On the existence of nonlinear Dirac-geodesics on compact manifolds

We show that for super-linear nonlinearity, there exists a “non-trivial” nonlinear Dirac-geodesic on ${\mathbb{T}^n_{\Gamma}=\mathbb{R}^n/\Gamma}$ , a flat tori, in each “bosonic” sector. We also show that for any compact Riemannian manifold with “bumpy” metric, there exists a non-trivial nonlinear Dirac-geodesic in each bosonic sector if the nonlinearity is cubic or super-cubic and “large”. Our proof is based on critical point theory, in particular, a generalized linking argument applied to a strongly indefinite functional on a fibered Hilbert manifold.