A. A. Belavin

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.

Pseudoparticle Solutions of the Yang-Mills Equations

We find regular solutions of the four dimensional euclidean Yang-Mills equations. The solutions minimize locally the action integrals which is finite in this case. The topological nature of the solutions is discussed.

Algebraic geometry and the geometry of quantum strings

Abstract The p -loop amplitude of closed oriented bosonic strings in 26 dimensions is considered. The integration measure in this case is a measure on the moduli space M p of Riemann surfaces of genus p . It is proved that for p > 1 this measure is a squared modulus of a holomorphic function having a second order pole on degenerate surfaces, divided by (det N 1 ) 13 . N 1 is a matrix of scalar products of holomorphic one-forms on a surface. This property fixes the measure uniquely; this can be derived either by a direct calculation, or using a theorem of Mumford.

Dynamical symmetry of integrable quantum systems

Abstract We consider the equations of triangles (alias Yang-Baxter equations), which a factorized two-particle S -matrix obeys. These equations are shown to possess a symmetry which consists of discrete Lorentz transformations acting independently on states of particles with different momenta. It is this symmetry which ensures compatibility of the overconstrained equations of triangles. The use of it enables one to construct the factorized two-particle S -matrix requiring invariance (automorphity) with respect to discrete Lorentz transformations.

Infinite Conformal Symmetry of Critical Fluctuations in Two-Dimensions

We study the massless quantum field theories describing the critical points in two dimensional statistical systems. These theories are invariant with respect to the 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 the Virasoro algebra. Exactly solvable theories associated with degenerate representations are analized. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations.

Yang-Mills equations as inverse scattering problem

Non-linear self-duality equations F v = rpm are shown to be conditions of compatibility of two linear equations.rAll the N-instanton fields are constructed explicity.

Exact solution of the two-dimensional model with asymptotic freedom

Abstract An exact solution of the SU(2)-symmetric theory with four-fermion interaction in one spatial and one time dimension is derived.

Two- and three-loop amplitudes in the bosonic string theory

Abstract Explicit formulae are obtained for two- and three-loop vacuum amplitudes in the theory of closed oriented bosonic strings at d =26 n terms of the theta constants, with the moduli space being parametrized by period matrices.

AGT conjecture and Integrable structure of Conformal field theory for c=1

Abstract AGT correspondence gives an explicit expressions for the conformal blocks of d = 2 conformal field theory. Recently an explanation of this representation inside the CFT framework was given through the assumption about the existence of the special orthogonal basis in the module of algebra A = Vir ⊗ H . The basis vectors are the eigenvectors of the infinite set of commuting integrals of motion. It was also proven that some of these vectors take form of Jack polynomials. In this note we conjecture and verify by explicit computations that in the case of the Virasoro central charge c = 1 all basis vectors are just the products of two Jack polynomials. Each of the commuting integrals of motion becomes the sum of two integrals of motion of two noninteracting Calogero models. We also show that in the case c ≠ 1 it is necessary to use two different Feigin–Fuks bosonizations of the Virasoro algebra for the construction of all basis vectors which take form of one Jack polynomial.

Instantons and 2d superconformal field theory

A recently proposed correspondence between 4-dimensional