Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Boundary structure constants for the A-series Virasoro minimal models

Ingo Runkel B

Published Jan 1, 1998• 98 citations
OpenAlex

Abstract

We consider A-series modular invariant Virasoro minimal models on the upper half plane. From Lewellen's sewing constraints a necessary form of the bulk and boundary structure constants is derived. Necessary means that any solution can be brought to the given form by rescaling of the fields. All constants are expressed essentially in terms of fusing (F-) matrix elements and the normalisations are chosen such that they are real and no square roots appear. It is not shown in this paper that the given structure constants solve the sewing constraints, however random numerical tests show no contradiction and agreement of the bulk structure constants with Dotsenko and Fateev. In order to facilitate numerical calculations a recursion relation for the F-matrices is given.

Related papers

Boundary structure constants for the A-series Virasoro minimal models

Ingo Runkel B

arXiv:hep-th/9811178•Nuclear Physics B•Jun 1, 1999•120 citations

Lattice models from CFT on surfaces with holes II: Cloaking boundary conditions and loop models

Enrico M. Brehm, Ingo Runkel B

In this paper we continue to investigate the lattice models obtained from 2d CFTs via the construction introduced in [arXiv:2112.01563]. On the side of the 2d CFT we consider the cloaking boundary condition relative to a fixed fusion category F of topological line defects. The resulting lattice model realises the topological symmetry F exactly. We compute the state spaces and Boltzmann weights of these lattice model in the example of unitary Virasoro minimal models. We work directly with amplitudes, rather than with normalised correlators, and we provide a careful treatment of the Weyl anomaly factor in terms of the Liouville action. We numerically evaluate the Ising CFT on the torus with one hole and cloaking boundary condition in two channels, and illustrate in this example that the anomaly factors are essential to obtain matching results for the amplitudes. We show that lattice models obtained from Virasoro minimal models at lowest non-trivial cutoff can be exactly mapped to loop models. This provides a first non-trivial check that our lattice models can contain the 2d CFT they were constructed from in their phase diagram, and we propose a condition on the cloaking boundary condition for F under which we expect this to happen in general.

arXiv:2410.19938•Oct 25, 2024•0 citations

Observables of boundary RG flows from string field theory

Jaroslav Scheinpflug, Martin Schnabl, Jakub Vošmera

A bstract We present simple explicit formulae for the change of the g -function, boundary state, boundary spectrum and structure constants between the endpoints of short boundary RG flows at next-to-leading order. The formulae are derived using open string field theory and tested on integrable RG flows between conformal boundary states in unitary Virasoro minimal models.

arXiv:2510.07155•Journal of High Energy Physics•Apr 9, 2026•2 citations

Revisiting the Ramond sector of the N=1 superconformal minimal models

Chun Chen, Joseph Maciejko

Key to the exact solubility of the unitary minimal models in two-dimensional conformal field theory is the organization of their Hilbert space into Verma modules, whereby all eigenstates of the Hamiltonian are obtained by the repeated action of Virasoro lowering operators onto a finite set of highest-weight states. The usual representation-theoretic approach to removing from all modules zero-norm descendant states generated in such a way is based on the assumption that those states form a nested sequence of Verma submodules built upon singular vectors, i.e., descendant highest-weight states. We show that this fundamental assumption breaks down for the Ramond-sector Verma module with highest weight $c/24$ in the even series of $\mathcal{N}=1$ superconformal minimal models with central charge $c$. To resolve this impasse, we conjecture, and prove at low orders, the existence of a nested sequence of linear-dependence relations that enables us to compute the character of the irreducible $c/24$ module. Based on this character formula, we argue that imposing modular invariance of the torus partition function requires the introduction of a non-null odd-parity Ramond-sector ground state. This symmetrization of the ground-state manifold allows us to uncover a set of conformally invariant boundary conditions not previously discussed and absent in the odd series of superconformal minimal models, and to derive for the first time a complete set of fusion rules for the even series of those models.

arXiv:1905.02297•Physical review. D/Physical review. D.•Dec 1, 2020•2 citations

Kac boundary conditions of the logarithmic minimal models

Paul A. Pearce, Elena Tartaglia, Romain Couvreur

We develop further the implementation and analysis of Kac boundary conditions in the general logarithmic minimal models with 1 ⩽ p < p ′ and p , p ′ coprime. Specifically, working in a strip geometry, we consider the ( r , s ) Kac boundary conditions. These boundary conditions are organized into infinitely extended Kac tables labeled by the Kac labels r , s = 1, 2, 3, …. They are conjugate to Virasoro Kac representations with conformal dimensions Δ r , s given by the usual Kac formula. On a finite strip of width N , built from a square lattice, the associated integrable boundary conditions are constructed by acting on the vacuum (1, 1) boundary with an s -type seam of width s − 1 columns and an r -type seam of width ρ − 1 columns. The r -type seam contains an arbitrary boundary field ξ . While the usual fusion construction of the r -type seam relies on the existence of Wenzl–Jones projectors restricting its application to r ⩽ ρ < p ′, this limitation was recently removed by Pearce et al who further conjectured that the conformal boundary conditions labeled by r are realized, in particular, for . In this paper, we confirm this conjecture by performing extensive numerics on the commuting double row transfer matrices and their associated quantum Hamiltonian chains. Letting [ x ] denote the fractional part, we fix the boundary field to the specialized values if and otherwise. For these boundary conditions, we obtain the Kac conformal weights Δ r , s by numerically extrapolating the finite-size corrections to the lowest eigenvalue of the quantum Hamiltonians out to sizes N ⩽ 32 − ρ − s . Additionally, by solving local inversion relations, we obtain general analytic expressions for the boundary free energies allowing for more accurate estimates of the conformal data.

arXiv:1410.0103•Journal of Statistical Mechanics Theory and Experiment•Jan 19, 2015•5 citations

Unitary and nonunitary A - D - E minimal models: coset graph fusion algebras, defects and dilogarithm identities

Paul A. Pearce, Jared J. Heymann, Thomas Quella

Abstract We consider unitary and nonunitary ( A , G ) coset minimal models on the cylinder with G = A , D , E . Elementary topological defects are implemented as non-contractible Verlinde, Pasquier and Ocneanu lines around the cylinder. The decomposition of compound defects, formed by fusing elementary defects together, is described by the Verlinde, Pasquier and Ocneanu graph fusion algebras. The action of the compound defects on the left- or right-vacuum boundary state builds the known conformal cylinder boundary conditions. Fusing these defects, in the presence of vacuum boundary conditions, reproduces the known general formulas for the conformal cylinder partition functions. We define the coset graph A ⊗ G / Z 2 , argue that it is a universal object and show that it encodes (i) the coset graph fusion algebra, (ii) the Affleck–Ludwig boundary g -factors, (iii) the defect g -factors given by quantum dimensions and (iv) the relative symmetry resolved Affleck–Ludwig entropies. Additionally, it is shown that the boundary and defect g -factors are related to the asymptotic counting of fusion paths on the coset graph. On the lattice, the defects are constructed as Yang–Baxter integrable seams including special braid and graph automorphism transfer matrices. Remarkably, many of the boundary conformal field theory (CFT) discrete structures, such as fusion matrices, modular matrices, quantum dimensions and defects, appear at the level of lattice Yang–Baxter integrable A - D - E restricted solid-on-solid models and these structures and their properties carry over to the CFT in the continuum scaling limit. Importantly, in the continuum scaling limit, the lattice transfer matrix T - and Y -system functional equations carry over to produce the coset graph fusion algebra for the defect lines. Moreover, the effective central charges and conformal weights are expressed in terms of dilogarithms of the braid and bulk asymptotics of the Y -system expressed in terms of the quantum dimensions.

arXiv:2512.21808•Journal of Physics A Mathematical and Theoretical•Sep 7, 2026•0 citations
We consider A-series modular invariant Virasoro minimal models on the upper half plane. From Lewellen's sewing constraints a necessary form of the bulk and boundary structure constants is derived. Necessary means that any solution can be brought to the given form by rescaling of the fields. All constants are expressed essentially in terms of fusing (F-) matrix elements and the normalisations are chosen such that they are real and no square roots appear. It is not shown in this paper that the given structure constants solve the sewing constraints, however random numerical tests show no contradiction and agreement of the bulk structure constants with Dotsenko and Fateev. In order to facilitate numerical calculations a recursion relation for the F-matrices is given.
In this paper we continue to investigate the lattice models obtained from 2d CFTs via the construction introduced in [arXiv:2112.01563]. On the side of the 2d CFT we consider the cloaking boundary condition relative to a fixed fusion category F of topological line defects. The resulting lattice model realises the topological symmetry F exactly. We compute the state spaces and Boltzmann weights of these lattice model in the example of unitary Virasoro minimal models. We work directly with amplitudes, rather than with normalised correlators, and we provide a careful treatment of the Weyl anomaly factor in terms of the Liouville action. We numerically evaluate the Ising CFT on the torus with one hole and cloaking boundary condition in two channels, and illustrate in this example that the anomaly factors are essential to obtain matching results for the amplitudes. We show that lattice models obtained from Virasoro minimal models at lowest non-trivial cutoff can be exactly mapped to loop models. This provides a first non-trivial check that our lattice models can contain the 2d CFT they were constructed from in their phase diagram, and we propose a condition on the cloaking boundary condition for F under which we expect this to happen in general.
Key to the exact solubility of the unitary minimal models in two-dimensional conformal field theory is the organization of their Hilbert space into Verma modules, whereby all eigenstates of the Hamiltonian are obtained by the repeated action of Virasoro lowering operators onto a finite set of highest-weight states. The usual representation-theoretic approach to removing from all modules zero-norm descendant states generated in such a way is based on the assumption that those states form a nested sequence of Verma submodules built upon singular vectors, i.e., descendant highest-weight states. We show that this fundamental assumption breaks down for the Ramond-sector Verma module with highest weight c/24c/24c/24 in the even series of N=1\mathcal{N}=1N=1 superconformal minimal models with central charge ccc. To resolve this impasse, we conjecture, and prove at low orders, the existence of a nested sequence of linear-dependence relations that enables us to compute the character of the irreducible c/24c/24c/24 module. Based on this character formula, we argue that imposing modular invariance of the torus partition function requires the introduction of a non-null odd-parity Ramond-sector ground state. This symmetrization of the ground-state manifold allows us to uncover a set of conformally invariant boundary conditions not previously discussed and absent in the odd series of superconformal minimal models, and to derive for the first time a complete set of fusion rules for the even series of those models.
We develop further the implementation and analysis of Kac boundary conditions in the general logarithmic minimal models with 1 ⩽ p < p ′ and p , p ′ coprime. Specifically, working in a strip geometry, we consider the ( r , s ) Kac boundary conditions. These boundary conditions are organized into infinitely extended Kac tables labeled by the Kac labels r , s = 1, 2, 3, …. They are conjugate to Virasoro Kac representations with conformal dimensions Δ r , s given by the usual Kac formula. On a finite strip of width N , built from a square lattice, the associated integrable boundary conditions are constructed by acting on the vacuum (1, 1) boundary with an s -type seam of width s − 1 columns and an r -type seam of width ρ − 1 columns. The r -type seam contains an arbitrary boundary field ξ . While the usual fusion construction of the r -type seam relies on the existence of Wenzl–Jones projectors restricting its application to r ⩽ ρ < p ′, this limitation was recently removed by Pearce et al who further conjectured that the conformal boundary conditions labeled by r are realized, in particular, for . In this paper, we confirm this conjecture by performing extensive numerics on the commuting double row transfer matrices and their associated quantum Hamiltonian chains. Letting [ x ] denote the fractional part, we fix the boundary field to the specialized values if and otherwise. For these boundary conditions, we obtain the Kac conformal weights Δ r , s by numerically extrapolating the finite-size corrections to the lowest eigenvalue of the quantum Hamiltonians out to sizes N ⩽ 32 − ρ − s . Additionally, by solving local inversion relations, we obtain general analytic expressions for the boundary free energies allowing for more accurate estimates of the conformal data.
Abstract We consider unitary and nonunitary ( A , G ) coset minimal models on the cylinder with G = A , D , E . Elementary topological defects are implemented as non-contractible Verlinde, Pasquier and Ocneanu lines around the cylinder. The decomposition of compound defects, formed by fusing elementary defects together, is described by the Verlinde, Pasquier and Ocneanu graph fusion algebras. The action of the compound defects on the left- or right-vacuum boundary state builds the known conformal cylinder boundary conditions. Fusing these defects, in the presence of vacuum boundary conditions, reproduces the known general formulas for the conformal cylinder partition functions. We define the coset graph A ⊗ G / Z 2 , argue that it is a universal object and show that it encodes (i) the coset graph fusion algebra, (ii) the Affleck–Ludwig boundary g -factors, (iii) the defect g -factors given by quantum dimensions and (iv) the relative symmetry resolved Affleck–Ludwig entropies. Additionally, it is shown that the boundary and defect g -factors are related to the asymptotic counting of fusion paths on the coset graph. On the lattice, the defects are constructed as Yang–Baxter integrable seams including special braid and graph automorphism transfer matrices. Remarkably, many of the boundary conformal field theory (CFT) discrete structures, such as fusion matrices, modular matrices, quantum dimensions and defects, appear at the level of lattice Yang–Baxter integrable A - D - E restricted solid-on-solid models and these structures and their properties carry over to the CFT in the continuum scaling limit. Importantly, in the continuum scaling limit, the lattice transfer matrix T - and Y -system functional equations carry over to produce the coset graph fusion algebra for the defect lines. Moreover, the effective central charges and conformal weights are expressed in terms of dilogarithms of the braid and bulk asymptotics of the Y -system expressed in terms of the quantum dimensions.