Two-dimensional conformal field theory and beyond. Lessons from a continuing fashion. (Q5945518)

Two-dimensional conformal field theory has many faces, for examples, description of surface of critical phenomena or the study of (super)string vacua. The paper is a review elucidating the role of CFT in the appearing new mathematical concepts and methods, that was generated by the development of quantum field theory and that in its turn make influence to theoretical physics. There are rational conformal field theory and the notions of anomalous dimensions, braid group statistics. The author describes consecutive steps of interaction between physics and mathematics, in particular, he explains the present state of investigations in solution of the problem of putting together quantum mechanics and general relativity into quantum gravity and sketch dramatic history of development of the theory of current algebras. Concerning anomalous dimensions, the author points out that such a dimension was discovered twice before it was being taken in the physical community, and then he gives their mathematical treatment as labels of irreducible positive representations of the conformal group. He connects these to the supersymmetric Yang-Mills theory. In the section devoted to the rational conformal field theories the problem of existence of such theories is discussed: the role in its solution of operator product expansions and of the result of Kac (computaion of determinant). Some future problems in such context are touched. In the section concerning fractional statistics the physical models with such statistics are counted and it is pointed out that such statistics arise as monodromy representations of the generalized braid groups. In conclusion the mathematical notions, methods and theories (vertex algebras, knot invariants, low-dimensional topology, mirror manifolds etc.) that have obtained new sounds at the last years and the influence of modern physics to the solution of the long-standing mathematical problems are discussed.