Complete pluriclosed metrics with vanishing Bismut Ricci form on the affine quadric

We study a one-parameter family of SO(4)-invariant pluriclosed Hermitian metrics $g_c$ with vanishing Bismut Ricci form on the affine quadric $Q_3\cong TS^3$, previously described in the physics literature. For every real parameter $c$, we prove local existence and uniqueness of a real-analytic solution to the defining singular initial-value problem, together with positivity of the associated metric near the singular orbit $S^3$. We then establish global existence and completeness for $|c|\leq1$ and finite-time degeneration for $|c|>1$. The complete family exhausts all SO(4)-invariant Bismut Hermitian-Einstein metrics on the affine quadric $Q_3$, and joins Stenzel's Kähler Ricci-flat to a metric that was first considered by Chamseddine-Volkov/Maldacena-Nuñez. We also determine the leading asymptotics and derive an explicit scalar curvature formula, proving strict positivity for the non-Kähler members and identifying the change in asymptotic scalar curvature at the endpoint $c=1$. Finally, we show that all these metrics are not Bismut flat and have full Bismut holonomy SU(3), providing new examples of such manifolds.