(1) electroweak W ±,Z µ,ν+higgs SU() U(1) strong gluon+quarks SU(3) SU(3) SU() U(1) erturbative strong force electroweak force Energy only analytical tool: erturbation theory 18

93 CFT: maximally suersymmetric gauge theory Feynman: If you want to learn about nature, to areciate nature, it is necessary to understand the language she seaks in. Fundamental interactions: gauge theory interaction articles gauge theory electromagnetic hoton+electron U(1) electroweak W ±,Z µ,ν+higgs SU() U(1) strong gluon+quarks SU(3) only analytical tool: erturbation theory maximally suersymmetric gauge theory (harmonic oscillator) interaction articles gauge theory N = 4 suersymm. gluon+quarks+scalars SU(N) all fields N 1 comonent matrix = g YM Ψ 1,,3,4 ր ց A µ Φ 1,,3,4,5,6 ց ր Ψ 1,,3,4 d 4 xtr [ 1 4 F 1 (DΦ) +iψd/ψ+v ] V(Φ,Ψ) = 1 4 [Φ,Φ] +Ψ[Φ,Ψ] Couling SU(3) SU() U(1) non erturbative no analytical tool erturbative strong force electroweak force Energy no running β = CFT no confinement suersymmetric heavy ion collision: finite T SUSY is broken quark-gluon lasma is not confined

94 CFT: Observables maximally suersymmetric gauge theory Ψ 1,,3,4 A Φ 1,,3,4,5,6 S = g YM Ψ 1,,3,4 fields SU(N) matrices d 4 xtr [ 1 4 F 1 (DΦ) +iψd/ψ+v ] V(Φ,Ψ) = 1 4 [Φ,Φ] +Ψ[Φ,Ψ] observables arameters: g YM,N observables: artition function gauge-invariant oerators O(x) = Tr(A 1Ψ Φ 3..),det() correlators: O 1 (x 1 )...O n (x n ) correlators: O 1 (x)o () = [da...]e is O 1 (x)o () = O 1 (x)o ()e iv erturbation: g YM g YM g YM N genus ex. g YM N3 = N λ, λ = g YM N artition func. (t Hooft) Z(λ, 1 N ) = N g( 1 N )g nα(g,n)λ n string interactions? 19

95 ositively curved sace AdS: string theory on Anti de Sitter

96 AdS: string theory on Anti de Sitter ositively curved sace Anti de Sitter: negatively curved sace

97 AdS: string theory on Anti de Sitter ositively curved sace Anti de Sitter: negatively curved sace

98 AdS: string theory on Anti de Sitter ositively curved sace Anti de Sitter: negatively curved sace relativistic oint article: ds = dx +dx S worldline ds = ẋ ẋdτ x x x( τ) x 1

99 AdS: string theory on Anti de Sitter ositively curved sace Anti de Sitter: negatively curved sace relativistic oint article: ds = dx +dx S worldline ds = ẋ ẋdτ relativistic string: ds = dx +dx x x x( τ) x 1 x S worldsheet da = (ẋ x ) ẋ x dτdσ x( τ,σ) x τ σ x 1

100 AdS: string theory on Anti de Sitter ositively curved sace Anti de Sitter: negatively curved sace relativistic oint article: ds = dx +dx S worldline ds = ẋ ẋdτ relativistic string: ds = dx +dx x x x( τ) x 1 x S worldsheet da = (ẋ x ) ẋ x dτdσ S 5 S 5 :Y +Y 1 +Y +Y 3 +Y 4 +Y 5 = R 5 AdS x( τ,σ) x τ σ x 1 AdS 5 : X +X 1 +X +X 3 +X 4 X 5 = R S = R dτdσ α 4π ( a X M a X M + a Y M a Y M ) +fermions suercoset PSU(, 4) SO(5) SO(1,4)

101 AdS/CFT corresondence (Maldacena 1998) II B suerstring on AdS 5 S 5 4D Minkowski sace time sace extra dimension 5D anti de Sitter sace 5D shere 6 1 Y i = R ++++ = R ( ) a X M a X M + a Y M a Y M +... R dτdσ α 4π g YM N = 4 D=4 SU(N) SYM d 4 xtr [ 1 4 F 1 (DΦ) +iψd/ψ+v ] V(Φ,Ψ) = 1 4 [Φ,Φ] +Ψ[Φ,Ψ] PSU(, 4) β = suerconformal SO(5) SO(1,4) gaugeinvariants:o = Tr(Φ ),det( ) 1

102 AdS/CFT corresondence (Maldacena 1998) II B suerstring on AdS 5 S 5 4D Minkowski sace time sace extra dimension 5D anti de Sitter sace 5D shere 6 1 Y i = R ++++ = R ( ) a X M a X M + a Y M a Y M +... R dτdσ α 4π Coul.: λ = R α, g s = λ N D QFT String energy levels: E(λ) E(λ) = E( )+ E 1 + E λ λ +... Dictionary g YM strong weak N = 4 D=4 SU(N) SYM d 4 xtr [ 1 4 F 1 (DΦ) +iψd/ψ+v ] V(Φ,Ψ) = 1 4 [Φ,Φ] +Ψ[Φ,Ψ] PSU(, 4) β = suerconformal SO(5) SO(1,4) gaugeinvariants:o = Tr(Φ ),det( ) λ = g YM N, N lanar O n (x)o m () = δ nm x n(λ) Anomalous dim (λ) (λ) = ()+λ 1 +λ +...

103 AdS/CFT corresondence (Maldacena 1998) II B suerstring on AdS 5 S 5 4D Minkowski sace time sace extra dimension 5D anti de Sitter sace 5D shere 6 1 Y i = R ++++ = R ( ) a X M a X M + a Y M a Y M +... R dτdσ α 4π Coul.: λ = R α, g s = λ N D QFT String energy levels: E(λ) E(λ) = E( )+ E 1 + E λ λ +... Dictionary g YM strong weak N = 4 D=4 SU(N) SYM d 4 xtr [ 1 4 F 1 (DΦ) +iψd/ψ+v ] V(Φ,Ψ) = 1 4 [Φ,Φ] +Ψ[Φ,Ψ] PSU(, 4) β = suerconformal SO(5) SO(1,4) gaugeinvariants:o = Tr(Φ ),det( ) λ = g YM N, N lanar O n (x)o m () = δ nm x n(λ) Anomalous dim (λ) (λ) = ()+λ 1 +λ +... D integrable QFT sectrum: Q = 1,,..., disersion: ǫ Q () = Q + λ π sin Exact scattering matrix: S Q1 Q ( 1,,λ)

104 Motivation: AdS/CFT

105 Motivation: AdS/CFT g J

106 Motivation: AdS/CFT g Classical string theory J

107 Motivation: AdS/CFT g Classical string theory Semiclassical string theory J

108 Motivation: AdS/CFT g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory J

109 Motivation: AdS/CFT g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory 1 loo erturbative gauge theory J

110 Motivation: AdS/CFT g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory J

111 Motivation: AdS/CFT g Classical string theory Semiclassical string theory Quantum String theory + string loo corrections S matrix Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory J

112 Motivation: AdS/CFT g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory + gauge loo corrections Asymtotic Bethe Ansatz S matrix 1 loo erturbative gauge theory J

113 Motivation: AdS/CFT Classical string theory g Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections 1 loo erturbative gauge theory Need finite J (volume) solution of the sectral roblem J

114 Classical integrability: AdS Classical string theory g AdS 5 S 5 Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections AdS σ model τ σ target = R α ( a X M a X M + a Y M a Y M ) +fermions 1 loo erturbative gauge theory J 3

115 Classical integrability: AdS Classical string theory g AdS 5 S 5 Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections AdS σ model τ σ target = R α ( a X M a X M + a Y M a Y M ) +fermions 1 loo erturbative gauge theory S 5 J Classical solutions are found, for examle magnon: J

116 Classical integrability: AdS Classical string theory g AdS 5 S 5 Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections AdS σ model τ σ target = R α ( a X M a X M + a Y M a Y M ) +fermions 1 loo erturbative gauge theory S 5 J Classical solutions are found, for examle magnon: J Coset Nσ model: g PSU(, 4) SO(4,1) SO(5) J = g 1 dg = J +J Z 4 graded structure: [Metsaev, Tseytlin 3] J,J 1,J,J 3 STr(J J ) STr(J 1 J 3 )

117 Classical integrability: AdS g Classical string theory AdS 5 S 5 Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix AdS σ model τ σ target = R α ( a X M a X M + a Y M a Y M ) +fermions + gauge loo corrections 1 loo erturbative gauge theory S 5 J Classical solutions are found, for examle magnon: J Coset Nσ model: g PSU(, 4) SO(4,1) SO(5) J = g 1 dg = J +J Z 4 graded structure: [Metsaev, Tseytlin 3] J,J 1,J,J 3 STr(J J ) STr(J 1 J 3 ) Integrability from flat connection: da A A = A(µ) = J +µ 1 J 1 +(µ +µ )J /+(µ +µ )J /+µj 3 Conserved charges from the trace of the monodromy matrix T(µ) = P ex A(x) µ dx µ No roof of quantum integrability: let us assume it! gtg 1 T

118 Bootstra rogram: AdS Nondiagonal scattering: S-matrix = scalar. Matrix g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections 1 loo erturbative gauge theory J 4

119 Bootstra rogram: AdS Nondiagonal scattering: S-matrix = scalar. Matrix Perturbative sectrum: 8 boson + 8 fermion global symmetry PSU( ) Q = 1 res b 1 b f 3 f 4 b 1 b f 3 f 4 g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz J S matrix (Q) [S, (Q)] = S matrix (Q)

120 Bootstra rogram: AdS Nondiagonal scattering: S-matrix = scalar. Matrix Perturbative sectrum: 8 boson + 8 fermion global symmetry PSU( ) Q = 1 res [S, (Q)] = b 1 b f 3 f 4 (Q) S matrix (Q) b 1 b f 3 f 4 g Quantum String theory Classical string theory Semiclassical string theory + string loo corrections Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory A.. b c.. c... d..... e b..d e A c.. c d..e d..... e.... f..... g f.. g a h.. h B.. i.... f g f..... g... h.. h i.. B a Wraing/uscher correction Asymtotic Bethe Ansatz J S matrix

121 Bootstra rogram: AdS Nondiagonal scattering: S-matrix = scalar. Matrix Perturbative sectrum: 8 boson + 8 fermion global symmetry PSU( ) Q = 1 res [S, (Q)] = b 1 b f 3 f 4 (Q) S matrix Unitarity S(z 1,z )S(z,z 1 ) = 1 (Q) b 1 b f 3 f 4 Crossing symmetry [Janik] [Volin] S(z 1,z ) = S c 1(z,z 1 +ω ) g Quantum String theory Classical string theory Semiclassical string theory + string loo corrections Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory A.. b c.. c... d..... e b..d e A c.. c d..e d..... e.... f..... g f.. g a h.. h B.. i.... f g f..... g... h.. h i.. B a S = u 1 u i u 1 u +i eiθ(z 1,z ) u = 1 cot E() Wraing/uscher correction [Beisert,Eden,Staudacher] = am(z) Asymtotic Bethe Ansatz J S matrix

122 Bootstra rogram: AdS Nondiagonal scattering: S-matrix = scalar. Matrix Perturbative sectrum: 8 boson + 8 fermion global symmetry PSU( ) Q = 1 res [S, (Q)] = b 1 b f 3 f 4 (Q) S matrix (Q) b 1 b f 3 f 4 Unitarity S(z 1,z )S(z,z 1 ) = 1 S 11 Crossing symmetry [Janik] [Volin] S(z 1,z ) = S c 1(z,z 1 +ω ) g Quantum String theory Classical string theory Semiclassical string theory + string loo corrections Nonerturbative gauge theory gauge loo corrections. A.. b c. 1 loo. erturbative gauge c theory... d..... e b..d e A c.. c d..e d..... e.... f..... g f.. g a h.. h B.. i.... f g f..... g... h.. h i.. B a 11 = u 1 u i u 1 u +i eiθ(z 1,z ) u = 1 cot E() Wraing/uscher correction [Beisert,Eden,Staudacher] = am(z) ω Asymtotic Bethe Ansatz J S matrix Maximal analyticity: boundstates aty symre: Q N anomalous thresholds [N.Dorey,Maldacena,Hofman,Okamura] Q 1 ω / Q 1 1 Q Physical domain ω 1/ ω 1/

123 Bethe-Yang=Asymtotic Bethe Ansatz Finite volume sectrum 1 n g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections 1 loo erturbative gauge theory J 5

124 Bethe-Yang=Asymtotic Bethe Ansatz Finite volume sectrum 1 n g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections Infinite volume sectrum: E( 1,..., n ) = ie( i ) E() = 1+(4gsin ) 1 loo erturbative gauge theory i [ π,π] J

125 Bethe-Yang=Asymtotic Bethe Ansatz Finite volume sectrum 1 n g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix Infinite volume sectrum: E( 1,..., n ) = ie( i ) E() = 1+(4gsin ) + gauge loo corrections 1 loo erturbative gauge theory J i [ π,π] Polynomial volume corrections: Asymtotic Bethe Ansatz; i quantized,. e i j S(j, 1 )...S( j, n )Ψ = Ψ S() = ˆP... 1 j... n

126 Bethe-Yang=Asymtotic Bethe Ansatz Finite volume sectrum 1 n g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections Infinite volume sectrum: E( 1,..., n ) = ie( i ) E() = 1+(4gsin ) 1 loo erturbative gauge theory i [ π,π] J Polynomial volume corrections: Asymtotic Bethe Ansatz; i quantized,. e i j S( j, 1 )...S( j, n )Ψ = Ψ S() = ˆP... 1 j... n Inhomogenous Hubbard sin-chain: e i js (u j) Q++ 4 (u j ) Q 4 (u j) T(u j)ṫ(u j ) = 1

127 Bethe-Yang=Asymtotic Bethe Ansatz Finite volume sectrum 1 n g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix Infinite volume sectrum: E( 1,..., n ) = ie( i ) E() = 1+(4gsin ) + gauge loo corrections 1 loo erturbative gauge theory J i [ π,π] Polynomial volume corrections: Asymtotic Bethe Ansatz; i quantized,. e i j S( j, 1 )...S( j, n )Ψ = Ψ S() = ˆP... 1 j... n Inhomogenous Hubbard sin-chain: e i js (u j) Q++ 4 (u j ) Q 4 (u j) T(u j)ṫ(u j ) = 1 T(u) = B 1 B+ 3 R (+) 4 B + 1 B 3 R ( ) 4 [ Q Q+ 3 Q Q 3 R ( ) 4 Q + 3 R (+) 4 Q 3 + Q++ Q 1 Q Q + 1 B+(+) 4 Q 1 B +( ) 4 Q + 1 ] [Beisert,Staudacher] Q j (u) = R j (u)b j (u) and R (±) j = k x(u) x j,k x j,k B (±) j = k 1 x(u) x j,k x j,k

128 Bethe-Yang=Asymtotic Bethe Ansatz Finite volume sectrum 1 n g Classical string theory Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix Infinite volume sectrum: E( 1,..., n ) = ie( i ) E() = 1+(4gsin ) + gauge loo corrections 1 loo erturbative gauge theory i [ π,π] J Polynomial volume corrections: Asymtotic Bethe Ansatz; i quantized,. e i j S( j, 1 )...S( j, n )Ψ = Ψ S() = ˆP... 1 j... n Inhomogenous Hubbard sin-chain: e i js (u j) Q++ 4 (u j ) Q T(u) = B 1 B+ 3 R (+) 4 B 1 + B 3 R ( ) 4 [ Q Q+ 3 Q Q 3 Q j (u) = R j (u)b j (u) and R (±) j Bethe Ansatz: Q+ B( ) 4 Q B(+) 4 R ( ) 4 Q + 3 R (+) 4 Q 3 = k 1 = 1 Q Q+ 1 Q+ 3 Q ++ Q 1 Q 3 x(u) x j,k x j,k + Q++ Q 1 Q Q + 1 B (±) j 4 (u j) T(u j)ṫ(u j ) = 1 B+(+) 4 Q 1 B +( ) 4 Q + 1 = k = 1 Q+ R( ) 4 Q R(+) 4 1 x(u) x j,k x j,k ] 3 = 1 [Beisert,Staudacher]

129 Thermodynamic Bethe Ansatz: AdS Classical string theory Ground-state energy exactly [Bombardelli,Tateo,Fioravanti,Frolov,Arutyunov, Gromov,Kazakov,Viera,Kozak] Euclidien E +(4gsin ) = 1 artition function: R g Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix J Z(,R) = R e E ()R (1+e ER ) e H()R Z(,R) = R Tr(e H(R) ) = R ne Ẽ n ()R e H(R) R 6

130 Thermodynamic Bethe Ansatz: AdS Classical string theory Ground-state energy exactly [Bombardelli,Tateo,Fioravanti,Frolov,Arutyunov, Gromov,Kazakov,Viera,Kozak] Euclidien E +(4gsin ) = 1 artition function: R g Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix J Z(,R) = R e E ()R (1+e ER ) e H()R Z(,R) = R Tr(e H(R) ) = R ne Ẽ n ()R e H(R) R Finite article/hole + Bethe root density ρ Q,ρ Q h,ρi,ρ i h : i no eriodicity

131 Thermodynamic Bethe Ansatz: AdS Classical string theory Ground-state energy exactly [Bombardelli,Tateo,Fioravanti,Frolov,Arutyunov, Gromov,Kazakov,Viera,Kozak] Euclidien E +(4gsin ) = 1 artition function: R g Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix J Z(,R) = R e E ()R (1+e ER ) e H()R Z(,R) = R Tr(e H(R) ) = R ne Ẽ n ()R e H(R) R Finite article/hole + Bethe root density ρ Q,ρ Q h,ρi,ρ i h : i Ẽ n (R) = i,qẽq( i ) Q ẼQ ( )ρ Q ( )d e i S Q ++ 4 Q 4 T Ṫ 4 = 1 Q+ B( ) 4 Q B(+) 4 1 = 1 Q Q+ 1 Q+ 3 Q ++ Q 1 Q 3 = 1 Q+ R( ) 4 Q R(+) 4 3 = 1 no eriodicity K m n (, )ρ n ( )d = π(ρ m +ρ m h )

132 Thermodynamic Bethe Ansatz: AdS Classical string theory Ground-state energy exactly [Bombardelli,Tateo,Fioravanti,Frolov,Arutyunov, Gromov,Kazakov,Viera,Kozak] Euclidien E +(4gsin ) = 1 artition function: R g Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix Z(,R) = R e E ()R (1+e ER ) e H()R J Z(,R) = R Tr(e H(R) ) = R ne Ẽ n ()R e H(R) R Finite article/hole + Bethe root density ρ Q,ρ Q h,ρi,ρ i h : i Ẽ n (R) = i,qẽq( i ) Q ẼQ ( )ρ Q ( )d e i S Q ++ 4 Q 4 T Ṫ 4 = 1 Q+ B( ) 4 Q B(+) 4 1 = 1 Q Q+ 1 Q+ 3 Q ++ Q 1 Q 3 = 1 Q+ R( ) 4 Q R(+) 4 3 = 1 no eriodicity K m n (, )ρ n ( )d = π(ρ m +ρ m h ) Z(,R) = d[ρ i,ρ i h ]e E(R) i ((ρ i +ρ i h )ln(ρi +ρ i h ) ρi lnρ i ρ i h lnρi h )d

133 Thermodynamic Bethe Ansatz: AdS Classical string theory Ground-state energy exactly [Bombardelli,Tateo,Fioravanti,Frolov,Arutyunov, Gromov,Kazakov,Viera,Kozak] Euclidien E +(4gsin ) = 1 artition function: R g Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory + gauge loo corrections 1 loo erturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix Z(,R) = R e E ()R (1+e ER ) Z(,R) = R Tr(e H(R) ) = R ne Ẽ n ()R e H()R e H(R) R J Finite article/hole + Bethe root density ρ Q,ρ Q h,ρi,ρ i h : i Ẽ n (R) = i,qẽq( i ) Q ẼQ ( )ρ Q ( )d e i S Q ++ 4 Q 4 T Ṫ 4 = 1 Q+ B( ) 4 Q B(+) 4 1 = 1 Q Q+ 1 Q+ 3 Q ++ Q 1 Q 3 = 1 Q+ R( ) 4 Q R(+) 4 3 = 1 no eriodicity K m n (, )ρ n ( )d = π(ρ m +ρ m h ) Z(,R) = d[ρ i,ρ i h ]e E(R) i ((ρ i +ρ i h )ln(ρi +ρ i h ) ρi lnρ i ρ i h lnρi h )d Saddle oint : ǫ i ( ) = ln ρi ( ) ρ i h ( ) ǫ j (θ) = δ j QẼQ( ) K j i (, )log(1+e ǫi ( ) )d Ground state energy exactly: E () = Q d π log(1+e ǫ Q( ) )d

134 Excited states TBA, Y-system: AdS Classical string theory Excited states exactly 1... j n... g Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections 1 loo erturbative gauge theory J 7

135 Excited states TBA, Y-system: AdS Classical string theory Excited states exactly 1... j n... i no eriodicity g Semiclassical string theory + string loo corrections Quantum String theory Nonerturbative gauge theory Wraing/uscher correction Asymtotic Bethe Ansatz S matrix + gauge loo corrections Y-system: AdS [Gromov,Kazakov,Viera] 1 loo erturbative gauge theory J Y a,s (θ+ i )Y a,s(θ i ) Y a+1,s Y a 1,s = (1+Y a,s 1)(1+Y a,s+1 ) (1+Y a+1,s )(1+Y a 1,s )