Quiver Varieties and Path Realizations arising from Adjoint Crystals of type $A_n^{(1)}$

Let $B(\Lambda_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(\Lambda_0)$ of $B(\Lambda_0)$ via quiver varieties and the path realization ${\mathcal P}^{\rm ad}(\Lambda_0)$ of $B(\Lambda_0)$ arising from the adjoint crystal $\adjoint$.