We reformulate the Thirring model in $D$ $(2 \le D < 4)$ dimensions as a gauge theory by introducing $U(1)$ hidden local symmetry (HLS) and study the dynamical mass generation of the fermion through the Schwinger-Dyson (SD) equation. By virtue of such a gauge symmetry we can greatly simplify the analysis of the SD equation by taking the most appropriate gauge (``nonlocal gauge'') for the HLS. In the case of even-number of (2-component) fermions, we find the dynamical fermion mass generation as the second order phase transition at certain fermion number, which breaks the chiral symmetry but preserves the parity in (2+1) dimensions ($D=3$). In the infinite four-fermion coupling (massless gauge boson) limit in (2+1) dimensions, the result coincides with that of the (2+1)-dimensional QED, with the critical number of the 4-component fermion being $N_{\rm cr} = \frac{128}{3\pi^{2}}$. As to the case of odd-number (2-component) fermion in (2+1) dimensions, the regularization ambiguity on the induced Chern-Simons term may be resolved by specifying the regularization so as to preserve the HLS. Our method also applies to the (1+1) dimensions, the result being consistent with the exact solution. The bosonization mechanism in (1+1) dimensional Thirring model is also reproduced in the context of dual-transformed theory for the HLS.Comment: 33 page

Based on the Schwinger-Dyson (SD) equation, the fermion mass generation is further studied in the D(2 < D < 4)-dimensional Thirring model as a gauge theory previously proposed. By using a certain approximation to the kernel, we analytically obtained explicit form of the dynamical mass of fermion and the critical line in (N, 1/g) space, where N is the number of fermions and g is the dimensionless vector-type fourfermion coupling constant. This analytical result is confirmed by the numerical solution for the SD equation with exact form of the kernel in (2+1) dimensions. * ) The original proof in Ref. 12) includes the case of D = 2, although the exact solution 16) seems to tell us no mass generation. In Appendix A we will show why this difference occurs. * ) Although we give only the critical value of g, as g cr (N ), we can read N cr (g) by inverting g = g cr (N cr (g)) with respect to N cr (g).