STRING FIELD THEORY AND QUANTUM GROUPS. I. QUANTUM GROUP STRUCTURES IN GEOMETRIC QUANTIZATION OF A SELF-INTERACTING STRING FIELD D.V.Juriev

The paper is devoted to a description of quantum group structures in the geometric quantization of a self-interacting string field, which appear under a transition from a tree-level of the theory to the account of loop effects in nonperturbative quantum field theory of strings. Theory of strings, which is the most perspective approach to the unification of Standard Model and its supergeneralizations with quantum gravity in the unified consistent theory of elementary particles and their interactions, exists in two forms: as a theory of the first quantized strings (string quantum mechanics) and as a string field theory (theory of the second quantized strings) [1,2]. One of the advantages of the first approach besides the technical circumstances is the clarity of a geometrical description of the particle interactions, whereas its essential deficiencies are, first, a difficulty of the consistent account of the nonperturbative effects, second, the explicit dependence of the formulation of theory on metric and topology of the background. Both difficulties obstruct to use the theory as for the problems of quantum gravity as for the unified quantum description of gravity and Yang-Mills fields of Standard Model and its supergeneralizations. The second approach combining string and quantum field ones is free of such disadvantages (and, moreover, is of interest for the quantum field theory of vortices in the quantum fluids), but, its comparative difficulty and awkwardness in the concerete computations as well as abundance of heterogeneous and unrelated directly concepts made its applications very inconvenient. In the author’s papers [3] (see also [4]) there was given a unified formalism for the string field theory based on the geometric quantization, which allowed to connect other known approaches, for example, the Witten’s polynomial (cubic) string field theory of 1986 [5], the Aref’eva-Volovich approach, the 1 This is an English translation of the original Russian version, which is located at the end of the article as an appendix. In the case of any differences between English and Russian versions caused by a translation the least has the priority as the original one.