Rod-structure classification of gravitational instantons with U(1)×U(1) isometry

The rod-structure formalism has played an important role in the study of black holes in D=4 and 5 dimensions with R×U(1)D−3 isometry. In this paper, we apply this formalism to the study of four-dimensional gravitational instantons with U(1)×U(1) isometry, which could serve as spatial backgrounds for five-dimensional black holes. We first introduce a stronger version of the rod structure with the rod directions appropriately normalised, and show how the regularity conditions can be read off from it. Requiring the absence of conical and orbifold singularities will in general impose periodicity conditions on the coordinates, and we illustrate this by considering known gravitational instantons in this class. Some previous results regarding certain gravitational instantons are clarified in the process. Finally, we show how the rod-structure formalism is able to provide a classification of gravitational instantons, and speculate on the existence of possible new gravitational instantons.