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

TL;DR

The paper proves existence and rigidity results for the $ ext{Sp}(1)$-Seiberg-Witten equations on hyperbolic 3-manifolds, linking the moduli space to conformal structures and cohomology, with implications for solutions on product manifolds.

Contribution

It establishes the existence of a canonical irreducible solution on hyperbolic 3-manifolds and relates the tangent space of the moduli space to conformal and cohomological structures.

Findings

Canonical irreducible solution exists on hyperbolic 3-manifolds.

The tangent space matches the space of trace-free Codazzi tensors.

If $H^1( ext{Gamma}, extbf{R}^{1,3})=0$, the solution is infinitesimally rigid.

Abstract

We prove that the $\mathrm{Sp}(1)$-Seiberg-Witten equation over a closed hyperbolic $3$-manifold ${\mathbb H}^3/\Gamma$ 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 $H^1(\Gamma,\mathbb R^{1,3})$, the first group cohomology of the $\Gamma$-module $\mathbb R^{1,3}$. In particular, if $H^1(\Gamma,\mathbb R^{1,3})=0$ then the canonical irreducible solution is infinitesimally rigid. We also prove that the $\mathrm{Sp}(1)$-Seiberg-Witten equation over $S^1\times \Sigma$ has no irreducible solutions and the moduli space of reducible solutions is same as the moduli space of flat $\mathrm{SU}(2)$-connections.