ZETA-REGULARIZATION OF THE O(N) NONLINEAR SIGMA-MODEL IN D-DIMENSIONS

The O(N) nonlinear sigma model in a D-dimensional space of the form R( D-M) X T-M, R(D-M) X S-M, or T-M X S-P is studied, where R(M), T-M, an d S-M correspond to flat space, a torus, and a sphere, respectively. U sing zeta-regularization and the 1/N expansion, the corresponding part ition functions-for deriving the free energy-and the gap equations are obtained, In particular, the free energy at the critical point on R(2 q+1) X S-2p+2 vanishes in accordance with the conformal equivalence to the flat space R(D). Numerical solutions of the gap equations at the critical coupling constants are given for several values of D. The pro perties of the partition function and its asymptotic behavior for larg e D are discussed. In a similar way, a higher-derivative nonlinear sig ma model is investigated, too. The physical relevance of our results i s discussed. (C) 1996 American Institute of Physics.