2024/08/02 by Gorapada Bera, Bera, Gorapada
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2408.01522
openalex publication_date 2024/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the Sp(1)-Seiberg-Witten equation over a closed hyperbolic 3-manifold \mathbb H3/Γ 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 H1(Γ,\mathbb R1,3), the first group cohomology of the Γ-module \mathbb R1,3. In particular, if H1(Γ,\mathbb R1,3)=0 then the canonical irreducible solution is infinitesimally rigid. We also prove that the Sp(1)-Seiberg-Witten equation over S1× Σ has no irreducible solutions and the moduli space of reducible solutions is same as the moduli space of flat SU(2)-connections.