Dynamical Symmetry Breaking in Curved Spacetime

This review deals with the theory of four-fermion interactions in curved spacetime. Starting with the D-dimensional Minkowski spacetime (2 ≤ D ≤ 4) the effective potential in the leading order of 1/N-expansion is calculated and the phase structure of the theory is investigated. Using the same technique the effective potential for composite operator ψψ in four-fermion models is calculated under the following circumstances: a) D-dimensional weakly curved spacetime (in linear curvature approximation), b) D-dimensional de Sitter and anti-de Sitter universe, c) D-dimensional Einstein universe. The phase structure of the theory is investigated analytically as well as numerically. Curvature induced phase transitions are discussed where fermion masses are dynamically generated. As an extension of four-fermion models we consider the gauged Nambu-Jona-Lasinio (NJL) model, higher derivative NJL model and supersymmetric NJL model in weakly curved spacetime where the effective potential is analytically evaluated. The phase structure of the models is again analyzed and the condition for the chiral symmetry breaking in the gauged NJL model is given in an analytical form. Finally the influence of two external effects (non-zero temperature and gravitational field, nontrivial topology and gravitational field as well as magnetic and gravitational field) to the phase structure of four-fermion models is analyzed. The possibility of curvature and temperature-induced or curvatureand topology-induced phase transitions is discussed. It is also argued that the chiral symmetry broken by a weak magnetic field may be restored due to the presence of gravitational field. Some applications of four-fermion models in quantum gravity are also briefly investigated. This paper was published in Prog. Theor. Phys. Suppl. 127 (1997) 93. 1e-mail : inagaki@ipc.hiroshima-u.ac.jp 2e-mail:muta@sci.hiroshima-u.ac.jp 3e-mail:sergei@ecm.ub.es, odintsov@quantum.univalle.edu.co