Virasoro Modes Research Articles

Staggered modules of N = 2 superconformal minimal models

We investigate a class of reducible yet indecomposable modules over the N=2 superconformal algebras. These so-called staggered modules exhibit a non-diagonalisable action of the Virasoro mode L0. Using recent results on the coset construction of N=2 minimal models, we explicitly construct such modules for central charges c=−1 and c=−6. We also describe spectral-flow orbits and symmetries of the families of staggered modules which arise via the coset.

The OPE of bare twist operators in bosonic SN orbifold CFTs at large N

In this work, we explore the twist operator OPEs of a generic bosonic symmetric product (SN) orbifold CFT. We conjecture that at large N the OPE of bare twist operators contains only bare twists and excitations of bare twists with fractional Virasoro modes. These fractionally excited operators are the only ones that depend exclusively on the lengths of the twists and the central charge, agreeing with the general structure of correlators of bare twists at large N found in the literature. To provide evidence for this, we study the coincidence limit of a four point function of bare twist operators to several non-leading orders. We show how the coefficients of these powers can be reproduced by considering bare twist operators excited by fractional Virasoro modes in the exchange channels.

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.

On staggered indecomposable Virasoro modules

In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L0 is assumed to be nondiagonalizable, possessing Jordan blocks of rank 2. Moreover, the module is further assumed to have a highest weight submodule, the “left module,” and that the quotient by this submodule yields another highest weight module, the “right module.” Such modules, which have been called staggered, have appeared repeatedly in the logarithmic conformal field theory literature, but their theory has not been explored in full generality. Here, such a theory is developed for the Virasoro algebra using rather elementary techniques. The focus centers on two different but related questions typically encountered in practical studies: How can one identify a given staggered module, and how can one demonstrate the existence of a proposed staggered module. Given just the values of the highest weights of the left and right modules, themselves subject to simple necessary conditions, invariants are defined which together with the knowledge of the left and right modules uniquely identify a staggered module. The possible values of these invariants form a vector space of dimension 0, 1, or 2, and the structures of the left and right modules limit the isomorphism classes of the corresponding staggered modules to an affine subspace (possibly empty). The number of invariants and affine restrictions is purely determined by the structures of the left and right modules. Moreover, in order to facilitate applications, the expressions for the invariants and restrictions are given by formulas as explicit as possible (they generally rely on expressions for Virasoro singular vectors). Finally, the text is liberally peppered throughout with examples illustrating the general concepts. These have been carefully chosen for their physical relevance or for the novel features they exhibit.

On the integrable structure of the Ising model

Starting from the lattice A3 realization of the Ising model defined on a strip with integrable boundary conditions, theexact spectrum (including excited states) of all the local integrals of motion is derivedin the continuum limit by means of thermodynamic Bethe ansatz techniques.It is also possible to follow the massive flow of this spectrum between the UVc = 1/2 conformal fixed point and the massive IR theory. The UV expression of the eigenstates ofsuch integrals of motion in terms of Virasoro modes is found to have only rationalcoefficients and their fermionic representation turns out to be simply related to thequantum numbers describing the spectrum.

Embedding of bases: From the [formula omitted] to the [formula omitted] models

A new quasi-particle basis of states is presented for all the irreducible modules of the M(3,p) models. It is formulated in terms of a combination of Virasoro modes and the modes of the field ϕ2,1. This leads to a fermionic expression for particular combinations of irreducible M(3,p) characters, which turns out to be identical with the previously known formula. Quite remarkably, this new quasi-particle basis embodies a sort of embedding, at the level of bases, of the minimal models M(2,2κ+1) into the M(3,4κ+2−δ) ones, with 0⩽δ⩽3.

Ghost Systems Revisited: Modified Virasoro Generators and Logarithmic Conformal Field Theories

We study the possibility of extending ghost systems with higher spin to a logarithmic conformal field theory. In particular we are interested in c=-26 which turns out to behave very differently to the already known c=-2 case. The energy momentum tensor cannot be built anymore by a combination of derivatives of generalized symplectic fermion fields. Moreover, the logarithmically extended theory is only consistent when considered on nontrivial Riemann surfaces. This results in a LCFT with some unexpected properties. For instance the Virasoro mode L_0 is diagonal and for certain values of the deformation parameters even the whole global conformal group is non-logarithmic.

Kaluza–Klein theory, AdS/CFT correspondence and black hole entropy

The asymptotic symmetries of the near-horizon geometry of a lifted (near-extremal) Reissner–Nordstrom black hole, obtained by inverting the Kaluza–Klein reduction, explain the deviation of the Bekenstein–Hawking entropy from extremality. We point out the fact that the extra dimension allows us to justify the use of a Virasoro mode decomposition along the time-like boundary of the near-horizon geometry, AdS2 × Sn, of the lower-dimensional (Reissner–Nordstrom) spacetime.

An open bosonic string with one end fixed

We study a bosonic string with one end free and the otherconfined to a D0-brane. Only the odd oscillator modes are allowed,which leads to a Virasoro algebra of even Virasoro modes only. The theoryis quantized in a gauge where worldsheet time and ordinary time areidentified. There are no negative or null norm states, and no tachyon. TheRegge slope is twice that of the open string;this can serve as a test of the usefulness of the model as asemi-quantitative description of mesons with one light and one extremelyheavy quark when such higher-spin mesons are found. TheVirasoro conditions select specific SO(D-1) irreps. Theasymptotic density of states can be estimated by adaptingthe Hardy-Ramanujan analysis to a partition of oddintegers; the estimate becomes exact as D goes toinfinity.