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Logarithmic correction to the entropy of extremal black holes in $\mathcal{N}=1$ Einstein-Maxwell supergravity

We study one-loop covariant effective action of \say{non-minimally coupled} $\mathcal{N}=1$, $d=4$ Einstein-Maxwell supergravity theory by heat kernel tool. By fluctuating the fields around the classical background, we study the functional determinant of Laplacian differential operator following Seeley-DeWitt technique of heat kernel expansion in proper time. We then compute the Seeley-DeWitt coefficients obtained through the expansion. A particular Seeley-DeWitt coefficient is used for determining the logarithmic correction to Bekenstein-Hawking entropy of extremal black holes using quantum entropy function formalism. We thus determine the logarithmic correction to the entropy of Kerr-Newman, Kerr and Reissner-Nordström black holes in {\say{non-minimally coupled}} $\mathcal{N}=1$, $d=4$ Einstein-Maxwell supergravity theory.