Conformal algebra: R-matrix and star-triangle relations.

Transcript

1 Conformal algebra: R-matrix and star-triangle relations. D. Chicherin, A. P. Isaev and S. Derkachov 3 Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia Chebyshev Laboratory, St.-Petersburg State University, 4th Line, 9b, Saint-Petersburg, 9978 Russia 3 St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 7, 903 St. Petersburg, Russia Supersymmetry in Integrable Models December 8-30, 03, Hannover, Germany /

2 Plan Introduction. Star-triangle relations (STR) and multiloop calculations R-operators and L-operators R- and L-operators for conformal algebra Conformal algebra conf(r p,q ) Spinor and differential representations of conf(r p,q ) 3 R-matrix and general R-operator /

3 Star-triangle relation (STR) Firstly considered in CFT (see e.g. E.S.Fradkin, M.Ya.Palchik, 978) d D z G(α, β) = (x z) α z (α+β) (z y) β (x) β (x y) ( D α β) (y), α where x, y, z R D, parameters α := D α, α +α+β +β = D, x β = (x µ x µ ) β, G(α,β) = a(α+β) a(α)a(β), a(β) = Γ(β ) π D/ β Γ(β). Graphic representation of STR (reconstruction of Feynman graphs): α x y = (x y) α 0 α+β β α z Operator version of STR: (API, 003) x y = G(α,β) x 0 β α (α+β) ˆp α ˆq (α+β) ˆp β = ˆq β ˆp (α+β) ˆq α!!! where we have used Heisenberg algebra: [ˆq µ, ˆp ν ] = δ µν. y 3 /

4 R-operators and L-operators Any STR is related to a solution R of the Yang-Baxter equation R (u)r 3 (u + v)r (v) = R 3 (v)r (u + v)r 3 (u) ( integrable models, e.g. Zamolodchikov s Fishnet diagram IM). Our aim is to find R which corresponds to the operator STR: ˆp u q (u+v) ˆp v = q v ˆp (u+v) q u. () Eq. () can be written in two equivalent forms ˆp u q (u+v) ˆp v = q v ˆp (u+v) q v ( ), where [q µ k, ˆp j ν ] = iδ kj δ µν and q µ = qµ qµ. Then one can prove that R-operator R (u v) = q (u v +) ˆp (u+ v+) ˆp (u v ) q (u+ v ) End(V V ) where V is the space of conformal fields with conf. dimension and u + = u + D, u = u, v + = v + D, v = v, satisfies YB equation. 4 /

5 The R-operator acts in the tensor product of two representation spaces of conformal algebra conf(r D ) = so(d +, ) Φ (x ) Φ (x ) V V, The meaning of R: it intertwines two representations or R (u v) : V V V V. R (u v) A B = B A R (u v). where A End(V ). 5 /

6 To demonstrate this we construct L-operator L ( ) : V V V V where V is the space of a matrix (e.g., spinor) representation T s of conf(r D ) conf. The L-operator satisfies RLL relations R 3 (u v)(l ( ) ) α β (u)(l( ) 3 ) β γ (v) = (L( ) ) α β (v)(l( ) 3 ) β γ (u)r 3(u v), where (L ( ) ) α β = T s(u(conf)) α β ρ (U(conf)) (L ( ) 3 ) α β = T s(u(conf)) α β ρ (U(conf)) R 3 (u v) = ρ (U(conf)) ρ (U(conf)) and U(conf) enveloping algebra of conf. 6 /

7 R p,q pseudoeuclidean space with the metric g µν = diag(,...,,,..., ). }{{}}{{} p q conf(r p,q ) Lie algebra of the conformal group in R p,q generated by {L µν, P µ, K µ, D} (µ,ν = 0,,...,p + q ): [L µν, L ρσ ] = i (g νρ L µσ + g µσ L νρ g µρ L νσ g νσ L µρ ) [K ρ, L µν ] = i (g ρµ K ν g ρν K µ ), [P ρ, L µν ] = i (g ρµ P ν g ρν P µ ), [D, P µ ] = i P µ, [D, K µ ] = i K µ, [K µ, P ν ] = i (g µν D L µν ), [P µ, P ν ] = 0, [K µ, K ν ] = 0, [L µν, D] = 0. L µν generators for the rotation group SO(p, q) in R p,q, P ν shift generators in R p,q, D dilatation operator, K ν conformal boost generators. 7 /

8 We have the well known isomorphism: and on generators it looks like conf(r p,q ) = so(p +, q + ) L µν = M µν, K µ = M n,µ M n+,µ, P µ = M n,µ + M n+,µ, D = M n,n+, (n = p+q). where M ab (a, b = 0,,...,n+) are generators of so(p+, q + ) [M ab, M dc ] = i(g bd M ac + g ac M bd g ad M bc g bc M ad ), g ab = diag(,...,,,...,,, ). }{{}}{{} p q Then the quadratic Casimir operator for conf(r p,q ) is C = M ab M ab = (L µνl µν + P µ K µ + K µ P µ ) D. 8 /

9 The first main result is that the explicit form of conf(r p,q ) = so(p+, q + )-type L-operator is: L(u +, u ) = u I+ T s(m ab ) ρ(m ab ). where M ab are generators of so(p +, q + ) and ρ(m ab ) = y a y b y b y a. Now we specify the spinor matrix representation T s of the algebra conf(r p,q ) = so(p+, q + ). 9 /

10 Spinor reps T s of conf(r p,q ) = so(p+, q + ) Let n = p+q(= D) be even integer and γ µ (µ = 0,..., n ) be n -dimensional gamma-matrices in R p,q : γ µ γ ν +γ ν γ µ = g µν I, γ n+ α γ 0 γ γ n, α = ( ) q+n(n )/, where α is such that γ n+ = I. Using gamma-matrices γ µ in R p,q one can construct representation T s of conf(r p,q ) = so(p+, q + ) T s (L µν ) = i 4 [γ µ, γ ν ] l µν, T s (K µ ) = γ µ ( γ n+ ) k µ, T s (P µ ) = γ µ (+γ n+ ) p µ, T s (D) = i γ n+ d. 0 /

11 We choose the representation for γ µ in R p,q as: γ µ = ( 0 σµ σ µ 0 ), γ n+ = ( 0 0 ), where σ µ σ ν +σ ν σ µ = g µν, σ µ σ ν +σ ν σ µ = g µν. Thus, the representation T s of conf(r p,q ) is ( i l µν = 4 (σ ) µσ ν σ ν σ µ ) 0 i 0 4 (σ = µσ ν σ ν σ µ ) ( ) ( ) p µ 0 0 = σ µ, k µ 0 σ µ =, d = i Recall that σ µν = (σ µν ) β α, σ µν = (σ µν ) α β, ( σµν 0 0 σ µν ( 0 0 are inequivalent spinor representations of so(p, q) = spin(p, q). ) )., /

12 Any element of conf(r p,q ) in the representation T s is A = i(ω µν l µν + a µ p µ + b µ k µ +βd) = ( β = +iωµν σ µν ib µ ) ( ) σ µ ε ε ia µ σ µ β. +iωµν σ µν ε ε It can be considered as the matrix of parameters ω µν, a µ, b µ, β R. /

13 Diff. representation of conf(r p,q ) = so(p +, q + ) The standard differential representation ρ of conf(r p,q ) can be obtained by the method of induced representations (G. Mack and A. Salam (969)) ρ (P µ ) = i xµ ˆp µ, ρ (D) = x µˆp µ i, ρ (L µν ) = ˆl µν + S µν, ρ (K µ ) = x ν (ˆl νµ + S νµ )+(x ν x ν )ˆp µ i x µ, ˆl µν (x νˆp µ x µˆp ν ), where x µ ˆq µ are coordinates in R p,q, R conformal parameter, S µν = S νµ are spin generators with commutation relations as for ˆl µν and [S µν, x ρ ] = 0 = [S µν,ˆp ρ ]. For the quadratic Casimir operator we have: ρ (C ) = ( S µν S µν ˆl µν ˆl µν) + ( n). The representations ρ and ρ n are contragradient to each other and in particular we have ρ (C ) = ρ n (C ). 3 /

14 In the representation ρ elements of conf(r p,q ) act on the fields Φ(x): ρ(ω µν L µν + a µ P µ + b µ K µ +βd)φ(x) = [( ) ε ε = Tr (Ts (M ε ε ab ) ρ(m )] ab) Φ(x). ( ) ε ε where is the matrix of parameters, and the matrix of ε ε generators is ( ) T s(m ab ) ρ (M ab ) = (T s ρ) Mab M ab = = ( n +S p x, p x S S x x p x+( n ) x, +S+x p Here we introduced p = σµˆp µ = i σµ xµ, x = i σ µ x µ, S = σµν S µν, S = σµν S µν. ), 4 /

15 The action of spin generators S µν on tensor fields of the type (l, l) is [S µν Φ] α α l α α l = (σ µν ) α α Φ α α l αα α l + +(σ µν ) α α l Φ α α l α α l α + +( σ µν ) α α Φ α α α l α α l + +( σ µν ) α l α Φ α α l α α α l. For symmetric representations it is convenient to work with the generating functions Φ(x,λ, λ) = Φ α α l α α l (x)λ α λ α l λ α λ α l, where λ and λ are auxiliary spinors and the action of S µν is given by differential operators over spinors S µν = λσ µν λ + λσ µν λ : [S µν Φ](x,λ, λ) = [ λσ µν λ + λ ] σ µν λ Φ(x,λ, λ), where λσ µν λ = λ α (σ µν ) α β λ β, λ σ µν λ = λ α ( σ µν ) β α λ β. 5 /

16 For 4-dimensional case R p,q = R,3 we have -component Weyl spinors λ, λ and tensor fields Φ α α l α α l (x) are automatically symmetric under permutations of dotted and undotted indices separately. Then for n = 4 we have σ µ = (σ 0,σ,σ,σ 3 ), σ µ = (σ 0, σ, σ, σ 3 ), where σ 0 = I and σ,σ,σ 3 are standard Pauli matrices. Consequently we obtain for the self-dual components of S µν S = ( σµν S µν = λ λ λ λ λ λ λ λ λ λ + λ λ ) and for anti-self-dual components of S µν ( S = σµν S µν = λ λ λ λ λ λ λ λ λ λ + λ λ ) 6 /

17 Consider conf(r p,q ) = so(p+, q + )-type operator: L ( ) (u) L ( ) (u +, u ) = u I+ T s(m ab ) ρ (M ab ) = = ( u + +S p x, x S S x x p x+(u + u ) x, u +S+x p p ), where T s is the spinor representation and ρ is the differential representation of the conformal algebra so(p +, q + ); u + = u + n, u = u, n = p+q, We have used the expression for the polarized Casimir operator T s(m ab ) ρ (M ab ) which was discussed in context of the differential representation of the conformal algebra. 7 /

18 Proposition. The operator L ( ) (u +, u ) satisfies the RLL relation R (u v)(l ( ) ) α β (u)(l( ) ) β γ(v) = = (L ( ) ) α β (v)(l( ) ) β γ(u)r (u v) End(V V V ), with R-operator End(V V ) R (u v) = q (u v +) ˆp (u+ v+) ˆp (u v ) q (u+ v ), if S = 0 = S (for any dimension n = p+q), i.e. in the case of the scalar propagators. 8 /

19 R-matrix and general R-operator Proposition. For two special cases the operator L ( ) (u) satisfies the RLL relation R (u v) L ( ) (u) L ( ) (v) = L ( ) (v) L ( ) (u) R (u v) End(V V V ) with the R-matrix R (u) End(V V), where V is the n -dimensional space of spinor representation T s of conf(r p,q ) and indices, are numbers of spaces V. The two cases are: Dimension n = p+q of the space R p,q is arbitrary and representation ρ of conf(r p,q ) is special and corresponds to the scalars: S = 0 and S = 0. Dimension n = p+q of the space R p,q is fixed by n = 4 and representation ρ of conf(r p,q ) is arbitrary: S 0 and S 0. Remark. The RLL relations look like defining relations for the Yangian Y(spin(p+, q + )) and L ( ) (u) the image of the evaluation repr. of this Yangian. 9 /

20 Let Γ a be n + -dim. gamma-matrices in R p+,q+ (n = p + q) which generate the Clifford algebra with the basis Γ a...a k = ( ) p(s) Γ k! s(a ) Γ s(ak ) (k n+), s S k where p(s) denote the parity of s. The SO(p+, q + )-invariant R-matrix is (it is necessary to take Weyl projection) R(u) = n+ k=0 R k (u) k! Γ a...a k Γ a...a k End(V V), where V is the n + -dimensional space of spinor representation T of SO(p +, q + ). To satisfy the Yang-Baxter equation the functions R k (u) have to obey the recurrent relations (R.Shankar and E.Witten (978), Al.B.Zamolodchikov (98), M.Karowsky and H.Thun (98)) R k+ (u) = u + k u + n k R k(u). 0 /

21 Proposition 3. For any spin S, S and n = p+q = 4 the operator L (,l, l) (u) satisfies the RLL relation R (u v) (L (,l, l ) ) α β (u) (L(,l, l ) ) β γ(v) = = (L (,l, l ) ) α β (v) (L(,l, l ) ) β γ(u) R (u v) End(V V,l, l V,l, l ), with special Yang-Baxter R-operator = [R Φ](x, λ, λ ; x, λ, λ ) = d 4 q d 4 k d 4 y d 4 z e i (q+k) x e i k (y z) q (u v ++) z (u+ v++) y (u v +) k (u+ v +) () Φ(x y, λ z k, λ q y; x z, λ q z, λ y k), where we have used compact notation x = σ µ x µ / x, x = σ µ x µ / x. Remark. The integrable model of the type of Zamolodchikov s Fishnet diagram Integrable System is not known for R given in (). /

22 Green function for two fields of the types (l, l) and ( l,l) in conformal field theory and the solution is well known (Φ(X),Φ(Y)) = (l)!( l)! ( λ(x y)η) l ( ) l λ(x y) η (x y) (4 ). In this Section for simplicity we shall use compact notation x µ x = σ µ x ; x = σ x µ µ x (3) /