Kac modules and boundary Temperley-Lieb algebras for logarithmic - PowerPoint PPT Presentation

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Kac modules and boundary Temperley-Lieb algebras for logarithmic minimal models Alexi Morin-Duchesne Universit´ e Catholique de Louvain Florence, 28/05/2015 Joint work with Jørgen Rasmussen and David Ridout arXiv:1503.07584 [hep-th]

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Outline • Loop models with boundary seams • Relation with the one-boundary Temperley-Lieb algebra • Virasoro Kac modules • Scaling limit of the loop models

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Dense loop model • Configuration of the dense loop model with a boundary seam: • Fugacity of closed loops: β = 2 cos λ λ = π ( p ′ − p ) p , p ′ ∈ Z + • Roots of unity: p < p ′ p ′

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Temperley-Lieb algebra TL n ( β ) Generators A connectivity ... ... ... I = e j = a = n j j + 1 n 1 2 3 1 = e 1 e 2 e 4 e 3 • Multiplication is by vertical concatenation: = β 2 a 3 = β 2 a 1 a 2 = Algebraic definition � � TL n ( β ) = I , e j ; j = 1 , . . . , n − 1 ( e j ) 2 = β e j e j e j ± 1 e j = e j e i e j = e j e i ( | i − j | > 1 )

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Temperley-Lieb algebra TL n ( β ) Generators A connectivity ... ... ... I = e j = a = n j j + 1 n 1 2 3 1 = e 1 e 2 e 4 e 3 • Multiplication is by vertical concatenation: ... ... ( e j ) 2 = = β ... ... = β e j ... ... j j + 1 n 1 1 j j + 1 n Algebraic definition � � TL n ( β ) = I , e j ; j = 1 , . . . , n − 1 ( e j ) 2 = β e j e j e j ± 1 e j = e j e i e j = e j e i ( | i − j | > 1 )

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Temperley-Lieb algebra TL n ( β ) Generators A connectivity ... ... ... I = e j = a = n j j + 1 n 1 2 3 1 = e 1 e 2 e 4 e 3 • Multiplication is by vertical concatenation: e j e j + 1 e j = ... ... = ... ... = e j j j + 1 n 1 j j + 1 n 1 Algebraic definition � � TL n ( β ) = I , e j ; j = 1 , . . . , n − 1 ( e j ) 2 = β e j e j e j ± 1 e j = e j e i e j = e j e i ( | i − j | > 1 )

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Transfer tangles with boundary seams • D ( u , ξ ) is an element of TL n + k ( β ) : (Pearce, Rasmussen, Zuber 2006) k . . . . . . . . . u u u u + ξ k u + ξ 2 u + ξ 1 D ( u , ξ ) = . . . . . . . . . u u u u − ξ k u − ξ 2 u − ξ 1 k � �� � n s k ( u ) = sin ( u + k λ ) = s 1 (− u ) + s 0 ( u ) ξ j = ξ + j λ u sin λ − 1 • Projectors: = = 1 2 β � � � � β 1 = − + + + 3 β 2 − 1 β 2 − 1 • YBE + BYBE [ D ( u , ξ ) , D ( v , ξ )] = 0 →

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Transfer tangles with boundary seams • D ( u , ξ ) is an element of TL n + k ( β ) : (Pearce, Rasmussen, Zuber 2006) 4 4 s k ( u ) = sin ( u + k λ ) = s 1 (− u ) + s 0 ( u ) ξ j = ξ + j λ u sin λ − 1 • Projectors: = = 1 2 β � � � � β 1 = − + + + 3 β 2 − 1 β 2 − 1 • YBE + BYBE [ D ( u , ξ ) , D ( v , ξ )] = 0 →

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Transfer tangles with boundary seams • D ( u , ξ ) is an element of TL n + k ( β ) : (Pearce, Rasmussen, Zuber 2006) 4 β 2 4 s k ( u ) = sin ( u + k λ ) = s 1 (− u ) + s 0 ( u ) ξ j = ξ + j λ u sin λ − 1 • Projectors: = = 1 2 β � � � � β 1 = − + + + 3 β 2 − 1 β 2 − 1 • YBE + BYBE [ D ( u , ξ ) , D ( v , ξ )] = 0 →

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Hamiltonian tangle � � • The Hamiltonian tangle H is obtained by taking d D ( u ,ξ ) � u = 0 : du n − 1 � 1 ( k ) ( k ) H = − E + s 0 ( ξ ) s k + 1 ( ξ ) E n j j = 1 where ... ... ... ( k ) E = ( j = 1 , . . . , n − 1 ) k j ... 1 j j + 1 n n + 1 n + k ... k = U k − 1 ( β ... ... ... ( k ) E 2 ) n k ... 1 2 n n + 1 n + k • U k ( x ) are Chebyshev polynomials of the second kind: U 0 ( β U 1 ( β U 2 ( β 2 ) = β 2 − 1 , U 3 ( β 2 ) = β ( β 2 − 2 ) , 2 ) = 1 , 2 ) = β, . . .

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Standard modules • Definition: V d n : vector space generated by link patterns n : number of nodes d : number of defects (vertical segments) Examples: � � V 0 6 = span , , , , � � V 4 6 = span , , , , • TL n ( β ) action on V d n : = β = 0 • Defines one representation of TL n ( β ) for each d .

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Lattice Kac modules • Projector – half-arc annihilation relation: = 0 k − 1 − β Example: = = = 0 2 β β

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Lattice Kac modules • Projector – half-arc annihilation relation: = 0 k − 1 − β Example: = = = 0 2 β β • D ( u , ξ ) and H act trivially on a subspace of V d n + k : 2 u u u u u + ξ 2 u + ξ 1 = 0 u u u u u − ξ 2 u − ξ 1 2

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Lattice Kac modules • Projector – half-arc annihilation relation: = 0 k − 1 − β Example: = = = 0 2 β β • D ( u , ξ ) and H act trivially on a subspace of V d n + k : 2 u u u u u + ξ 2 u + ξ 1 = 0 u u u u u − ξ 2 u − ξ 1 2 • Lattice Kac module K d n , k : quotient of V d n + k by the trivial subspace Examples for k = 2: � � � � � K 0 4 , 2 = span span , , , , , • Hamiltonians = realisations of H in K d n , k

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Lattice Kac modules • Projector – half-arc annihilation relation: = 0 k − 1 − β Example: = = = 0 2 β β • D ( u , ξ ) and H act trivially on a subspace of V d n + k : 2 u u u u u + ξ 2 u + ξ 1 = 0 u u u u u − ξ 2 u − ξ 1 2 • Lattice Kac module K d n , k : quotient of V d n + k by the trivial subspace Examples for k = 2: � � � � � K 4 4 , 2 = span span , , , , • Hamiltonians = realisations of H in K d n , k

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion ( 1 ) One-boundary TL algebra: TL n Generators A connectivity ... ... ... ... I = e j = e n = b = n j j + 1 n n 1 2 3 1 1 = e 2 e 6 e 3 e 5 e 4 e 6 • Multiplication is again by vertical concatenation: ( 2 ) ( 1 ) b 1 b 2 = = ββ 1 = ββ 1 b 3 , Algebraic definition � � ( 1 ) TL n ( β, β 1 , β 2 ) = I , e j ; j = 1 , . . . , n ( e j ) 2 = β e j e j e j ± 1 e j = e j e i e j = e j e i ( | i − j | > 1 ) e 2 n = β 2 e n e n − 1 e n e n − 1 = β 1 e n − 1 e i e n = e n e i ( i < n − 1 )

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion ( 1 ) One-boundary TL algebra: TL n Generators A connectivity I = ... e j = ... ... e n = ... b = n j j + 1 n n 1 2 3 1 1 = e 2 e 6 e 3 e 5 e 4 e 6 • Multiplication is again by vertical concatenation: ( 2 ) ( 1 ) b 1 b 2 = = β 2 = β 2 b 3 , ( 2 ) ( 1 ) Algebraic definition � � ( 1 ) TL n ( β, β 1 , β 2 ) = I , e j ; j = 1 , . . . , n ( e j ) 2 = β e j e j e j ± 1 e j = e j e i e j = e j e i ( | i − j | > 1 ) e 2 n = β 2 e n e n − 1 e n e n − 1 = β 1 e n − 1 e i e n = e n e i ( i < n − 1 )

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Boundary seam algebras � � • Definition: ( k ) , E ( k ) B n , k = I j ; j = 1 , . . . , n ( k ) = ... ... ( k ) I ... E = ... ... ... ... j ... = U k − 1 ( β ( k ) ... ... ... E 2 ) n ... • D ( u , ξ ) and H are elements of B n , k . • Algebraic relations, with β 1 = U k ( β β 2 = U k − 1 ( β 2 ) , 2 ) : j ) 2 = β E ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( E E j E j ± 1 E = E E i E = E j E ( | i − j | > 1 ) j j j j i n ) 2 = β 2 E ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( E E n − 1 E n E n − 1 = β 1 E E i E = E n E ( i < n − 1 ) n n n − 1 i • These algebraic relations are well-defined for all β . • B n , k is a quotient of TL ( 1 ) n . Its generators satisfy more relations.

Loop model Boundary algebras Virasoro modules Scaling limit Conclusion Extra relations: generic case • Extra relations for β generic: � k = 1: ( e n e n − 1 − 1 ) e n = 0 � k = 2: ( e n e n − 1 e n − 2 − β e n − 2 + 1 )( e n e n − 1 − β ) e n = 0 � k = 3: e n e n − 1 e n − 2 e n − 3 e n e n − 1 e n − 2 e n e n − 1 e n + lower order terms = 0 � � Polynomial of degree ( k + 1 )( k + 2 ) � Any k : in the e j = 0 2 • These are the full set of algebraic relations defining B n , k . • The lattice Kac modules K d n , k are really modules over B n , k .