BPS kinks in the Gross-Neveu model

We find the exact spectrum and degeneracies for the Gross-Neveu model in two dimensions. This model describes N interacting Majorana fermions; it is asymptotically free, and has dynamical mass generation and spontaneous chiral symmetry breaking. We show here that the spectrum contains $2^{N/2\mathrm{}}$ kinks for any N. The unusual $\sqrt{2}$ in the number of kinks for odd N comes from restrictions on the allowed multi-kink states. These kinks are the Bogomol’nyi-Prasad-Sommerfield states for a generalized supersymmetry where the conserved current is of dimension $N/2;$ the $N=3\mathrm{}$ case is the $\mathcal{N}=1\mathrm{}$ supersymmetric sine-Gordon model, for which the spectrum consists of $2\sqrt{2}$ kinks. We find the exact S matrix for these kinks, and the exact free energy for the model.