Models Of Conformal Field Theory Research Articles

Renormalization group and infinite algebraic structure inD-dimensional conformal field theory

We consider scalar field theory in the D-dimensional space with nontrivial metric and local action functional of most general form. It is possible to construct for this model a generalization of renormalization procedure and RG-equations. In the fixed point the diffeomorphism and Weyl transformations generate an infinite algebraic structure of D-Dimensional conformal field theory models. The Wilson expansion and crossing symmetry enable to obtain sum rules for dimensions of composite operators and Wilson coefficients.

Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center

The SL(2,Z) representation $\pi$ on the center of the restricted quantum group U_{q}sl(2) at the primitive 2p-th root of unity is shown to be equivalent to the SL(2,Z) representation on the extended characters of the logarithmic (1,p) conformal field theory model. The multiplicative Jordan decomposition of the U_{q}sl(2) ribbon element determines the decomposition of $\pi$ into a ``pointwise'' product of two commuting SL(2,Z) representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2,Z) representation on the (1,p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of U_{q}sl(2) at the primitive 2p-th root of unity is shown to coincide with the fusion algebra of the (1,p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of~U_{q}sl(2).

Vertex algebras and conformal field theory models in four dimensions

AbstractThe notion of global conformal invariance (GCI) in Minkowski space allows to prove rationality of correlation functions and to extend the concept of vertex algebra to any number D of space‐time dimensions. The case of even D, which includes a conformal stress‐energy tensor with a rational 3‐point function, is of particular interest. Recent progress, reviewed in the talk, includes a full account of Wightman positivity at the 4‐point level for D=4, and a study of modular properties of thermal expectation values of the conformal energy operator.

Toda Systems as Models of Conformal Classical Field Theory

Toda Systems as Models of Conformal Classical Field Theory

CFT Description of the Three-Dimensional Hawking-Page Transition

We construct a phenomenological conformal field theory (CFT) model of the three-dimensional Hawking-Page transition. We find that free fermion CFT models on the boundary torus give a description of the three-dimensional Hawking-Page transition. If modular invariance is respected, the free fermion model implies that the transition occurs continuously through the conical space phase and the small black hole phase. On the other hand, if we are allowed to break modular invariance, we can construct a free fermion model that reproduces the usual Euclidean semi-classical result, and in particular exhibits a first-order phase transition.

Three-point function in the minimal Liouville gravity

We revisit the problem of the structure constants of the operator product expansions in the minimal models of conformal field theory, rederiving these previously known constants and presenting them in a form particularly useful in Liouville gravity applications. We discuss the analytic relation between our expression and the structure constant in the Liouville field theory and also give the three- and two-point correlation numbers on the sphere in the minimal Liouville gravity in the general form.

Three-point function in the minimal Liouville gravity

The problem of the structure constants of the operator product expansions in the minimal models of conformal field theory is revisited. We rederive these previously known constants and present them in the form particularly useful in the Liouville gravity applications. Analytic relation between our expression and the structure constant in Liouville field theory is discussed. Finally we present in general form the three- and two-point correlation numbers on the sphere in the minimal Liouville gravity.

Three-point function in the minimal Liouville gravity

We revisit the problem of the structure constants of the operator product expansions in the minimal models of conformal field theory, rederiving these previously known constants and presenting them in a form particularly useful in Liouville gravity applications. We discuss the analytic relation between our expression and the structure constant in the Liouville field theory and also give the three- and two-point correlation numbers on the sphere in the minimal Liouville gravity in the general form.

Logarithmic limits of minimal models

It is discussed how a limiting procedure of (super)conformal field theories may result in logarithmic (super)conformal field theories. The construction is illustrated by logarithmic limits of (unitary) minimal models in conformal field theory and in N = 1 superconformal field theory.

Twisted modules over vertex algebras on algebraic curves

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let V be a vertex algebra, H a finite group of automorphisms of V, and C an algebraic curve such that H⊂Aut( C). We show that a suitable collection of twisted V-modules gives rise to a section of a certain sheaf on the quotient X= C/ H. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac–Moody vertex algebras. We also give a chiral algebra interpretation of twisted modules.

Torus structure on graphs and twisted partition functions for minimal and affine models

Using the Ocneanu quantum geometry of ADE diagrams (and of other diagrams belonging to higher Coxeter–Dynkin systems), we discuss the classification of twisted partition functions for affine and minimal models in conformal field theory and study several examples associated with the WZW, Virasoro and W 3 cases.

Excited state TBA for the φ2,1 perturbed [formula omitted] model

We examine some excited state energies in the non-unitary integrable quantum field theory (IQFT) obtained from the perturbation of the minimal conformal field theory (CFT) model M 3,5 by its operator φ 2,1. Using the correspondence of this IQFT to the scaling limit of the dilute- A 2 lattice model (in a particular regime) we derive the functional equations for the QFT commuting transfer matrices. These functional equations can be transformed to a closed set of TBA-like integral equations which determine the excited state energies in the finite-size system. In particular, we explicitly construct these equations for the ground state and two lowest excited states. Numerical results for the associated energy gaps are compared with those obtained by the truncated conformal space approach (TCSA).

Four-dimensional conformal field theory models with rational correlation functions

Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in four-dimensional Minkowski spacetime. We consider in detail a model of a neutral scalar field of dimension two. It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of is shown to coincide with a simpler one, generated by a bilocal scalar field V(x1,x2) of dimension (1,1). The modes of V together with the unit operator span an infinite-dimensional Lie algebra V whose vacuum (i.e. zero-energy lowest-weight) representations only depend on the central charge c. Wightman positivity (i.e. unitarity of the representations of V) is proven to be equivalent to c.

A non-rational CFT with c = 1 as a limit of minimal models

We investigate the limit of minimal model conformal field theories where the central charge approaches one. We conjecture that this limit is described by a non-rational CFT of central charge one. The limiting theory is different from the free boson but bears some resemblance to Liouville theory. Explicit expressions for the three point functions of bulk fields are presented, as well as a set of conformal boundary states. We provide analytic and numerical arguments in support of the claim that this data forms a consistent CFT.

Modular transformation and boundary states in logarithmic conformal field theory

We study the c=−2 model of logarithmic conformal field theory in the presence of a boundary using symplectic fermions. We find boundary states with consistent modular properties. A peculiar feature of this model is that the vacuum representation corresponding to the identity operator is a sub-representation of a “reducible but indecomposable” larger representation. This leads to unusual properties, such as the failure of the Verlinde formula. Despite such complexities in the structure of modules, our results suggest that logarithmic conformal field theories admit bona fide boundary states.

New families of flows between two-dimensional conformal field theories

We present evidence for the existence of infinitely-many new families of renormalisation group flows between the nonunitary minimal models of conformal field theory. These are associated with perturbations by the $\phi_{21}$ and $\phi_{15}$ operators, and generalise a family of flows discovered by Martins. In all of the new flows, the finite-volume effective central charge is a non-monotonic function of the system size. The evolution of this effective central charge is studied by means of a nonlinear integral equation, a massless variant of an equation recently found to describe certain massive perturbations of these same models. We also observe that a similar non-monotonicity arises in the more familiar $\phi_{13}$ perturbations, when the flows induced are between nonunitary minimal models.

The Perturbation ϕ2, 1 of the M(p, p + 1) Models of Conformal Field Theory and Related Polynomial-Character Identities

Using q-trinomial coefficients of Andrews and Baxter along with the technique of telescopic expansions, we propose and prove a complete set of polynomial identities of Rogers-Ramanujan type for M(p, p + 1) models of conformal field theory perturbed by the operator ϕ2, 1. The bosonic form of our polynomials is closely related to corner transfer matrix sums which arise in the computation of the order parameter in the regime 1+ of Ap−1 dilute models. In the limit where the degree of the polynomials tends to infinity our identities provide new companion fermionic representations for all Virasoro characters of unitary minimal series.

Intertwining operator algebras and vertex tensor categories for affine Lie algebras

We apply the general theory of tensor products of modules for a vertex operator algebra developed in our papers hep-th/9309076, hep-th/9309159, hep-th/9401119, q-alg/9505018, q-alg/9505019 and q-alg/9505020 to the case of the Wess-Zumino-Novikov-Witten models and related models in conformal field theory. We show that for the category of modules for a vertex operator algebra containing a subalgebra isomorphic to a tensor product of rational vertex operator algebras associated to affine Lie algebras, the intertwining operators among the modules have the associativity property, the category has a natural structure of vertex tensor category, and a number of related results hold. We obtain, as a corollary and special case, a construction of the previously-studied braided tensor category structure on the category of finite direct sums of standard (integrable highest weight) modules of a fixed positive integral level for an affine Lie algebra.

QUANTUM NOETHER'S THEOREM AND CONFORMAL FIELD THEORY: A STUDY OF SOME MODELS

We study the problem of recovering Wightman conserved currents from the canonical local implementations of symmetries which can be constructed in the algebraic framework of quantum field theory, in the limit in which the region of localization shrinks to a point. We show that, in a class of models of conformal quantum field theory in space-time dimension 1+1, which includes the free massless scalar field and the SU(N) chiral current algebras, the energy-momentum tensor can be recovered. Moreover we show that the scaling limit of the canonical local implementation of SO(2) in the free complex scalar field is zero, a manifestation of the fact that, in this last case, the associated Wightman current does not exist.

Logarithmic conformal field theory and Seiberg-Witten models

The periods of arbitrary abelian forms on hyperelliptic Riemann surfaces, in particular the periods of the meromorphic Seiberg-Witten differential λ SW, are shown to be in one-to-one correspondence with the conformal blocks of correlation functions of the rational logarithmic conformal field theory with central charge c= c 2,1=−2. The fields of this theory precisely simulate the branched double covering picture of a hyperelliptic curve, such that generic periods can be expressed in terms of certain generalised hypergeometric functions, namely the Lauricella functions of type F D .