Seiberg–Witten theory is used to obtain new obstructions to the existence of Einstein metrics on 4-manifolds with conical singularities along an embedded surface. In the present article, the cone angle is required to be of the form $2\pi /p$, $p$ a positive integer, but we conjecture that similar results will also hold in greater generality.
"Edges, orbifolds, and Seiberg–Witten theory." J. Math. Soc. Japan 67 (3) 979 - 1021, July, 2015. https://doi.org/10.2969/jmsj/06730979