Asymptotically hyperbolic 4-manifolds — Riemannian manifolds that look like hyperbolic space near their conformal boundary — can have singularities at the boundary that prevent smooth extension. Detecting these singularities is a problem in conformal geometry: the boundary data (conformal class of the boundary metric) determines the bulk geometry, but not all boundary data produces smooth bulk metrics.
Herfray introduces the 0-instanton obstruction tensor — a conformal invariant of the boundary that vanishes if and only if the bulk extends smoothly. The tensor is built from the self-dual Weyl curvature of the bulk metric, evaluated at the boundary. It's the conformal geometry analogue of an instanton number: it counts the topological obstruction to smooth extension, but in a conformally invariant way.
The name “0-instanton” is precise. In gauge theory, instantons are self-dual connections with topological charge. The obstruction tensor measures the self-dual curvature of the Levi-Civita connection at the boundary — the gravitational analogue of an instanton. The “0” indicates it's the leading-order term in the asymptotic expansion near the boundary.
The tensor is computable from boundary data alone. Given a conformal class on the boundary 3-manifold, the obstruction tensor can be calculated without solving the bulk Einstein equations. If it vanishes, a smooth bulk exists. If it doesn't, no smooth bulk exists with that boundary data.
A single tensor, defined on the boundary, that determines smoothness of the interior. The conformal boundary remembers whether its bulk is smooth.