the cigar-shaped background metric with supersymmetry approaches and thermal properties of system

In this paper, we introduce the cigar- shaped metric . such metric can be obtained by the schwarzschild black hole. here, we write the dirac equation in the corresponding background. we take some change of variable for the above equation and re- write the mentioned equation similar to the hypergeometric equation. we take advantage from the known polynomial( hypergeometric) and factorized the dirac equation in cigar- shaped background in terms of first order equations and achieve the wave function. also, this first order equations lead us to use super symmetry(susy) approach and obtain the energy spectrum. susy quantum mechanics can be used to solve ordinary differential equation. using this method, we obtained the energy spectrum of the fermions on the aforementioned background. then, by computing the partition function, we obtained the statistical quantities of the system. finally, we take information from thermodynamic and investigate the thermal properties of the system.