We use state-of-the art GW-BSE formalism to study electronic structure and optical properties of oxygen vacancies in $alpha$-alumina. Many body perturbation theory within GW approximation in recent years have been used extensively to study excited state properties of a wide range of systems. Moreover, solving Bethe-Salpeter equation (BSE) enable us to capture excitonic effects in a material. We compute the charge transition levels (CTLs) for oxygen vacancies using DFT+GW formalism. We propose an alternative approach to calculate these CTLs, which provides a more efficient way to perform electrostatic correction required because of finite supercell sizes and periodic boundary condition used in first principles calculations. We find that oxygen vacancy in this material has deep donor levels, (+2/+1) at 2.5 eV and a (+1/0) level at 3.8 eV above the VBM. We also study F-center absorption and emission processes using constrained--DFT and BSE. Our calculated absorption and emission energies are in excellent agreement with experimental results.
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