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Abstract: It is a classical result that on a compact 3-manifold $Y$, the set of metrics for which there exists a harmonic spinor of the spin Dirac operator is a codimension 1 subset in the space of all metrics. In this talk, I will discuss an extension of this result to the case of the $mathbb Z_2$-Dirac operator, which is defined as the Dirac operator on the complement of a codimension 2 submanifold $mathcal Zsubseteq Y$ twisted by a flat connection on $Y-mathcal Z$ whose holonomy lies in $mathbb Z_2$. The deformation problem for solutions of this operator carries an infinite-dimensional obstruction for a fixed $mathcal Z$. Coupling the operator to the geometry of $mathcal Z$  by considering the infinite-dimensional family of Dirac operators parameterized by embedded submanifolds gives rise to a Fredholm problem up to a loss of regularity phenomenon. The proof of the result then requires the use of the Nash-Moser Implicit Function Theorem or related techniques.