Dirac equation in Euclidean Newman-Penrose formalism with applications to instanton metrics

We derive the Dirac equation in the Euclidean version of the Newman-Penrose formalism and show that it splits into two sets of equations, particle and anti-particle equations, under the swapping symmetry and these equations are coupled, respectively, with the self-dual and anti-self-dual parts of the gauge in the gravity. We also solve it for Eguchi-Hanson and Bianchi VII0 gravitational instanton metrics. The solutions are obtained for the Bianchi VII0 gravitational instanton metric as exponential functions by using complex variable and for the Eguchi-Hanson gravitational instanton metric as the product of two hypergeometric functions. In addition, we discuss the regularity and the swapping symmetry of the solutions and show that the topological index of the Dirac equation is zero for both of these metrics.