A. M. Perelomov

QUASI-EXACTLY-SOLVABLE QUANTAL PROBLEMS: ONE-DIMENSIONAL ANALOGUE OF RATIONAL CONFORMAL FIELD THEORIES

The class of quasi-exactly-solvable problems in ordinary quantum mechanics discovered recently shows remarkable parallels with rational two-dimensional conformal field theories. This fact suggests that investigation of the quasi-exactly-solvable models may shed light on rational conformal field theories. We discuss a relation between these two theoretical schemes and propose a mathematical formulation for the procedure of constructing quasi-exactly solvable systems. This discussion leads us to a kind of generalization of the Sugawara construction.

Exact Gell-Mann-Low function of supersymmetric Kähler sigma models

Abstract We consider a broad class of Kahler supersymmetric sigma models in two-dimensional space-time. The exact Gell-Mann-Low function is found within the framework of the method proposed earlier [1, 2] and based on analysis of classical solutions. It is shown that the exact beta function accounting for all orders in the coupling constant actually coincides with the one-loop result.

Hyperkählerian manifolds and exact β-functions of two-dimensional N = 4 supersymmetric σ-models

Two-dimensional supersymmetric sigma-models on cotangent bundles over CP n are investigated. These manifolds are supplied with hyperkahlerian metrics, and the corresponding σ-models possess N = 4 supersymmetry. Also they admit instantonic solutions, which allows us to apply the NSVZ method to calculate exact β-functions. β( g 2 ) = 0, as was expected.

Complex Geometry and String Theory

String theory is a rather new field of theoretical physics. It appeared only twenty years ago to describe phenomenology of strong interactions of elementary particles, and until recently, it has been developing rather slowly. This is because string theory has encountered a number of difficulties that were not easy to overcome; in particular, this theory contained the so-called anomalies that hindered construction of self-consistent string theory. Especially, anomalies led to the breakdown of symmetry properties of this theory after its quantization.

SOME EXAMPLES OF INSTANTONS IN SIGMA MODELS II: K3 MANIFOLDS

In this paper the authors describe instantons on some manifolds of type K3. The count of zero modes in two-dimensional sigma-models on such manifolds was produced previously. The paper can be divided into two parts. In the first the authors expose the necessary facts on K3 manifolds, essentially following the monographs, and in the second they describe instanton configurations.

Finiteness of N = 4 supersymmetric sigma models

Abstract The two-dimensional supersymmetric K3 σ-model is not renormalized in all orders of perturbation theory.