We find exact solutions of the Dirac equation in the 2+1 dimensional curved background by separation of variables. These solutions are given in terms of hypergeometric functions. We also perform the Gordon decomposition for the Dirac current to discuss the time dependence of the polarization densities and the magnetization density, and to show that the polarization densities are more effective than the magnetization density in the pair production in finite time intervals.

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We investigate the Generalized Uncertainty Principle (GUP) effect on the Hawking radiation of the 2+1 dimensional Martinez-Zanelli black hole by using the Hamilton-Jacobi method. In this connection, we discuss the tunnelling probabilities and Hawking temperature of the spin-1/2 and spin-0 particles for the black hole. Therefore, we use the modified Klein-Gordon and Dirac equations based on the GUP. Then, we observe that the Hawking temperature of the scalar and Dirac particles depend on not only the black hole properties, but also the properties of the tunnelling particle, such as angular momentum, energy and mass. And, in this situation, we see that the tunnelling probability and the Hawking radiation of the Dirac particle is different from that of the scalar particle.

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 VII 0 gravitational instanton metrics. The solutions are obtained for the Bianchi VII 0 gravitational instanton metric as exponential functions by using complex variable ξ 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.