Geometric realization of Khovanov-Lauda-Rouquier algebras associated with Borcherds-Cartan data

We construct a geometric realization of the Khovanov-Lauda-Rouquier algebra $R$ associated with a symmetric Borcherds-Cartan matrix $A=(a_{ij}{)}_{i,j\in I}$ via quiver varieties. As an application, if $a_{ii}\mathrm{\ne }0$ for any $i\in I$, we prove that there exists a 1-1 correspondence between Kashiwara’s lower global basis (or Lusztig’s canonical basis) of $U_{\text{A}}^{-}(\text{g})$ (resp.\ $V_{\text{A}}(\lambda )$) and the set of isomorphism classes of indecomposable projective graded modules over $R$ (resp.\ $R^{\lambda }$).