Conformal Minimal Models Research Articles

Matrix models, topological strings, and supersymmetric gauge theories

We show that B-model topological strings on local Calabi–Yau threefolds are large- N duals of matrix models, which in the planar limit naturally give rise to special geometry. These matrix models directly compute F-terms in an associated N=1 supersymmetric gauge theory, obtained by deforming N=2 theories by a superpotential term that can be directly identified with the potential of the matrix model. Moreover by tuning some of the parameters of the geometry in a double scaling limit we recover ( p, q) conformal minimal models coupled to 2d gravity, thereby relating non-critical string theories to type II superstrings on Calabi–Yau backgrounds.

Random defect lines in conformal minimal models

We analyze the effect of adding quenched disorder along a defect line in the 2D conformal minimal models using replicas. The disorder is realized by a random applied magnetic field in the Ising model, by fluctuations in the ferromagnetic bond coupling in the Tricritical Ising model and Tricritical Three-state Potts model (the $\phi_{12}$ operator), etc.. We find that for the Ising model, the defect renormalizes to two decoupled half-planes without disorder, but that for all other models, the defect renormalizes to a disorder-dominated fixed point. Its critical properties are studied with an expansion in $\eps \propto 1/m$ for the mth Virasoro minimal model. The decay exponents $X_N=\frac{N}{2}(1-\frac{9(3N-4)}{4(m+1)^2}+ \mathcal{O}(\frac{3}{m+1})^3)$ of the Nth moment of the two-point function of $\phi_{12}$ along the defect are obtained to 2-loop order, exhibiting multifractal behavior.This leads to a typical decay exponent $X_{\rm typ}={1/2} (1+\frac{9}{(m+1)^2}+\mathcal{O}(\frac{3}{m+1})^3)$. One-point functions are seen to have a non-self-averaging amplitude. The boundary entropy is larger than that of the pure system by order 1/m^3. As a byproduct of our calculations, we also obtain to 2-loop order the exponent $\tilde{X}_N=N(1-\frac{2}{9\pi^2}(3N-4)(q-2)^2+\mathcal{O}(q-2)^3)$ of the Nth moment of the energy operator in the q-state Potts model with bulk bond disorder.

The scaling supersymmetric Yang–Lee model with boundary

We define the scaling supersymmetric Yang-Lee model with boundary as the (1,3) perturbation of the superconformal minimal model SM(2/8) (or equivalently, the (1,5) perturbation of the conformal minimal model M(3/8)) with a certain conformal boundary condition. We propose the corresponding boundary S matrix, which is not diagonal for general values of the boundary parameter. We argue that the model has an integral of motion corresponding to an unbroken supersymmetry, and that the proposed S matrix commutes with a similar quantity. We also show by means of a boundary TBA analysis that the proposed boundary S matrix is consistent with massless flow away from the ultraviolet conformal boundary condition.

ADE spectra in conformal field theory

We demonstrate that certain Virasoro characters (and their linear combinations) in minimal and non-minimal conformal models which admit factorized forms are manifestly related to the ADE series. This permits to extract quasi-particle spectra of a Lie algebraic nature which resembles the features of Toda field theory. These spectra possibly admit a construction in terms of the W n -generators. In the course of our analysis we establish interrelations between the factorized characters related to the parafermionic models, the compactified boson and the minimal models.

Boundary operators and touching of loops in 2D gravity

We investigate the correlators in unitary minimal conformal models coupled to two-dimensional gravity from the two-matrix model. We show that simple fusion rules for all of the scaling operators exist. We demonstrate the role played by the boundary operators and discuss its connection to how loops touch each other.

Form factors of exponential operators and exact wave function renormalization constant in the Bullough-Dodd model

We compute the form factors of exponential operators e kgϕ( x) in the two-dimensional integrable Bullough-Dodd model ( a 2 (2) affine Toda field theory). These form factors are selected among the solutions of general non-derivative scalar operators by their asymptotic cluster property. Through analytical continuation to complex values of the coupling constant these solutions permit to compute the form factors of scaling relevant primary fields in the lightest-breather sector of integrable / gf 1,2 and / gf 1,5 deformations of conformal minimal models. We also obtain the exact wave-function renormalization constant Z( g) of the model and the properly normalized form factors of the operators ϕ( x) and : ϕ 2( x) :.

Green's function for the [formula omitted] operator on genus-2 surfaces

We explicitly evaluate the Green function for the ∂ operator acting on the quadratic differentials on genus-2 hyperelliptic surfaces. As a consequence we express the second-order null vector equations for the conformal blocks of the minimal conformal models in terms of coordinates that naturally exist on the hyperelliptic surfaces.

Randomly coupled minimal models

Using 1-loop renormalisation group equations, we analyze the effect of randomness on multi-critical unitary minimal conformal models. We study the case of two randomly coupled M p models and found that they flow in two decoupled M p−1 models, in the infrared limit. This result is then extend to the case with M randomly coupled M p models, which will flow toward M decoupled M p−1 .

Generalized KdV and quantum inverse scattering description of conformal minimal models

We propose an alternative description of two-dimensional conformal field theory in terms of quantum inverse scattering. It is based on the generalized KdV systems attached to A 2 (2), yielding the classical limit of Virasoro as Poisson bracket structure. The corresponding T-system is shown to coincide with the one recently proposed by Kuniba and Suzuki. We classify the primary operators of the minimal models that commute with all the integrals of motion, and that are therefore candidates to perturb the model by keeping the conservation laws. For our A 2 (2) structure these happen to be Φ 1,2, Φ 2,1, Φ 1,5, in contrast to the A 1 (1) case, studied by Bazhanov, Lukyanov and Zamolodchikov [preprint CLNS 94/1316, RU-94-98, hep-th/9412229], related to Φ 1,3.

Hamiltonian reduction of SL (2) theories at the level of correlators

Since the work of Bershadsky and Ooguri and Feigin and Frenkel it is well known that correlators of SL(2) current algebra for admissible representations should reduce to correlators for conformal minimal models. A precise proposal for this relation has been given at the level of correlators: When SL(2) primary fields are expressed as φ j ( z n , x n ) with x n being a variable to keep track of the SL(2) representation multiplet (possibly infinitely dimensional for admissible representations), then the minimal model correlator is supposed to be obtained simply by putting all x n = z n . Although strong support for this has been presented, to the best of our understanding a direct, simple proof seems to be missing so in this paper we present one based on the free field Wakimoto construction and our previous study of that in the present context. We further verify that the explicit SL(2) correlators we have published in a recent preprint reduce in the above way, up to a constant which we also calculate. We further discuss the relation to more standard formulations of hamiltonian reduction.

Quantum symmetry and fundamental S-matrix for vector perturbed WDnmodels

Perturbations of WDn and W3 conformal theories which generalize the (1,2) perturbations of conformal minimal models are shown to be integrable by 'counting arguments'. The A2n-l,q2 and D4,q3 of corresponding S-matrices are conjectured and proved by explicit construction of conserved non-local charges in the WD3 case with the proper quantum symmetry group. The gradation change of the known R-matrix with A52 symmetry from homogeneous to spin, which turns out to be relevant for the perturbation considered, is shown to make the R-matrix crossing invariant and fixes the effective coupling constant as a function of the initial one. Using these results the fundamental S-matrix for vector perturbed minimal WD3 models is constructed on the basis of the RSOS model with the corresponding symmetry. The S-matrix is checked to reproduce the known results in two particular cases of minimal model number: k=l and k to infinity .

On integrable c < 1 open-closed string theory

The integrable structure of open-closed string theories in the ( p, q) conformal minimal model backgrounds is presented. The relation between the τ-function of the closed string theory and that of the open-closed string theory is uncovered. The resulting description of the open-closed string theory is shown to fit very naturally into the framework of the sl(q, C) KdV hierarchies. In particular, the twisted bosons which underlie and organise the structure of the closed string theory play a similar role here and may be employed to derive loop equations and correlation function recursion relations for the open-closed strings in a simple way.

Is 2D turbulence a conformal turbulence?

A critical analysis of the conformal approach to the theory of 2d turbulence is delivered. It is shown, in particular, that conformal minimal models cannot give a general turbulent solution, which should provide for constant fluxes of all vorticity integrals of motion.

Weak quasi quantum group symmetries in all minimal conformal models

We give a systematic proof that the fusion rules of all minimal conformal models permit only weak quasitriangular quasi Hopf algebras as symmetry algebras. On the other hand we construct a weak quasi tensor functor for every rational conformal field theory which leads to a weak quasitriangular quasi Hopf algebra via the (generalized) reconstruction theorem of Majid.

Fermionic Thirring theories as minimal model actions

The action describing the SU (2) massless fermionic Thirring model at the conformal points which coincide with the Goddard-Kent-Olive solutions of the affine-Virasoro construction is proposed as the lagrangian for the minimal conformal models.

Solving Virasoro constraints on integrable hierarchies via the Kontsevich-Miwa transform

We solve Virasoro constraints on the KP hierarchy in terms of minimal conformal models. The constraints we start with are implemented by the Virasoro generators depending on a background charge Q. Then the solutions to the constraints are given by the theory which has the same field content as the David-Distler-Kawai theory: it consists of a minimal matter scalar with background charge Q, dressed with an extra “Liouville” scalar. In particular, the Virasoro-constrained τ-function is related to the correlator of a product of (dressed) “21” operators. The construction is based on a generalization of the Kontsevich parametrization of the KP times achieved by introducing into it Miwa parameters which depend on the value of Q. Under the thus defined Kontsevich-Miwa transformation, the Virasoro constraints are proven to be equivalent to a master equation depending on the parameter Q. The master equation is further identified with the decoupling equation corresponding to a null vector at level 2. We conjecture that higher-level decoupling equations are similarly related to constraints on the KP τ-function. We also consider the master equation for the N-reduced KP hierarchies. Several comments are made on a possible relation between the generalized master equation and scaled Kontsevich-type matrix integrals, and on the form the equation takes in higher genera.

SL(2, [formula omitted]) Chern-Simons theories with rational charges and two-dimensional conformal field theories

We present a hamiltonian quantization of the SL(2, R ) three-dimensional Chern-Simons theory with fractional coupling constant k = s/ r on a space manifold with torus topology in the “constrain-first” framework. By generalizing the “Weyl-odd” projection to the fractional charge case, we obtain multi-components holomorphic wave functions whose components are the Kac-Wakimoto characters of the modular-invariant admissible representations of  1 current algebra with fractional level. The modular representations carried by the quantum Hilbert space satisfy both Verlinde's and Vafa's constraints coming from conformal field theory. They are the “square roots” of the representations associated to the conformal ( r, s) minimal models. Our results imply that Chern-Simons theory with SO(2, 2) as gauge group, which describes (2 + 1)-dimensional gravity with negative cosmological constant, has the modular properties of the Virasoro discrete series. On the way, we show that the two-dimensional counterparts of Chern-Simons SU(2) theories with half-integer charge k = 1 2 p are the modular invariant D p+1 series of  1 current algebra of level 2 p − 2.

Stable non-perturbative minimal models coupled to 2D quantum gravity

A generalisation of the non-perturbatively stable solutions of string equations which respect the KdV flows, obtained recently for the (2 m − 1, 2) conformal minimal models coupled to two-dimensional quantum gravity, is presented for the ( p, q) models. These string equations are the most general string equations compatible with the qth generalised KdV flows. They exhibit a close relationship with the bi-hamiltonian structure in these hierarchies. The Ising model is studied as a particular example, for which a real non-singular numerical solution to the string susceptibility is presented.

INTEGRABLE DEFORMATIONS OF THE NONUNITARY MINIMAL CONFORMAL MODEL ℳ3,5

The scaling region of the nonunitary minimal conformal model M3,5 is described by three different integrable massive field theories. We propose the scattering theory for the integrable deformation of M3,5 by the field ψ with anomalous dimensions [Formula: see text]. The spectrum of this theory is confirmed by the Truncation Conformal Space Approach. We also consider the thermodynamics of the one-dimensional quantum system defined by the transfer matrix relative to the deformation of M3,5 by the field φ with anomalous dimensions [Formula: see text]. This deformation drives the original conformal model into a region of the phase diagram where there are plasma oscillations.

INTEGRABLE DEFORMATIONS OF CONFORMAL THEORIES AND BOOTSTRAP TREES

Recent results in the study of massive integrable quantum field theories in (1+1) dimensions considered as perturbed conformal minimal models are presented. The on mass-shell properties of such theories, with a particular emphasis on the bootstrap principle, are investigated.