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Let $M$ be a closed spin manifold and let $N$ be a closed manifold. For maps$f\colon M\to N$ and Riemannian metrics $g$ on $M$ and $h$ on $N$, we considerthe Dirac operator $D^f_{g,h}$ of the twisted Dirac bundle $\SigmaM\otimes_{\mathbb{R}} f^*TN$. To this Dirac operator one can associate an indexin $KO^{-dim(M)}(pt)$. If $M$ is $2$-dimensional, one gets a lower bound forthe dimension of the kernel of $D^f_{g,h}$ out of this index. We investigatethe question whether this lower bound is obtained for generic tupels $(f,g,h)$.

Minimal kernels of Dirac operators along maps