New asymptotically flat gravitational instanton

Abstract

A new two-parameter asymptotically flat (AF) toric gravitational instanton is identified as a special case of the Euclidean double Kerr-NUT solution, by imposing certain symmetry and regularity conditions on its rod structure. These conditions are solved explicitly, except for one which takes the form of a fifth-order polynomial. This gravitational instanton has Euler number $χ=4$ and Hirzebruch signature $τ=0$, and its global topology is $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$ with a circle $S^1$ removed appropriately. It is the third of an infinite sequence of AF toric gravitational instantons that was proved to exist by Li and Sun, the first two being the Kerr and Chen--Teo instantons. It is also the first known example of a Ricci-flat gravitational instanton that is not Hermitian.