Discrete parafermions and quantum-group symmetries Yacine Ikhlef - PowerPoint PPT Presentation

Discrete parafermions and quantum-group symmetries Yacine Ikhlef LPTHE (CNRS/Paris-6) joint work with R. Weston (Edinburgh), M. Wheeler (Melbourne), P. Zinn-Justin (LPTHE). Florence, 13/05/2015

Outline 1. Introduction 2. The Bernard-Felder construction 3. Mapping to loop models

1. Introduction

Discretely holomorphic functions ◮ Discrete function: F ( z ) on midpoints of square lattice L z 3 z 2 z 4 z 1 ◮ Discrete “Cauchy-Riemann” equation: i π 4 F ( z 1 ) − e − i π i π 4 F ( z 3 ) + e − i π 4 F ( z 2 ) − e 4 F ( z 4 ) = 0 e � ◮ Short-hand notation: F ( z ) δ z = 0 ⋄

Loop models in Statistical Mechanics The Temperley-Lieb loop model ◮ Plaquette configurations: x y ◮ Lattice configurations: W ( C ) = x N x ( C ) y N y ( C ) n N ℓ ( C ) ◮ Boltzmann weights: � ◮ Partition function: Z = W ( C ) config . C

Loop models in Statistical Mechanics Correlation functions ◮ Averaging on Boltzmann weights: � � f ( C ) � := 1 W ( C ) f ( C ) . Z C ◮ Two-leg correlation function: � G ( z 1 , z 2 ) := 1 W ( C ) Z C | z 1 , z 2 ∈ same loop ◮ Phases in scaling limit: ◮ Non-critical phase: G ( z 1 , z 2 ) ∼ exp( −| z 1 − z 2 | /ξ ) ◮ Critical phase: G ( z 1 , z 2 ) ∼ | z 1 − z 2 | − 2 X 2 ◮ “Coulomb-gas” studies ⇒ TL model is critical for 0 < n ≤ 2.

Discretely holomorphic observables in loop models b z a ◮ Pick a pair of boundary points ( a , b ) − → define BC ab . ◮ Define correlation function: � 1 W ( C ) e i s θ a → z ( C ) F s ( z ) := Z ab C | z ∈ open path [ θ a → z := winding angle of red arc from a to z ] � ◮ Theorem: if n = 2 sin π s then ∀⋄ ∈ Ω , F s ( z ) δ z = 0. 2 ⋄

Algebraic structure behind discrete holomorphicity? ◮ Discretely holomorphic observables like F s exist in various models: TL, O ( n ), Z N clock models . . . ◮ Rhombic lattice ⇒ additional parameter α z 3 z 2 α α z 4 z 1 Modified Cauchy-Riemann equation: e − i α i α 2 F ( z 2 ) − e − i α i α 2 F ( z 1 )+ e 2 F ( z 3 ) − e 2 F ( z 4 ) = 0 ( CR α ) ◮ Observations : 1. F s satisfies CR α when W ≡ integrable Boltzmann weights 2. α ≡ spectral parameter ◮ Q: general relation discrete holomorphicity ↔ integrability?

Discrete holomorphicity in Physics and Mathematics ◮ [Dotsenko,Polyakov 88] : Linear relations for fermions in Ising ◮ [Smirnov 01–06] : Conf. inv. for interfaces in perco+Ising ◮ [Cardy,Riva,Rajabpour,YI 06–09] : Discr. holo. in various lattice models, obs. relation to integrability ◮ [Smirnov,Chelkak,Hongler,Izyurov,Kyt¨ ol¨ a 09–12] : Scaling limit of interfaces+corr. func. in Ising ◮ [Duminil-Copin,Smirnov 10] : Proof of connectivity constant for SAW on honeycomb ◮ [Beaton,de Gier,Guttmann,Jensen 11–12] : Critical boundary parameter for SAW on honeycomb ◮ [Fendley 12] : Discr. holo. from topological QFT ◮ [Alam,Batchelor 12] : CR eq ↔ star-triangle in Z N models ◮ [Hongler,Kyt¨ ol¨ a,Zahabi 12] : Discr. holo. for non-local currents in Ising, transfer-matrix formalism

2. The Bernard-Felder construction

Hopf algebras Bi-algebra structure � A ⊗ A → A ◮ Product m : a ⊗ b �→ a . b � � A → A ⊗ A ∆ ◮ Coproduct ∆ : a �→ � i a ′ i ⊗ a ′′ i i a ′ a ′′ a i i ◮ ∆( a . b ) = ∆( a ) . ∆( b ) , ∆( a + λ b ) = ∆( a ) + λ ∆( b ) ◮ (∆ ⊗ id ) ◦ ∆ = ( id ⊗ ∆) ◦ ∆ ◮ Example: enveloping algebra of a Lie algebra g ◮ g Lie algebra, with bracket [ X a , X b ] = i f abc X c ◮ A := U ( g ) = span(words on alphabet { X a } ) ◮ bracket ≡ commutator ([ a , b ] = ab − ba ) ◮ Trivial coproduct ∆( X a ) = X a ⊗ 1 + 1 ⊗ X a

Hopf algebras Tensor-product representations ◮ V finite-dimensional vector space Map π : A → End ( V ) is a representation of A iff: ◮ π is linear and surjective, ◮ π is a morphism: π ( ab ) = π ( a ) π ( b ). ◮ Coproduct = tool to construct higher-dim. representations: � � a ′ i ⊗ a ′′ π 1 ( a ′ i ) ⊗ π 2 ( a ′′ ∆( a ) = − → π 12 ( a ) := i ) i i i � ∆ L − 1 . . . ◮ Iterate: i a (1) a (2) a (3) a ( L ) a i i i i ◮ Example: A = U ( g ), for a Lie algebra g L � π ( L ) ( X a ) = 1 ⊗ · · · ⊗ 1 ⊗ π ( X a ) ⊗ 1 ⊗ · · · ⊗ 1 ↑ m =1 m − th

Hopf algebras The R -matrix ◮ The two representations V 1 ⊗ V 2 and V 2 ⊗ V 1 are isomorphic. ◮ Intertwiner R 12 : V 1 ⊗ V 2 → V 2 ⊗ V 1 such that: ∀ a ∈ A , R 12 π 12 ( a ) = π 21 ( a ) R 12 ◮ Expand coproduct [ π 12 ( a ) = � i π 1 ( a ′ i ) ⊗ π 2 ( a ′′ i )]: V 2 V 1 V 2 V 1 a ′ a ′′ � � i i R 12 R 12 = i i a ′ a ′′ i i V 1 V 2 V 1 V 2 ◮ Consistency condition = Yang-Baxter equation: ( R 23 ⊗ 1 ) . ( 1 ⊗ R 13 ) . ( R 12 ⊗ 1 ) = ( 1 ⊗ R 12 ) . ( R 13 ⊗ 1 ) . ( 1 ⊗ R 23 )

Non-local conserved currents [Bernard-Felder, 91] ◮ Generators of A : { J 1 , J 2 . . . } and { µ 1 , µ 2 . . . } . Assume the coproduct of A has the following form: ∆ ∆( J k ) = J k ⊗ 1 + µ k ⊗ J k + ∆ ∆( µ k ) = µ k ⊗ µ k ◮ Iteration of coproduct ⇒ “conserved charges”: � L Q k := ∆ L − 1 ( J k ) = µ k ⊗ · · · ⊗ µ k ⊗ J k ⊗ 1 ⊗ · · · ⊗ 1 ↑ m =1 m ◮ Non-local currents: ψ k ( m ) := µ k ⊗ · · · ⊗ µ k ⊗ J k ⊗ 1 ⊗ · · · ⊗ 1 ↑ m ψ k ( m ) = . . . V 1 V m V L

Commutation relations ◮ From intertwining relations [ R 12 π 12 ( a ) = π 21 ( a ) R 12 ]: ◮ For a = J k : + = + ◮ For a = µ k : = ◮ Transfer matrix: T = V V ′ V V ′ . . . V V ′ ◮ Conservation laws: T .π ( L ) ( a ) = π ( L ) ( a ) . T ∀ a ∈ A ,

The affine quantum group A = U q ( � s ℓ 2 ) ◮ Generators: E 0 , E 1 , F 0 , F 1 , T 0 , T 1 { E 0 , E 1 , F 0 , F 1 } =raising/lowering ops, { T 0 , T 1 } =diag. ops. ◮ Product rules: T i − T − 1 i [ T 0 , T 1 ] = 0 [ E i , F j ] = δ ij q − q − 1 = q 2( − 1) δ ij E j = q 2( − 1) δ ij +1 F j T i E j T − 1 T i F j T − 1 i i (+ higher order rules . . . ) ◮ Coproduct rules: ∆( F i ) = F i ⊗ T − 1 ∆( E i ) = E i ⊗ 1 + T i ⊗ E i + 1 ⊗ F i i ∆( T i ) = T i ⊗ T i ◮ Introduce ¯ ∆(¯ E i ) = ¯ E i ⊗ 1 + T i ⊗ ¯ E i := qT i F i ⇒ E i ◮ BF structure: { J k } = { E 0 , E 1 , ¯ E 0 , ¯ E 1 } { µ k } = { T 0 , T 1 } .

Evaluation representations of A = U q ( � s ℓ 2 ) ◮ Representations are labelled by a complex number u Explicit form: � � � � � �  u − 1 q − 1  0 0 0 0  ¯  E 0 �→ E 0 �→ T 0 �→    u 0 0 0 0 q  π u : � � � � � �     0 u 0 0 q 0  ¯  E 1 �→ E 1 �→ T 1 �→  u − 1 q − 1 0 0 0 0 ◮ Intertwiner: R ( u / v ) π u , v = π v , u R ( u / v )   [ qu / v ] 0 0 0   [ z ] = z − z − 1 0 [ u / v ] 1 0   R ( u / v ) =  ,  0 1 [ u / v ] 0 q − q − 1 0 0 0 [ qu / v ]

Application to the six-vertex model ◮ Use basis for V u : {↑ , ↓} . Plaquette configurations: ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ◮ Boltzmann weights:   ω 1 0 0 0   0 ω 5 ω 4 0   R 6V =   0 ω 3 ω 6 0 0 0 0 ω 2 ◮ When R 6V ≡ R U q ( � s ℓ 2 ) , the 6V model is integrable.

3. Mapping to loop models

From the TL model to the 6V model [Baxter, Kelland, Wu 73] ◮ Orient each loop independently: = + e 2 i πλ e − 2 i πλ n = 2 cos 2 πλ ◮ Partition function: � x N x ( C ) y N y ( C ) e 2 i πλ [ N + ℓ ( C ) − N − ℓ ( C )] Z = C ◮ Distribute phase factors locally: α α e i αλ e − i αλ

From the TL model to the 6V model (2) ◮ Vertex configurations: + + ◮ Six-vertex weights arising from loop model: � ω 5 = e +2 i λα x + e − 2 i λ ( π − α ) y ω 1 = ω 2 = x , ω 3 = ω 4 = y , ω 6 = e − 2 i λα x + e +2 i λ ( π − α ) y ◮ Set q = − e 2 i λπ , w = e − 2 i λα : ω 1 = ω 2 = [ qw ] , ω 3 = ω 4 = [ w ] ⇒ ω 5 = ω 6 = 1 .

Conserved currents in the 6V model � ∆( E 0 ) = E 0 ⊗ 1 + T 0 ⊗ E 0 ⇒ BF current ψ 0 ◮ ∆( T 0 ) = T 0 ⊗ T 0 ψ 0 ( m ) = T 0 ⊗ T 0 ⊗ · · · ⊗ T 0 ⊗ E 0 ⊗ 1 ⊗ · · · ⊗ 1 ↑ m − th ◮ Commutation with R -matrix ⇒ linear relation: ψ 0 ( z 1 ) − ψ 0 ( z 2 ) − ψ 0 ( z 3 ) + ψ 0 ( z 4 ) = 0 . z 3 z 2 z 4 z 1 V ′ V ◮ Similar construction for E 1 , ¯ E 0 , ¯ E 1 → ψ 1 , ¯ ψ 0 , ¯ ψ 1 .

Mapping of conserved currents What is the meaning of � ψ 0 ( z ) � in terms of loops? γ b a ψ 0 ( z ) cannot sit alone on a closed loop ψ 0 = u × = 0 � ⇒ � ψ 0 ( z ) � = u W ( C ) × ( phase factor ) Z C | z ∈ γ

Mapping of conserved currents (2) Identification of phase factors q = e i π (2 λ − 1) ◮ θ b → z = θ a → z + π , b a ◮ phase factor: θ a → z + θ b → z − π e i λ ( θ a → z + θ b → z ) = A e i (4 λ − 1) θ a → z × q 2 π ↑ ↑ turns T 0 ⊗ · · · ⊗ T 0 � ◮ ⇒ � ψ 0 ( z ) � = uA W ( C ) e i (4 λ − 1) θ a → z = uA × F s ( z ) Z C | z ∈ γ spin: s = 4 λ − 1 (remember Theorem in Intro)