Chiral symmetry breaking in the Nambu-Jona-Lasinio model in curved spacetime with a nontrivial topology

We discuss the phase structure (in the $\frac{1}{N}$ expansion) of the Nambu-Jona-Lasinio model in curved spacetime with nontrivial topology ${\mathcal{M}}^3\times S^1$. The evaluation of the effective potential of the composite field $\overline{\psi }\psi$ is presented in the linear curvature approximation (topology is treated exactly) and in the leading order of the $\frac{1}{N}$ expansion. The combined influence of topology and curvature to the phase transitions is investigated. It is shown, in particular, that at zero curvature and for small radius of the torus there is a second order phase transition from the chiral symmetric to the chiral nonsymmetric phase. When the curvature grows and (or) the radius of $S^1$ decreases, then the phase transition is in general of first order. The dynamical fermionic mass is also calculated in a number of different situations.