Minimal kernels of Dirac operators along maps

Abstract

Let $M$ be a closed spin manifold and let $N$ be a closed manifold. For maps $f:M\to N$ and Riemannian metrics $g$ on $M$ and $h$ on $N$, we consider the Dirac operator $D_{g,h}^f$ of the twisted Dirac bundle $\Sigma M{\otimes }_{\mathbb{R}}{f}^{\ast }TN$. To this Dirac operator one can associate an index in $K{O}^{-dim(M)}(pt)$. If $M$ is $2$-dimensional, one gets a lower bound for the dimension of the kernel of $D_{g,h}^f$ out of this index. We investigate the question whether this lower bound is obtained for generic tupels $(f,g,h)$.