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 operatorψψ 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 curvature-and 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.

The phase structure of four-fermion theories is thoroughly investigated with varying temperature and chemical potential for arbitrary space-time dimensions (2 ≤ D < 4) by using the 1/N expansion method. It is shown that the chiral symmetry is restored in the theory under consideration for sufficiently high temperature and/or chemical potential. The critical line dividing the symmetric and broken phase is given explicitly. It is found that for space-time dimension 2 ≤ D < 3 both the first-order and second-order phase transition occur depending on the value of temperature and chemical potential while for 3 ≤ D < 4 only the second-order phase transition exists. *

The phase structure of Nambu-Jona-Lasinio model with N-component fermions in curved space-time is studied in the leading order of the 1/N expansion. The effective potential for composite operatorψψ is calculated by using the normal coordinate expansion in the Schwinger proper-time method. The existence of the first-order phase transition caused by the change of the space-time curvature is confirmed and the dynamical mass of the fermion is calculated as a simultaneous function of the curvature and the four-fermion coupling constant. The phase diagram in the curvature and the coupling constant is obtained.

We investigate the influence of an external magnetic field on chiral symmetry breaking in a four-fermion interaction model at finite temperature and chemical potential. Using the Fock-Schwinger proper time method, we calculate the effective potential for the four-fermion interaction model to leading order in the 1/Nc expansion. The phase structure of the chiral symmetry breaking is determined in the T -µ, H-T and µ-H planes. The external magnetic field modifies the phase structure. It is found that a new phase appears for a large chemical

We investigate the influence of an external magnetic field on chiral symmetry breaking in the Nambu-Jona-Lasinio (NJL) model at finite temperature and chemical potential. According to the Fock-Schwinger proper-time method, we calculate the effective potential in the leading order of the 1/N c expansion. The phase boundary dividing the symmetric phase and the broken phase is illustrated numerically. A complex behavior of the phase boundary is found for large chemical potential.