Logarithmic Minimal Models, W -Extended Fusion and Verlinde Formulas 24 September 2008 GGI Florence Paul A. Pearce Department of Mathematics and Statistics, University of Melbourne • PAP, J.Rasmussen, J.-B.Zuber, Logarithmic minimal models , J.Stat.Mech. P11017 (2006) • J.Rasmussen, PAP, Fusion algebras of logarithmic minimal models , J.Phys. A40 13711–33 (2007) • PAP, J.Rasmussen, P.Ruelle, Integrable boundary conditions and W -extended fusion of the logarithmic minimal models LM (1 , p ), arXiv:0803.0785, J. Phys. A (2008) • J.Rasmussen, PAP, W -extended fusion of critical percolation , arXiv:0804.4335, J. Phys. A (2008) • J.Rasmussen, W -extended logarithmic minimal models , arXiv:0805.299, Nucl. Phys. B (2008) • PAP, J.Rasmussen, Verlinde formula and the projective Grothendieck ring of logarithmic minimal models , in preparation (2008) 0-1
Some Background Broadbent & Hammersley: Percolation 1957– de Gennes, des Cloizeaux: Polymers 1972– Saleur, Duplantier: Conformal theory of polymers, percolation 1986– Gurarie: Logarithmic operators in CFT 1993– Kausch: Symplectic fermions 1995– Rozansky, Read, Saleur, Schomerus, . . . Supergroup Approach to Log CFT 1992– Gaberdiel, Kausch, Flohr, Runkel, Feigin et al, Mathieu, Ridout,. . . Algebraic Log CFT 1996– Pearce, Rasmussen, Ruelle, Zuber: Lattice Approach to Log CFT 2006– Lattice Approach: • Statistical systems with local degrees of freedom yield rational CFTs. • Polymers, percolation and related lattice models do not have local degrees of freedom only nonlocal degrees of freedom (polymers, connectivities, SLE paths) and are associated with Logarithmic CFTs . . . nonlocal logarithmic lattice degrees of ⇒ CFT freedom 0-2
Logarithmic Minimal Models LM ( p, p ′ ) • Face operators defined in planar Temperley-Lieb algebra (Jones 1999) = sin( λ − u ) + sin u X j ( u ) = sin( λ − u ) I + sin u X ( u ) = ; sin λ e j u sin λ sin λ sin λ λ = ( p ′ − p ) π 1 ≤ p < p ′ coprime integers , = crossing parameter p ′ u = spectral parameter , β = 2 cos λ = fugacity of loops Planar Algebra (Temperley-Lieb Algebra) YBE Nonlocal Statistical Mechanics (Yang-Baxter Integrable Link Models) continuum lattice limit realization Logarithmic CFTs Nonlocal Degrees of Freedom (Logarithmic Minimal Models) 0-3
Polymers and Percolation on the Lattice λ = π • Critical Dense Polymers: ( p, p ′ ) = (1 , 2), 2 κ = 4 p ′ d SLE path = 2 − 2∆ p,p ′ − 1 = 2 , = 8 p ∆ 1 , 1 = 0 lies outside rational M (1 , 2) Kac table β = 0 ⇒ no loops ⇒ space filling dense polymer λ = π u = λ 2 = π • Critical Percolation: ( p, p ′ ) = (2 , 3), 3, (isotropic) 6 κ = 4 p ′ path = 2 − 2∆ p,p ′ − 1 = 7 d SLE 4 , = 6 p ∆ 2 , 2 = 1 8 lies outside rational M (2 , 3) Kac table Bond percolation on the blue square lattice: Critical probability = p c = sin( λ − u ) = sin u = 1 2 β = 1 ⇒ local stochastic process 0-4
Boundary Yang-Baxter Equation • The Boundary Yang-Baxter Equation (BYBE) is the equality of boundary 2-tangles v u u − v = λ − u − v λ − u − v u − v u v • For r, s = 1 , 2 , 3 , . . . , the ( r, s ) = ( r, 1) ⊗ (1 , s ) BYBE solution is built as the fusion product of ( r, 1) and (1 , s ) integrable seams acting on the vacuum (1 , 1) triangle: ( r,s ) ( r, 1) (1 ,s ) (1 , 1) = ⊗ ⊗ . . − u − ξ ρ 2 − u − ξ ρ − u − ξ 0 − − 3 = u u − ξ ρ u − ξ ρ u − ξ 1 − 1 − 2 . . � �� � � �� � ρ − 1 columns s − 1 columns • The column inhomogeneities are: ξ k = ( k + k 0 + 1 2 ) λ • There is at least one choice of the integers ρ and k 0 for each r . 0-5
Double-Row Transfer Matrices • For a strip with N columns, the double-row transfer “matrix” is the N -tangle . . . . λ − u λ − u λ − u u D ( u ) = u u u . . . . • Using the Yang-Baxter (YBE) and Boundary Yang-Baxter Equations (BYBE) in the planar Temperley-Lieb (TL) algebra, it can be shown that, for any ( r, s ), these commute and are crossing symmetric D ( u ) D ( v ) = D ( v ) D ( u ) , D ( u ) = D ( λ − u ) • Multiplication is vertical concatenation of diagrams, equality is the equality of N -tangles. • In the case of one non-trivial boundary condition, the transfer matrices are found to be diagonalizable. For fusion, we take non-trivial boundary conditions on the left and right ( r ′ , s ′ ) ⊗ ( r, s ). In this case, the transfer matrices can exhibit Jordan cells and are not in general diagonalizable. • It is necessary to act on a vector space of states to obtain matrix representatives and spectra . 0-6
Planar Link Diagrams • The planar N -tangles act on a vector space V N of planar link diagrams . The dimension of V N is given by Catalan numbers. For N = 6, there is a basis of 5 link diagrams: 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 • The first link diagram is the reference state. Other states are generated by the action of the TL generators by concatenation from below = = etc. β 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 • The action of the TL generators on the states is nonlocal. It leads to matrices with entries 0 , 1 , β that represent the TL generators. For N = 6, the action of e 1 and e 2 on V 6 is 0 1 0 1 0 0 0 0 0 β 0 β 0 1 0 0 0 0 0 0 e 1 = , e 2 = , 0 0 0 0 0 1 0 0 0 β etc. 0 0 0 0 0 0 1 0 β 1 0 0 0 0 0 0 0 0 0 0 • The transfer matrices are built from the TL generators. 0-7
Defects • More generally, the vector space of states V ( ℓ ) can contain ℓ defects : N N = 4 , ℓ = 2 : 1 2 3 4 1 2 3 4 1 2 3 4 • The ℓ defects can be closed on the right or the left. In this way, the number of defects propagating in the bulk is controlled by the boundary conditions. In particular, for (1 , s ) boundary conditions, the ℓ = s − 1 defects simply propagate along a boundary. • Defects in the bulk can be annihilated in pairs but not created under the action of TL = etc. 1 2 3 4 5 6 1 2 3 4 5 6 • The transfer matrices are thus block-triangular with respect to the number of defects. 0-8
Dense Polymer Kac Table • Central charge: ( p, p ′ ) = (1 , 2) . . . . . . . . . . . . ... . . . . . . s c = 1 − 6( p − p ′ ) 2 = − 2 63 35 15 3 − 1 3 · · · 10 pp ′ 8 8 8 8 8 8 · · · 9 6 3 1 0 0 1 • Infinitely extended Kac table of conformal weights: 35 15 3 − 1 3 15 · · · 8 8 8 8 8 8 8 ( p ′ r − ps ) 2 − ( p − p ′ ) 2 · · · 7 3 1 0 0 1 3 ∆ r,s = 4 pp ′ 15 3 − 1 3 15 35 · · · 6 (2 r − s ) 2 − 1 8 8 8 8 8 8 = , r, s = 1 , 2 , 3 , . . . 8 · · · 5 1 0 0 1 3 6 • Kac representation characters: 3 − 1 3 15 35 63 · · · 4 8 8 8 8 8 8 · · · χ r,s ( q ) = q − c/ 24 q ∆ r,s (1 − q rs ) 3 0 0 1 3 6 10 � ∞ n =1 (1 − q n ) − 1 3 15 35 63 99 · · · 2 8 8 8 8 8 8 • Irreducible Representations: · · · 1 0 1 3 6 10 15 There is an irreducible representation for r 1 2 3 4 5 6 each distinct conformal weight. The Kac representations which happen to be irre- ducible are marked with a red quadrant. 0-9
Critical Percolation Kac Table • Central charge: ( p, p ′ ) = (2 , 3) . . . . . . . . . . . . ... . . . . . . s c = 1 − 6( p − p ′ ) 2 = 0 65 21 1 · · · 10 12 5 1 pp ′ 8 8 8 28 143 10 35 1 − 1 · · · 9 • Infinitely extended Kac table 3 24 3 24 3 24 of conformal weights: 33 5 1 · · · 8 7 2 0 8 8 8 ( p ′ r − ps ) 2 − ( p − p ′ ) 2 21 1 5 · · · 7 5 1 0 8 8 8 ∆ r,s = 4 pp ′ 10 35 1 − 1 1 35 · · · 6 (3 r − 2 s ) 2 − 1 3 24 3 24 3 24 = , r, s = 1 , 2 , 3 , . . . 5 1 21 24 · · · 5 2 0 1 8 8 8 • Kac representation characters: 1 5 33 · · · 4 1 0 2 8 8 8 1 − 1 1 35 10 143 · · · χ r,s ( q ) = q − c/ 24 q ∆ r,s (1 − q rs ) 3 3 24 3 24 3 24 � ∞ n =1 (1 − q n ) 1 21 65 · · · 2 0 1 5 8 8 8 • Irreducible Representations: 5 33 85 · · · 1 0 2 7 8 8 8 There is an irreducible representation for r 1 2 3 4 5 6 each distinct conformal weight. The Kac representations which happen to be irre- ducible are marked with a red quadrant. 0-10
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