Virasoro Modules Research Articles

Tensor product of generalized polynomial modules for the Virasoro algebra

In this paper, we construct a class of modules T over the Virasoro algebra by taking tensor products of the modules N(Ω(λ,b),V) defined in [24,19] and the irreducible modules defined in [31]. This provides a unified description of many known examples of Virasoro modules, for example, in [28,34,35,12,20,5]. We obtain the necessary and sufficient conditions for T to be irreducible and study their submodule structure when they are reducible. Then we also determine the conditions for two such modules to be isomorphic. In the last part of the paper, we compare the tensor product modules with other known Virasoro modules, concluding that they provide new simple modules in general, while admit some interesting isomorphisms with modules defined in [40] and [33].

Classification of simple smooth modules over the Heisenberg–Virasoro algebra

In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra ${\mathfrak {D}}$ , and simple smooth modules over the twisted Heisenberg–Virasoro algebra $\bar {\mathfrak {D}}$ with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak {D}}$ . A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg–Virasoro vertex algebras. We also present a few examples of simple smooth ${\mathfrak {D}}$ -modules and $\bar {\mathfrak {D}}$ -modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak {D}$ and $\bar {\mathfrak {D}}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.

New irreducible non-weight Virasoro modules from tensor products

The tensor product non-weight Virasoro modules are studied in this paper, where and (with and V being an irreducible module over a certain finite dimensional Lie algebra related to the Virasoro algebra) are irreducible Virasoro modules, respectively, defined in Liu and Zhao [Generalized polynomial modules over the Virasoro algebra. Proc Amer Math Soc. 2016;144:5103–5112] and Lü and Zhao [Irreducible Virasoro modules from irreducible Weyl modules. J Algebra. 2014;414:271–287]. The necessary and sufficient condition for to be irreducible is obtained. Then we determine the necessary and sufficient condition for two such irreducible modules to be isomorphic. At last, we show that the irreducible -modules are new for m>1.

The Restriction of Polynomial Modules for the Virasoro Algebra to sl2(C)

The Lie algebra [Formula: see text] may be regarded in a natural way as a subalgebra of the infinite-dimensional Virasoro Lie algebra, so it is natural to consider connections between the representation theory of the two algebras. In this paper, we explore the restriction to [Formula: see text] of certain induced modules for the Virasoro algebra. Specifically, we consider Virasoro modules induced from so-called polynomial subalgebras, and we show that the restriction of these modules results in twisted versions of familiar modules such as Verma modules and Whittaker modules.

New simple Virasoro modules from Weyl algebra modules

In this paper, we construct a class of modules N(P,V) over the Virasoro algebra from modules P over the degree-1 Weyl algebra K and modules V over the positive part of the Virasoro algebra. This generalizes a construction in [22] and provides a unified description of many known examples of Virasoro modules, for example, in [24,22,3,23]. We obtain the necessary and sufficient conditions for N(P,V) to be simple and also determine the isomorphism class of these modules. At the end of the paper, we compare the modules N(P,V) with other known simple Virasoro modules, concluding that they are new simple modules in general.

Characters, coadjoint orbits and Duistermaat-Heckman integrals

The asymptotics of characters χkλ(exp⁡(h/k)) of irreducible representations of a compact Lie group G for large values of the scaling factor k are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits of G. This phenomenon generalises to coadjoint orbits of central extensions of loop groups LGˆ and of diffeomorphisms of the circle Diffˆ(S1). We show that the asymptotics of characters of integrable modules of affine Kac-Moody algebras and of the Virasoro algebra factorize into a divergent contribution of the standard form and a convergent contribution which can be interpreted as a formal DH orbital integral.For some Virasoro modules, our results match formal DH integrals recently computed by Stanford and Witten. In this case, the k-scaling has the same origin as the one which gives rise to classical conformal blocks. Furthermore, we consider reduced spaces of Virasoro coadjoint orbits and we suggest a new invariant which replaces symplectic volume in the infinite dimensional situation. We also consider other modules of the Virasoro algebra (in particular, the modules corresponding to minimal models) and we obtain DH-type expressions which do not correspond to any Virasoro coadjoint orbits.We introduce volume functions V(x) corresponding to formal DH integrals over coadjoint orbits of the Virasoro algebra and show that they are related by the Hankel transform to spectral densities ρ(E) recently studied by Saad, Shenker and Stanford.

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

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}} 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} . 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.

The Conway Moonshine Module is a reflected K3 theory

Recently, Duncan and Mack-Crane established an isomorphism, as Virasoro modules at central charges c = 12, between the space of states of the Conway Moonshine Module and the space of states of a special K3 theory that was extensively studied some time ago by Gaberdiel, Volpato and the two authors. In the present work, we lift this result to the level of modules of the extensions of these Virasoro algebras to N = 4 super Virasoro algebras. Moreover, we relate the super vertex operator algebra and module structure of the Conway Moonshine Module to the operator product expansion of this special K3 theory by a procedure we call reflection. This procedure can be applied to certain superconformal field theories, transforming all fields to holomorphic ones. It also allows to describe certain superconformal field theories within the language of super vertex operator algebras. We discuss reflection and its limitations in general, and we argue that through reflection, the Conway Moons hine Module inherits from the K3 theory a richer structure than anticipated so far. The comparison between the Conway Moonshine Module and the K3 theory is considerably facilitated by exploiting the free fermion description as well as the lattice vertex operator algebra descript ion of both theories. We include an explicit construction of cocycles for the relevant charge lattice s, which are half integral. The transition from the K3 theory to the Conway Moonshine Module via reflection promotes the latter to the role of a medium that collects the symmetries of K3 theories from distinct points of the moduli space, thus uncovering a version of symmetry surfing in this context.

Staggered and affine Kac modules over [formula omitted

This work concerns the representation theory of the affine Lie algebra A1(1) at fractional level and its links to the representation theory of the Virasoro algebra. We introduce affine Kac modules as certain finitely generated submodules of Wakimoto modules. We conjecture the existence of several classes of staggered A1(1)-modules and provide evidence in the form of detailed examples. We extend the applicability of the Goddard-Kent-Olive coset construction to include the affine Kac and staggered modules. We introduce an exact functor between the associated category of A1(1)-modules and the corresponding category of Virasoro modules. At the level of characters, its action generalises the Mukhi-Panda residue formula. We also obtain explicit expressions for all irreducible A1(1)-characters appearing in the decomposition of Verma modules, re-examine the construction of Malikov-Feigin-Fuchs vectors, and extend the Fuchs-Astashkevich theorem from the Virasoro algebra to A1(1).

Simple restricted modules for Neveu-Schwarz algebra

In this paper, we give a construction of simple modules generalizing both highest weight and Whittaker modules for the Neveu-Schwarz algebra, in the spirit of the work of Mazorchuk and Zhao on simple Virasoro modules. We establish a 1-1 correspondence between simple restricted Neveu-Schwarz modules and simple modules of a family of finite dimensional solvable Lie superalgebras associated to the Neveu-Schwarz algebra. Moreover, for two of these superalgebras all simple modules are classified.

A new class of irreducible Virasoro modules from tensor product

In this paper, we obtain a class of Virasoro modules by taking tensor products of the irreducible Virasoro modules Ω(λ,α,h) and Ω(μ,β) with irreducible highest weight modules V(θ,h) or with irreducible Virasoro modules Indθ(N) defined in [20]. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for two of them to be isomorphic. We also compare these modules with other known non-weight Virasoro modules, showing that most of these simple modules are new.

Logarithmic correlation functions for critical dense polymers on the cylinder

We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter nn. In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity \alpha \in (0,\infty)α∈(0,∞). These correlators are defined as ratios Z(x)/Z_0Z(x)/Z0 of partition functions, where Z_0Z0 is a reference partition function wherein only simple half-arcs are attached to the boundary of the cylinder. For Z(x)Z(x), the boundary of the cylinder is also decorated with simple half-arcs, but it also has two special positions 11 and xx where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite nn using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as n\to \inftyn→∞ are expressed as simple integrals that depend on the scaling parameter \tau = \frac {x-1} n \in (0,1)τ=x−1n∈(0,1). For small \tauτ, the leading behaviours are proportional to \tau^{1/4}τ1/4, \tau^{1/4}\log \tauτ1/4logτ, \log \taulogτ and \log^2 \taulog2τ. We interpret the lattice results in terms of ratios of conformal correlation functions. We assume that the corresponding boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules, with central charge c=-2c=−2 and conformal dimensions \Delta = -\frac18Δ=−18 or \Delta = 0Δ=0. With these assumptions, we obtain differential equations of order two and three satisfied by the conformal correlation functions, solve these equations in terms of hypergeometric functions, and find a perfect agreement with the lattice results. We use the lattice results to compute structure constants and ratios thereof which appear in the operator product expansions of the boundary condition changing fields. The fusion of these fields is found to be non-abelian.

Fock space realisations of staggered modules in 2D logarithmic CFTs

We utilise bosonic Fock spaces, considered as Virasoro modules, to make free field realisations of the so-called staggered modules of two-dimensional logarithmic conformal field theories. A general formula for the β-invariant of a staggered Fock module is derived, and found to agree with values previously known in the literature. In this way a large class of staggered modules is produced; one which provides an explicit free-field construction for many of those previously studied. We show how these modules can arise algebraically by including the modes of weight-0 fields into the algebra.

Generalized polynomial modules over the Virasoro algebra

Let $\mathcal{B}_r$ be the $(r+1)$-dimensional quotient Lie algebra of the positive part of the Virasoro algebra $\mathcal{V}$. Irreducible $\mathcal{B}_r$-modules were used to construct irreducible Whittaker modules in [MZ2] and irreducible weight modules with infinite dimensional weight spaces over $\mathcal{V}$ in [LLZ].In the present paper, we construct non-weight Virasoro modules $F(M, \Omega(\lambda,\beta))$ from irreducible $\mathcal{B}_r$-modules $M$ and $(\mathcal{A},\mathcal{V})$-modules $\Omega(\lambda,\beta)$. We give necessary and sufficient conditions for the Virasoro module $F(M, \Omega(\lambda,\beta))$ to be irreducible. Using the weighting functor introduced by J. Nilsson, we also we also give the isomorphism criterion for two $F(M, \Omega(\lambda,\beta))$.

A new family of modules over the Virasoro algebra

In this paper, we constructed a new family of Virasoro modules Ω(λ,α,h) parameterized by complex numbers λ≠0, α∈C and a polynomial h(t). They are a special type of modules on which the non-central Cartan part acts freely of infinite rank. These modules are natural generalizations of the modules Ω(λ,α) defined and studied in [5], [11] and [19]. We determine the sufficient and necessary conditions for them to be irreducible. When they are reducible, we also determine all their maximal submodules and corresponding simple quotients explicitly. For some special cases, we can determine all submodules.

Integrability and conformal data of the dimer model

The central charge of the dimer model on the square lattice is still being debated in the literature. In this paper, we provide evidence supporting the consistency of a description. Using Lieb’s transfer matrix and its description in terms of the Temperley–Lieb algebra at , we provide a new solution of the dimer model in terms of the model of critical dense polymers on a tilted lattice and offer an understanding of the lattice integrability of the dimer model. The dimer transfer matrix is analyzed in the scaling limit, and the result for is expressed in terms of fermions. Higher Virasoro modes are likewise constructed as limits of elements of and are found to yield a realization of the Virasoro algebra, familiar from fermionic bc ghost systems. In this realization, the dimer Fock spaces are shown to decompose, as Virasoro modules, into direct sums of Feigin–Fuchs modules, themselves exhibiting reducible yet indecomposable structures. In the scaling limit, the eigenvalues of the lattice integrals of motion are found to agree exactly with those of the conformal integrals of motion. Consistent with the expression for obtained from the transfer matrix, we also construct higher Virasoro modes with c = 1 and find that the dimer Fock space is completely reducible under their action. However, the transfer matrix is found not to be a generating function for the c = 1 integrals of motion. Although this indicates that Lieb’s transfer matrix description is incompatible with the c = 1 interpretation, it does not rule out the existence of an alternative, c = 1 compatible, transfer matrix description of the dimer model.

Irreducible Virasoro modules from tensor products

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega(\lambda,b)$ defined in [LZ], with irreducible highest weight modules $V(\theta,h)$ or with irreducible Virasoro modules Ind$_{\theta}(N)$ defined in [MZ2]. We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $\Omega(\lambda,b)$ with a classical Whittaker module is isomorphic to the module $\mathrm{Ind}_{\theta,\lambda}(\mathbb{C_\mathbf{m}})$ defined in [MW]. As a by-product we obtain the necessary and sufficient conditions for the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to be irreducible. We also generalize the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $ n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are studied and an open problem in [GLZ] is solved. Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.

Virasoro representations with central charges 1/2 and 1 on the real neutral fermion Fock space F⊗1/2

We study a family of fermionic oscillator representations of the Virasoro algebra via 2-point-local Virasoro fields on the Fock space F⊗1/2 of a neutral (real) fermion. We obtain the decomposition of F⊗1/2 as a direct sum of irreducible highest weight Virasoro modules with central charge c = 1. As a corollary we obtain the decomposition of the irreducible highest weight Virasoro modules with central charge c = 1/2 into irreducible highest weight Virasoro modules with central charge c = 1. As an application we show how positive sum (fermionic) character formulas for the Virasoro modules of charge c = 1/2 follow from these decompositions.

A class of simple weight Virasoro modules

For a simple module M over the positive part of the Virasoro algebra (actually for any simple module over some finite dimensional solvable Lie algebras ar) and any α∈C, a class of weight modules N(M,α) over the Virasoro algebra is constructed. These weight modules have infinite dimensional weight spaces if dim⁡M>1. The necessary and sufficient conditions for N(M,α) to be simple are obtained. We also determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided. In particular the irreducible weight Virasoro modules Γ(α1,α2,λ1,λ2) are defined on the polynomial algebra C[x]⊗C[t,t−1] for any α1,α2,λ1,λ2∈C with λ1 or λ2 nonzero. By twisting the weight modules N(M,α) we also obtain nonweight simple Virasoro modules N(M,β) for any nonconstant β∈C[t,t−1].

Simple Virasoro modules induced from codimension one subalgebras of the positive part

Simple Virasoro modules induced from codimension one subalgebras of the positive part