Alexander B. Zamolodchikov

Infinite conformal symmetry in two-dimensional quantum field theory

We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of Virasoro algebra, and that the correlation functions are built up of the “conformal blocks” which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the degenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations.

Factorized S-Matrices in Two Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Theory Models

Abstract The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N ⩾ 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N ⩾ 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N ⩾ 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.

Current algebra and Wess-Zumino model in two dimensions

We investigate quantum field theory in two dimensions invariant with respect to conformal (Virasoro) and non-abelian current (Kac-Moody) algebras. The Wess-Zumino model is related to the special case of the representations of these algebras, the conformal generators being quadratically expressed in terms of currents. The anomalous dimensions of the Wess-Zumino fields are found exactly, and the multipoint correlation functions are shown to satisfy linear differential equations. In particular, Wittens non-abelean bosonisation rules are proven.

Fractal Structure of 2D Quantum Gravity

We resolve renormalization problems, indicated in Ref. 1 and find explicit formulae for the spectrum of anomalous dimensions in 2d—quantum gravity. Comparison with combinatorial approximation of random surfaces and its numerical analyses shows complete agreement with all known facts.

CONFORMAL BOOTSTRAP IN LIOUVILLE FIELD THEORY

Attention to the two-dimensional Liouville Field Theory (LFT) is drawn basically for two reasons. First it was recognized [1] as an effective field theory of the 2d. quantum gravity. In particular it is very relevant in the string theory [1–4]. Second, it is an example of non-rational conformai field theory (CFT) which is very likely exactly solvable (e.g., the classical equations of motion are integrable).The interest to LFT was renewed recently with the development of the matrix model approach to 2d gravity [5, 6]. It was shown that LFT is able to reproduce some of the predictions of the matrix model approach, in particular the scaling behavior [7–9], the genus one partition functions [10] and some of the integrated correlation functions [11–15]. It is very plausible therefore that LFT describes the same 2d quantum gravity as the matrix models do (at least in the “weak coupling region” C L > 25).An analytic expression is proposed for the three-point function of the exponential fields in the Liouville field theory on a sphere. In the classical limit it coincides with what the classical Liouville theory predicts. Using this function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and verify numerically that it satisfies the conformal bootstrap equations, i.e., that the operator algebra thus defined is associative. We consider also the Liouville reflection amplitude which follows explicitly from the structure constants.

Conformal quantum field theory models in two dimensions having Z3 symmetry

Abstract An infinite set of conformal quantum field theory models is constructed in two dimensions. The models possess an infinite-dimensional symmetry generated by a local spin-3 current in addition to the conformal symmetry, and exhibit Z3 invariance. The simplest model of this set describes the critical point of a 2-state Potts model.

Integrable structure of conformal field theory, quantum KdV theory and Thermodynamic Bethe Ansatz

AbstractWe construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as “T-operators,” act in highest weight Virasoro modules. TheT-operators depend on the spectral parameter λ and their expansion around λ=∞ generates an infinite set of commuting Hamiltonians of the quantum KdV system. TheT-operators can be viewed as the continuous field theory versions of the commuting transfermatrices of integrable lattice theory. In particular, we show that for the values

Integrable Field Theory from Conformal Field Theory

c = 1 - 3\frac{{3(2n + 1)^2 }}{{2n + 3}}