Remarks on $\mathrm{S}\mathrm{p}(1)$-Seiberg-Witten equation over $3$-manifolds

We prove that the $\mathrm{S}\mathrm{p}(1)$-Seiberg-Witten equation over a closed hyperbolic $3$-manifold ${\mathbb{H}}^3\mathrm{/}\mathrm{\Gamma }$ always admits a canonical irreducible solution induced by the hyperbolic metric. We also prove that the Zariski tangent space of the moduli space at this canonical solution is same as the Zariski tangent space of the moduli space of locally conformally flat structures at the hyperbolic metric. This space is again same as the space of trace-free Codazzi tensors and carries an injection to $H^1(\mathrm{\Gamma },{\mathbb{R}}^{1,3})$, the first group cohomology of the $\mathrm{\Gamma }$-module ${\mathbb{R}}^{1,3}$. In particular, if $H^1(\mathrm{\Gamma },{\mathbb{R}}^{1,3})=0$ then the canonical irreducible solution is infinitesimally rigid. We also prove that the $\mathrm{S}\mathrm{p}(1)$-Seiberg-Witten equation over $S^1\times \mathrm{\Sigma }$ has no irreducible solutions and the moduli space of reducible solutions is same as the moduli space of flat $\mathrm{S}\mathrm{U}(2)$-connections.