Integrable Conformal Field Theory Research Articles

Strongly γ-Deformed N=4 Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory.

We demonstrate by explicit multiloop calculation that γ-deformed planar N=4 supersymmetric Yang-Mills (SYM) theory, supplemented with a set of double-trace counterterms, has two nontrivial fixed points in the recently proposed double scaling limit, combining vanishing 't Hooft coupling and large imaginary deformation parameter. We provide evidence that, at the fixed points, the theory is described by an integrable nonunitary four-dimensional conformal field theory. We find a closed expression for the four-point correlation function of the simplest protected operators and use it to compute the exact conformal data of operators with arbitrary Lorentz spin. We conjecture that both conformal symmetry and integrability should survive in γ-deformed planar N=4 SYM theory for arbitrary values of the deformation parameters.

Integration of the SL(2, [formula omitted])/U(1) gauged WZNW model with periodic boundary conditions

Gauged WZNW models are integrable conformal field theories. We integrate the classical SL(2, R )/U(1) theory with periodic boundary conditions, which describes closed strings moving in a curved target-space geometry. We calculate its Poisson bracket structure by solving an initial state problem. The results differ from previous field-theoretic calculations due to zero-modes. For a future exact canonical quantization the physical fields are (non-locally) transformed onto canonical free fields.

The Poisson Bracket Structure of the SL(2, R)/U(1) Gauged WZNW Model with Periodic Boundary Conditions

The gauged SL(2, R)/U(1) Wess-Zumino-Novikov-Witten (WZNW) model is classically an integrable conformal field theory. A second-order differential equation of the Gelfand-Dikii type defines the Poisson bracket structure of the theory. For periodic boundary conditions zero modes imply non-local Poisson brackets which, nevertheless, can be represented by canonical free fields.

Analytical solution of the SL(2, [formula omitted])/U(1) WZNW black hole model

The gauged SL(2, R )/U(1) Wess-Zumino-Novikov-Witten (WZNW) model is classically an integrable conformal field theory. We have found a Lax pair representation for the non-linear equations of motion, and a Bäcklund transformation. A second-order differential equation of the Gelfand-Dikii type defines the Poisson bracket structure of the theory, and its fundamental solutions describe the general solution of the WZNW model as well. The physical and free fields are related by non-local transformations. The (anti-)chiral component of the energy-momentum tensor which factorizes into conserved quantities satisfying a non-linear algebra characteristic for parafermions assumes the canonical free filed form. So the black hole model is expected to be prepared for an exact canonical quantization.

HIGHER SPIN SYMMETRY AND SU(2) WZNW MODEL

Generators of an infinite number of conserved charges of the integrable SU(2) WZNW model are identified with the generators of a W1+∞ symmetry algebra. This integrable conformal field theory also describes the ZN symmetric Ising model at the critical point in the large-N limit.

INTEGRABLE CONFORMAL FIELD THEORY IN FOUR DIMENSIONS AND FOURTH-RANK GEOMETRY

We consider the conformal properties of geometries described by higher-rank line elements. A crucial role is played by the conformal Killing equation (CKE). We introduce the concept of null-flat spaces in which the line element can be written as dsr=r!dζ1 … dζr. We then show that, for null-flat spaces, the critical dimension, for which the CKE has infinitely many solutions, is equal to the rank of the metric. Therefore, in order to construct an integrable conformal field theory in four dimensions we need to rely on fourth-rank geometry. We consider the simple model ℒ = ¼ Gμνλρ ∂μ ϕ ∂ν ϕ ∂λ ϕ ∂ρϕ and show that it is an integrable conformal model in four dimensions. Furthermore, the associated operators satisfy a Vir4 algebra.

A quantum group structure in integrable conformal field theories

We discuss a quantization prescription of the conformal algebras of a class ofd=2 conformal field theories which are integrable. We first give a geometrical construction of certain extensions of the classical Virasoro algebra, known as classicalW algebras, in which these algebras arise as the Lie algebra of the second Hamiltonian structure of a generalized Korteweg-de Vries hierarchy. This fact implies that theW algebras, obtained as a reduction with respect to the nilpotent subalgebras of the Kac-Moody algebra, describe the integrability of a Toda field theory. Subsequently we determine the coadjoint operators of theW algebras, and relate these to classical Yang-Baxter matrices. The quantization of these algebras can be carried out using the concept of a so-called quantum group. We derive the condition under which the representations of these quantum groups admit a Hilbert space completion by exploring the relation with the braid group. Then we consider a modification of the Miura transformation which we use to define a quantumW algebra. This leads to an alternative interpretation of the coset construction for Kac-Moody algebras in terms of nonlinear integrable hierarchies. Subsequently we use the connection between the induced braid group representations and the representations of the mapping class group of Riemann surfaces to identify an action of theW algebras on the moduli space of stable curves, and we give the invariants of this action. This provides a generalization of the situation for the Virasoro algebra, where such as invariant is given by the so-called Mumford form which describes the partition function of the bosonic string.