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Representations of the Virasoro algebra from lattice models

Koo, W M, Hubert Saleur

Published Dec 17, 1993• 54 citations
OpenAlex

Abstract

We investigate in details how the Virasoro algebra appears in the scaling limit of the simplest lattice models of XXZ or RSOS type. Our approach is straightforward but to our knowledge had never been tried so far. We simply formulate a conjecture for the lattice stress-energy tensor motivated by the exact derivation of lattice global Ward identities. We then check that the proper algebraic relations are obeyed in the scaling limit. The latter is under reasonable control thanks to the Bethe-ansatz solution. The results, which are mostly numerical for technical reasons, are remarkably precise. They are also corroborated by exact pieces of information from various sources, in particular Temperley-Lieb algebra representation theory. Most features of the Virasoro algebra (like central term, null vectors, metric properties...) can thus be observed using the lattice models. This seems of general interest for lattice field theory, and also more specifically for finding relations between conformal invariance and lattice integrability, since basis for the irreducible representations of the Virasoro algebra should now follow (at least in principle) from Bethe-ansatz computations.

Related papers

The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

Linnea Grans-Samuelsson, Jesper Lykke Jacobsen, Hubert Saleur

Abstract We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of $$ {W}_{0,{\mathfrak{q}}^{\pm 2}} $$ W 0 , q ± 2 contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in $$ {\mathfrak{q}}^{\pm 2} $$ q ± 2 . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.

arXiv:2010.12819•Journal of High Energy Physics•Feb 16, 2021•10 citations

Lattice W-algebras and logarithmic CFTs

Azat M. Gainutdinov, Hubert Saleur, I Yu Tipunin

This paper is part of an effort to gain further understanding of 2D logarithmic conformal field theories (LCFTs) by exploring their lattice regularizations. While all work so far has dealt with the Virasoro algebra (or the product ), the best known (although maybe not the most relevant physically) LCFTs in the continuum are characterized by a W-algebra symmetry, whose presence is powerful, but whose role as a ‘symmetry’ remains mysterious. We explore here the origin of this symmetry in the underlying lattice models. We consider XXZ spin chains for a root of unity, and argue that the centralizer of the ‘small’ quantum group goes over the W-algebra in the continuum limit. We justify this identification by representation theoretic arguments, and give, in particular, lattice versions of the W-algebra generators. In the case , which corresponds to symplectic fermions at central charge , we provide a full analysis of the scaling limit of the lattice Virasoro and W generators, and show in details how the corresponding continuum Virasoro and W-algebras are obtained. Striking similarities between the lattice W algebra and the Onsager algebra are observed in this case.

arXiv:1212.1378•Journal of Physics A Mathematical and Theoretical•Nov 12, 2014•29 citations

The Temperley–Lieb algebra and its generalizations in the Potts andXXZmodels

Alexander Nichols

We discuss generalizations of the Temperley–Lieb algebra in the Potts and XXZ models. These can be used to describe the addition of integrable boundary terms of different types. We use the Temperley–Lieb algebra and its one-boundary, two-boundary, and periodic extensions to classify different integrable boundary terms in the two-, three-, and four-state Potts models. The representations always lie at critical points where the algebras becomes non-semisimple and possess indecomposable representations. In the one-boundary case we show how to use representation theory to extract the Potts spectrum from an XXZ model with particular boundary terms and hence obtain the finite size scaling of the Potts models. In the two-boundary case we find that the Potts spectrum can be obtained by combining several XXZ models with different boundary terms. As in the Temperley–Lieb case, there is a direct correspondence between representations of the lattice algebra and those in the continuum conformal field theory.

arXiv:hep-th/0509069•Journal of Statistical Mechanics Theory and Experiment•Jan 10, 2006•33 citations

Anyonic Interpretation of Virasoro Characters and the Thermodynamic Bethe Ansatz

Berlin Sfb Preprint, A. G. Bytsko, A. Fring

Employing factorized versions of characters as products of quantum dilogarithms corresponding to irreducible representations of the Virasoro algebra, we obtain character formulae which admit an anyonic quasi-particle interpretation in the context of minimal models in conformal field theories. We propose anyonic thermodynamic Bethe ansatz equations, together with their corresponding equation for the Virasoro central charge, on the base of an analysis of the classical limit for the characters and the requirement that the scattering matrices are asymptotically phaseless.

•Jan 1, 1997•8 citations

Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models

Marcin Radosław Piątek, R. G. Nazmitdinov, Antonio Puente +1 more

Abstract Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs’ classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, $$ \mathcal{N} $$ N = 2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the “Dotsenko-Fateev matrix model action”. The latter is the “DF action” evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. For instance, analogous quantities as the “DF on-shell action” appear in 2d CFT realization of the so-called Richardson’s solution of the reduced BCS model describing physics of ultra-small superconducting grains. Precisely, the Richardson solution and its particular limit characterizing the rational Gaudin model have known implementation in 2d CFT in terms of a free field representation of certain (perturbed and unperturbed) WZW blocks. The WZW analogue of the “DF on-shell action” appears here and plays a crucial role, it is a generating function for eigenvalues of quantum integrals of motion. An exploration of relationships between power expansions and Coulomb gas representations of the aforementioned WZW blocks could pave a way for new analytical tools useful in the study of models of the Richardson-Gaudin type.

arXiv:2110.15009•Journal of High Energy Physics•Apr 15, 2022•3 citations

Survey on the Conformal Field Theories in two dimensions and Minimal Models

Atlas Çelik

The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of $n$ point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.

Työväentutkimus Vuosikirja•May 20, 2026•0 citations
We investigate in details how the Virasoro algebra appears in the scaling limit of the simplest lattice models of XXZ or RSOS type. Our approach is straightforward but to our knowledge had never been tried so far. We simply formulate a conjecture for the lattice stress-energy tensor motivated by the exact derivation of lattice global Ward identities. We then check that the proper algebraic relations are obeyed in the scaling limit. The latter is under reasonable control thanks to the Bethe-ansatz solution. The results, which are mostly numerical for technical reasons, are remarkably precise. They are also corroborated by exact pieces of information from various sources, in particular Temperley-Lieb algebra representation theory. Most features of the Virasoro algebra (like central term, null vectors, metric properties...) can thus be observed using the lattice models. This seems of general interest for lattice field theory, and also more specifically for finding relations between conformal invariance and lattice integrability, since basis for the irreducible representations of the Virasoro algebra should now follow (at least in principle) from Bethe-ansatz computations.
Abstract We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of W0,q±2 {W}_{0,{\mathfrak{q}}^{\pm 2}} W0,q±2​ W 0 , q ± 2 contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in q±2 {\mathfrak{q}}^{\pm 2} q±2 q ± 2 . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.
This paper is part of an effort to gain further understanding of 2D logarithmic conformal field theories (LCFTs) by exploring their lattice regularizations. While all work so far has dealt with the Virasoro algebra (or the product ), the best known (although maybe not the most relevant physically) LCFTs in the continuum are characterized by a W-algebra symmetry, whose presence is powerful, but whose role as a ‘symmetry’ remains mysterious. We explore here the origin of this symmetry in the underlying lattice models. We consider XXZ spin chains for a root of unity, and argue that the centralizer of the ‘small’ quantum group goes over the W-algebra in the continuum limit. We justify this identification by representation theoretic arguments, and give, in particular, lattice versions of the W-algebra generators. In the case , which corresponds to symplectic fermions at central charge , we provide a full analysis of the scaling limit of the lattice Virasoro and W generators, and show in details how the corresponding continuum Virasoro and W-algebras are obtained. Striking similarities between the lattice W algebra and the Onsager algebra are observed in this case.
We discuss generalizations of the Temperley–Lieb algebra in the Potts and XXZ models. These can be used to describe the addition of integrable boundary terms of different types. We use the Temperley–Lieb algebra and its one-boundary, two-boundary, and periodic extensions to classify different integrable boundary terms in the two-, three-, and four-state Potts models. The representations always lie at critical points where the algebras becomes non-semisimple and possess indecomposable representations. In the one-boundary case we show how to use representation theory to extract the Potts spectrum from an XXZ model with particular boundary terms and hence obtain the finite size scaling of the Potts models. In the two-boundary case we find that the Potts spectrum can be obtained by combining several XXZ models with different boundary terms. As in the Temperley–Lieb case, there is a direct correspondence between representations of the lattice algebra and those in the continuum conformal field theory.
Employing factorized versions of characters as products of quantum dilogarithms corresponding to irreducible representations of the Virasoro algebra, we obtain character formulae which admit an anyonic quasi-particle interpretation in the context of minimal models in conformal field theories. We propose anyonic thermodynamic Bethe ansatz equations, together with their corresponding equation for the Virasoro central charge, on the base of an analysis of the classical limit for the characters and the requirement that the scattering matrices are asymptotically phaseless.
Abstract Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs’ classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, N \mathcal{N} N N = 2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the “Dotsenko-Fateev matrix model action”. The latter is the “DF action” evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. For instance, analogous quantities as the “DF on-shell action” appear in 2d CFT realization of the so-called Richardson’s solution of the reduced BCS model describing physics of ultra-small superconducting grains. Precisely, the Richardson solution and its particular limit characterizing the rational Gaudin model have known implementation in 2d CFT in terms of a free field representation of certain (perturbed and unperturbed) WZW blocks. The WZW analogue of the “DF on-shell action” appears here and plays a crucial role, it is a generating function for eigenvalues of quantum integrals of motion. An exploration of relationships between power expansions and Coulomb gas representations of the aforementioned WZW blocks could pave a way for new analytical tools useful in the study of models of the Richardson-Gaudin type.
The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of nnn point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.