Conformal group: Yang-Baxter equation and star-triangle relation

Transcription

1 Conformal group: Yang-Baxter equation and star-triangle relation S. Derkachov PDMI, St.Petersburg CONFORMAL SYMMETRY IN FOUR-DIMENSIONAL FIELD THEORIES 04, July 4-6, Regensburg

2 Plan Conformal algebra differential representation intertwining operators Star-triangle relation D-dimension: scalars A.N.Vasil ev, Y.M.Pis mak, J.Honkonen, D.Kazakov,... Generalized star-triangle relation Coulomb problem D.Chicherin two-dimension: tensors L.N.Lipatov G.Korchemsky, A.Manashov 4-dimension: tensors A.P.Isaev, D.Chicherin Star-triangle relation Yang-Baxter equation R.Baxter, V.Bazhanov, A.Zamolodchikov, L.Faddeev,... G.Korchemsky, A.Manashov,A.P.Isaev, D.Chicherin

3 Lie algebra of the conformal group in Euclidean D-dimensional space Generators {L µν, P µ, K µ, D} µ, ν =,..., D Commutation relations [D, P µ] = i P µ, [D, K µ] = i K µ, [L µν, L ρσ] = i δ νρ L µσ + δ µσ L νρ δ µρ L νσ δ νσ L µρ, [K ρ, L µν] = i δ ρµ K ν δ ρν K µ, [P ρ, L µν] = i δ ρµ P ν δ ρν P µ, [K µ, P ν] = i δ µν D L µν, [P µ, P ν] = 0, [K µ, K ν] = 0, [L µν, D] = 0 L µν generate the Lie algebra sod of the rotation group SOD in R D Differential representation P µ = i xµ ˆp µ D = x µˆp µ i L µν = x νˆp µ x µˆp ν + S µν K µ = x ν x µˆp ν x νˆp µ + S νµ+x ν x νˆp µ i x µ x µ are coordinates in R D, R scaling dimension, S µν = S νµ are spin generators [S µν, x ρ] = [S µν,ˆp ρ] = 0 with the same commutation relations as for generators L µν [S µν, S ρσ] = iδ νρs µσ + δ µσs νρ δ µρs νσ δ νσs µρ

4 D-dimensional space: scalars S µν = 0 equivalent representations D intertwining operator S {L µν, P µ, K µ, D} = {L µν, P µ, K µ, D} D S [SΦ]x = translation d D y Sx, y Φy ; Sx, y = x µ + y µ Sx, y = 0, rotation yν y µ Sx, y = y µ y ν c x y D S = ˆp D x µ x ν Sx, y, x ν x µ dilatation xµ + y µ Sx, y = D Sx, y, x µ y µ special conformal transformation x x µx ν + y y µy ν Sx, y = D x µ+y µ Sx, y. x µ x ν y µ y ν

5 tensor fields of the type l, l [S µνφ] α α l α α l = σ µν α 4-dimensional space α + σ µν α α Φ α α α l α α l Φ α α l αα α l + +σ µν α α l Φ α α l α α l α σ µν α l α Φ α α l α α α l σ µ α α = σ 0, iσ, iσ, iσ 3, σ µ αα = σ 0, iσ, iσ, iσ 3 σ µν α β = i 4 σµσν σνσµ, σµν α β = i σµσν σνσµ 4 Dotted and undotted indexes compose symmetric sets separately generating function λ and λ are auxiliary spinors Φx, λ, λ = Φ α α l α α l x λ α λ α l λ α λ α l [S µνφ]x, λ, λ = [ λ σ µν λ + λ σ µν λ] Φx, λ, λ λ σ µν λ = λ ασ µν α β λ β ; λ σµν λ = λ α σ µν β α λ β

6 Intertwining operator equivalent representations, l, l D, l, l intertwining operator S {L µν, P µ, K µ, D} =,l, l {Lµν, Pµ, Kµ, D} D, l,l S [SΦ]x, λ, λ = d 4 y Sx, λ, λ; y, η, ηφy, η, η Translation X = x, λ, λ, Y = y, η, η + SX, Y = 0 x µ y µ η=0, η=0 Lorentz rotations [ ] i y ν y µ + η σ T µν η + η σ T µν η SX, Y = y µ y ν [ ] = i x µ x ν +λ σ µν λ + λ σ µν λ SX, Y x ν x µ Dilatation xµ + y µ SX, Y = 4 SX, Y x µ y µ

7 Intertwining operator Special conformal transformation iy + iy µy ν + y ν η σ T νµ η + η σ T νµ η+i4 y µ SX, Y y µ y ν = ix ix µx ν + x ν λ σ νµ λ + x µ x λ σ νµ λ i4 xµ SX, Y ν x Solution x = σ µ x µ ; x = σ µ x µ x Integral operator SX, Y = l! l! [SΦ]x, λ, λ = operator form ˆp = i x [SΦ]x, λ, λ = l l λx y η λx y η x y 4 d 4 y Φ y, λx y, λx y x y 4 d 4 y y 4 Φx y, λ y, λ y = G.M. Sotkov, R.P. Zaikov d 4 y e iyˆp y 4 Φ x, λ y, λ y

8 Euclidean D-dimensional space Uniqueness condition α+β + γ = D Integral identity d D w x w α y w β z w = Vα, β, γ γ y z α x z β x y γ Feynman diagram y β x α γ z = x α V α, β, γ γ β x y = x-y -α y z α Vα, β, γ = π D VαVβVγ ; Vα Γα Γα Operator identity ˆp µ = i µ ˆp a x a+b ˆp b = x b ˆp a+b x a ; α = D α A.P.Isaev

9 Operator reformulation of the star-triangle relation Fourier transformation Integral operator ˆp a x z a+b ˆp b = x z b ˆp a+b x z a Simple transformation d D y [ˆp α Φ]x ˆp= i x x α p α ; α = D α e i yp y D α = π D 4 α Vα p α ˆp α Φx d D y e y x = d D y x y Φy D α y α Φ x = d D y x y α Φy d D y y α Φx y =

10 Digression: one step from uniqueness, two steps... Uniqueness condition Integral identity = n=0 z n n! α+β + γ = D d D w x w α w + z γ w y = β x y α +β +n D x + z β +n y + z π D Γα + nγβ + nγγ n α +n ΓαΓβΓγ Operator identity ˆp µ = i µ x a ˆp + z a+b x = b n=0 A na, b = ˆp + z 4zn A na, b b+n x a+b+n ˆp + z a+n Γa+nΓb + nγa+b + n ΓaΓbΓa + bn!

11 Digression: Coulomb problem in D-dimensional space Hamiltonian H = ˆp λ x ; x = x +...+x D ; ˆp = ˆp ˆp D, Green function resolvent H + z Gz = Perturbative expansion λ ˆp + z λ x Gz = ˆp + z + λ ˆp + z λ 3 x Gz = l Gz = ˆp + z λ x ˆp + z + λ ˆp + z x ˆp + z x ˆp + z +... = ˆp + z + λ ˆp + z x ˆp + z + Γ n+ λ Γ n+ + λ z z 4z n + Γ λ Γ + λ z ˆp + z n+ 3 x n+ ˆp + z p+n+ 3 z Γ n+ λ Γ n++ λ z z 4z n n=0 Γ λ Γ + λ ˆp z + z n+ x n+ 3 ˆp + z n+ z n=0

12 More general star-triangle relation in D = Magic of the complex numbers x = x µ x µ = x +...+x D D= x = x + x = x + ix x ix = z z x α = x + x α = z α z α z α zᾱ [z] α ; α ᾱ Z Uniqueness condition α D= α, ᾱ ; α ᾱ Z α+β + γ = ᾱ+ β + γ = Integral identity d w [z w] α [x w] β [y w] = Vα, β, γ γ [z x] γ [z y] β [y x], α Vα, β, γ = π Vα Vβ Vγ ; Vα = Operator identity ˆp = i z ; ˆ p = i z [ˆp] a [z] a+b [ˆp] b = [z] b [ˆp] a+b [z] a Γ ᾱ Γα

13 More general star-triangle relation in D = 4 Integral identity α+β + γ = 4+m, m = 0,,,... Operator identity d 4 z x Az y m z x z α z γ z y = β = π Γα + mγβ + mγγ Γα Γβ Γγ xay m x β +m x y γ y α +m α = α ; A µν A µλ = 0, A νµ A λµ = 0 ˆp µ ˆp µm ˆp a+m = x µ x µm x b+m A µ ν A µmν m x a+b+m A µ ν A µmν m ˆp a+b+m ˆp ν ˆp νm ˆp b+m = x ν x νm x a+m

14 Yang-Baxter equation Yang-Baxter equation R u v R 3u R v = R 3v R u R 3u v EndV V V 3 R iju : V i V j V i V j Operator construction of two-parametric solution D-dimensional scalar case x = x x R u = x u+ D ˆp u+ ˆp u+ x u+ + D two-dimensional general case = l+ ; = l+ R u = [z ] u l l [ˆp ] u+l l [ˆp ] u l +l [z ] u+l +l

15 Integral operator R u = x Feynman diagrams u+ D ˆp u+ ˆp [R u Φ]x, x = 4 u π D V u + u+ x V u + d D y d D y Φy, y x + D u x y +D +u x y +D Kernel of integral operator u + = u + D, u = u, v+ = v + D u+ + D +u y D, v = v u x u -v +n/ - - y Ru-v = c v + -u - v - -u + x u+ -v + +n/ y

16 Feynman diagrams Proof of the Yang-Baxter equation Rv 3 Ru-v Ru Ru Ru-v 3 Rv 3 3 3

17 General operator structure R-operator D-dimensional space, scalars R u = S u + D S u + S u + S u + + D Star-triangle relation in operator form S α S α+β S β = S β S α+β S α S α S α+β S β = S β S α+β S α Basic operators S and S act on the function Φx ; x in a similar manner d D y e iyˆp [S αφ]x ; x = Φx ; x ˆp α y D Φx ; x +α d D y e iyˆp [S αφ]x ; x = Φx ; x ˆp α y D Φx ; x +α duality transformation S α : ˆp x x x S α S α [S d D y e iy x αφ] x ; x = Φ x ; x y D x α +α Φx ; x

18 General operator structure R-operator 4-dimensional space, tensors R u = S u + 4 S u + S u + S u Star-triangle relation in operator form S α S α+β S β = S β S α+β S α S α S α+β S β = S β S α+β S α Basic operators S and S act on the function ΦX ; X in a similar manner d 4 y e iyˆp [S αφ]x ; X = Φx y +α, λ y, λ y;x d 4 y e iyˆp [S αφ]x ; X = ΦX ; x y +α, λ y, λ y duality transformation λ λ S α : ˆp x λ λ S α S α [S d 4 y e iy x αφ] X ; X = Φ x y +α, λ y, λ ; x, λ, λ y.

19 R-operator R u = S u + 4 General operator structure S u + S u + S u [R uφ]x ; X = q u+4 + d 4 q d 4 k d 4 y d 4 z e i q+k x e i k y z z y k u++ u++ u+ + Star-triangle relation Φx y, λ z k, λ q y; x z, λ q z, λ y k. S α S α+β S β = S β S α+β S α d 4 z d 4 k d 4 y e i z ˆp e i k x e i y ˆp Φx z α+ k α+β+ y β+, λ z y, λ k y; x, λ, λ z k = d 4 q d 4 y d 4 k e i q x e i y ˆp e i k x = Φx q β+ y α+β+ k α+, λ q k, λ q y; x, λ, λ y k

20 Summary Intertwining operators Star-triangle relations Yang-Baxter operators