We study the carbon dimer defect in a hexagonal boron-nitride monolayer using the GW and Bethe-Salpeter many-body perturbation theories within a finite size cluster approach. While quasiparticle energies converge very slowly with system size due to missing long-range polarization effects, optical excitations converge much faster, with a $1/R^3$ scaling law with respect to cluster average radius. We obtain a luminescence zero-phonon energy of 4.36 eV, including significant 0.13 eV zero-point vibrational energy and 0.15 eV reorganization energy contributions. Inter-layer screening decreases further the emission energy by about 0.3 eV. These results bring support to the recent identification of the substitutional carbon dimer as the likely source of the zero-phonon 4.1 eV luminescence line. Finally, the GW quasiparticle energies are extrapolated to the infinite h-BN monolayer limit, leading to a predicted defect HOMO-LUMO photoemission gap of 7.6 eV. Comparison with the optical gap yields a very large excitonic binding energy of 3 eV for the associated localized Frenkel exciton.
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