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Abstract
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We study the propagation of massless fermionic
fields, implementing a family of special functions: Heun
functions, in solving the wave equation in three three-
dimensional backgrounds, including the BTZ black hole in
string theory and Lifshitz black hole solutions in conformal
gravity and Hu–Sawicki F(R) theory. The main properties of
the selected black hole solutions is that their line elements are
Weyl related to that of a homogeneous spacetime, whose spa-
tial part possesses Lie symmetry, described by Lobachevsky-
type geometry with arbitrary negative Gaussian curvature.
Using the Weyl symmetry of massless Dirac action, we con-
sider the perturbation equations of fermionic fields in rela-
tion to those of the homogeneous background, which having
definite singularities, are transformed into Heun’s equation.
We point out the existence of quasinormal modes labeled by
the accessory parameter of the Heun function. The distribu-
tion of the quasinormal modes has been clarified to satisfy
the boundary conditions that require ingoing and decaying
waves at the event horizon and conformal infinity, respec-
tively. It turned out that the procedure based on the Heun
function, beside reproducing the previously known results
obtained via hypergeometric function for the BTZ and Lif-
shitz black hole solution in conformal gravity, brings up new
families of quasinormal frequencies, which can also contain
purely imaginary modes. Also, the analysis of the quasinor-
mal modes shows that with the negative imaginary part of
complex frequencies ω = ωRe + iωI m , the fermionic pertur-
bations are stable in this background.
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