Minimal kernels of Dirac operators along maps

TL;DR

This paper studies the minimal kernel dimension of Dirac operators twisted by maps between manifolds, exploring whether the lower bounds given by index theory are generically attained.

Contribution

It investigates the generic attainment of lower bounds for Dirac operator kernels in the context of twisted Dirac operators on closed spin manifolds.

Findings

Lower bounds for kernel dimensions are established via index theory.

The paper analyzes conditions under which these bounds are generically achieved.

Results are specific to 2-dimensional manifolds and twisted Dirac operators.

Abstract

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 consider the Dirac operator $D^f_{g,h}$ of the twisted Dirac bundle $\Sigma M\otimes_{\mathbb{R}} f^*TN$. To this Dirac operator one can associate an index in $KO^{-dim(M)}(pt)$. If $M$ is $2$-dimensional, one gets a lower bound for the dimension of the kernel of $D^f_{g,h}$ out of this index. We investigate the question whether this lower bound is obtained for generic tupels $(f,g,h)$.