Uniqueness and Nondegeneracy of Ground State for the $L^2$-Critical Boson Star Equation

We prove uniqueness, up to phase rotations and translations, of the ground state for the $L^2$-critical boson star equation \[ (\sqrt{-Δ}+1)Q =\bigl(|x|^{-1}* |Q|^2\bigr)Q, \qquad x\in\mathbb R^3. \] We also determine the kernels of the real and imaginary linearized operators. As a consequence