Kac boundary conditions of the logarithmic minimal models

Pearce, Paul A.; Tartaglia, Elena; Couvreur, Romain

doi:10.1088/1742-5468/2015/01/p01018

Cited by 5 publications

(27 citation statements)

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“…In particular, in this paper, we only consider boundary conditions on the strip corresponding to conformal Kac modules with highest weight ∆ 1,s . However, the methods of this paper should extend to the more general Kac modules with highest weight ∆ r,s , which are realised on the lattice by including a seam on the boundary [37,63,64]. These methods should also extend to boundary conditions described by the one-and two-boundary Temperley-Lieb algebra [38,39,115,116].…”

Section: Resultsmentioning

confidence: 99%

“…The value d SLE path = 7 4 was proved by Beffara [55]. The incorporation of critical dense polymers LM (1,2) [38,[56][57][58][59][60] and critical percolation LM (2,3) into the framework of the family of logarithmic minimal models LM(p, p ′ ) [37,39,41,[61][62][63][64] establishes that these models are Yang-Baxter integrable. The transfer matrices of the logarithmic minimal models are built from so called transfer tangles of the planar Temperley-Lieb algebra [14,15], which we respectively denote by D(u) and T (u) for the boundary and the periodic cases.…”

Section: Introductionmentioning

confidence: 92%

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Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations

Morin-Duchesne¹,

Klümper²,

Pearce³

2017

J. Stat. Mech.

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Using the planar Temperley-Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as LM(2, 3) in the Yang-Baxter integrable family of logarithmic minimal models LM(p, p ′ ). We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite Y -system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a D 3 Dynkin diagram. Following the methods of Klümper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge c = 0 and conformal weights ∆ 1,s for s ∈ Z 1 with ∆ r,s = (3r − 2s) 2 − 1 /24. For the periodic case, the finite-size corrections agree with the conformal weights ∆ 0,s , ∆ 1,s with s ∈ 1 2 Z 0 . These are obtained analytically using Rogers dilogarithm identities. We incorporate all finite excitations by formulating empirical selection rules for the patterns of zeros of all the eigenvalues of the standard modules. We thus obtain the conformal partition functions on the cylinder and the modular invariant partition function (MIPF) on the torus. By applying q-binomial and q-Narayana identities, it is shown that our refined finitized characters on the strip agree with those of Pearce, Rasmussen and Zuber. For percolation on the torus, the MIPF is a non-diagonal sesquilinear form in affine u(1) characters given by the u(1) partition function Z 2,3 (q) = Z

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 92%

Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations

Morin-Duchesne¹,

Klümper²,

Pearce³

2017

J. Stat. Mech.

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“…together with the crossing symmetry κ(λ − u, ξ) = κ(u, ξ). The last inversion relation is related to κ PTC ρ (u, ξ) and (3.21) of [28] by the trivial rescaling…”

Section: Bulk and Boundary Free Energiesmentioning

confidence: 99%

“…The O(1) inversion relation for the boundary free energy f 0 (u) of the left Kac vacuum has been similarly solved in [28] −f 0 (u) = log κ 0 (u) = log cos u cos(λ − u)…”

Section: Bulk and Boundary Free Energiesmentioning

confidence: 99%

“…In the logarithmic context, it is no longer true in general that each representation is conjugate to a conformal boundary condition. For the logarithmic minimal models, however, it is known [17,[26][27][28][29] that there are a countably infinite number of Virasoro Kac representations with conjugate (r, s) boundary conditions organized into infinitely extended Kac tables. The central charges and conformal weights, given by the Kac formula, are…”

Section: Introductionmentioning

confidence: 99%

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Logarithmic minimal models with Robin boundary conditions

2016

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We consider general logarithmic minimal models LM(p, p ′ ), with p, p ′ coprime, on a strip of N columns with the (r, s) Robin boundary conditions introduced by Pearce, Rasmussen and Tipunin. On the lattice, these models are Yang-Baxter integrable loop models that are described algebraically by the one-boundary Temperley-Lieb algebra. The (r, s) Robin boundary conditions are a class of integrable boundary conditions satisfying the boundary Yang-Baxter equations which allow loop segments to either reflect or terminate on the boundary. The associated conformal boundary conditions are organized into infinitely extended Kac tables labelled by the Kac labels r ∈ Z and s ∈ N. The Robin vacuum boundary condition, labelled by (r, s − 1 2 ) = (0, 1 2 ), is given as a linear combination of Neumann and Dirichlet boundary conditions. The general (r, s) Robin boundary conditions are constructed, using fusion, by acting on the Robin vacuum boundary with an (r, s)-type seam consisting of an r-type seam of width w columns and an s-type seam of width d = s − 1 columns. The r-type seam admits an arbitrary boundary field which we fix to the special value ξ = − λ 2 where λ = (p ′ −p)π p ′ is the crossing parameter. The s-type boundary introduces d defects into the bulk. We consider the commuting double-row transfer matrices and their associated quantum Hamiltonians and calculate analytically the boundary free energies of the (r, s) Robin boundary conditions. Using finite-size corrections and sequence extrapolation out to system sizes N + w + d ≤ 26, the conformal spectrum of boundary operators is accessible by numerical diagonalization of the Hamiltonians. Fixing the parity of N for r = 0 and restricting to the ground state sequences w = |r|p ′ p , r ∈ Z with the inverse r = (−1) N +w+d pw p ′ , we find that the conformal weights take the values ∆ p,p ′ r,s− 1 2 where ∆ p,p ′ r,s is given by the usual Kac formula. The (r, s) Robin boundary conditions are thus conjugate to scaling operators with half-integer values for the Kac label s − 1 2 . Level degeneracies support the conjecture that the characters of the associated (reducible or irreducible) representations are given by Virasoro Verma characters.

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Boundary algebras and Kac modules for logarithmic minimal models

2015

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Virasoro Kac modules were initially introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley-Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl-Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley-Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley-Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley-Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin-Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.Comment: 71 pages. v3: version published in Nucl. Phys.

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Kac boundary conditions of the logarithmic minimal models

Cited by 5 publications

References 57 publications

Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations

Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations

Logarithmic minimal models with Robin boundary conditions

Boundary algebras and Kac modules for logarithmic minimal models

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Kac boundary conditions of the logarithmic minimal models

Cited by 5 publications

References 57 publications

Conformal partition functions of critical percolation fromD3thermodynamic Bethe Ansatz equations

Conformal partition functions of critical percolation fromD3thermodynamic Bethe Ansatz equations

Logarithmic minimal models with Robin boundary conditions

Boundary algebras and Kac modules for logarithmic minimal models

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Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations

Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations